Abstract
We derive conditions for well-posedness of semilinear evolution equations with unbounded input operators. Based on this, we provide sufficient conditions for such properties of the flow map as Lipschitz continuity, bounded-implies-continuation property, boundedness of reachability sets, etc. These properties represent a basic toolbox for stability and robustness analysis of semilinear boundary control systems. We cover systems governed by general \(C_0\)-semigroups, and analytic semigroups that may have both boundary and distributed disturbances. We illustrate our findings on an example of a Burgers’ equation with nonlinear local dynamics and both distributed and boundary disturbances.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
1.1 Semilinear evolution equations
In this work, we analyze the well-posedness and properties of the flow for semilinear evolution equations of the form
Here, A generates a strongly continuous semigroup over a Banach space X, the operators B and \(B_2\) are admissible with respect to some function space, and f is Lipschitz continuous in the first variable (see Assumption 3.3 for precise requirements on f). This class of systems is rather general:
-
If B and \(B_2\) are bounded operators, (1) corresponds to the classic semilinear evolution equations covering broad classes of semilinear PDEs with distributed inputs. If A is a bounded operator, such a theory was developed in [7]. In the case of unbounded generators A, we refer to [15, 43], [Chapter 11][3, 6], etc.
-
If \(B_2=0\), and B is an admissible operator, then (1) reduces to the class of general linear control systems that fully covers linear boundary control systems (see [6, 11, 23, 56, 57] for an overview). In particular, this class includes linear evolution PDEs with boundary inputs.
-
Consider a linear system
$$\begin{aligned} \dot{x} = Ax + Bv, \end{aligned}$$(2)with admissible B. Let us apply a feedback controller \(v(x)=f(x,u_1)+u_2\) that is subject to additive actuator disturbance \(u_2\) and further disturbance input \(u_1\). Substituting this controller into (2), we arrive at systems (1), with \(B_2=B\).
-
In [17], it was shown that the class of systems (1) includes 2D Navier–Stokes equations (under certain boundary conditions) with in-domain inputs and disturbances. Furthermore, in [17] the authors have designed an error feedback controller that guarantees approximate local velocity output tracking for a class of reference outputs. Viscous Burgers’ equation with nonlinear local terms and boundary inputs of Dirichlet or Neumann type falls into the class (1) as well.
-
In [51], it was shown that semilinear boundary control systems with linear boundary operators could be considered a special case of systems (1). In this case, it suffices to consider \(B_2\) as the identity operator. Furthermore, in [51], the well-posedness and input-to-state stability of a class of analytic boundary control systems with nonlinear dynamics and a linear boundary operator were analyzed with the methods of operator theory.
1.2 ISS for infinite-dimensional systems
Our main motivation to analyze the systems (1) stems from the robust stability theory. During the last decade, we have witnessed tremendous progress in robust stability analysis of nonlinear infinite-dimensional systems subject to unknown unstructured disturbances. Input-to-state stability (ISS) framework admits a significant place in this development, striving to become a unifying paradigm for robust control and observation of PDEs and their interconnections, including ODE–PDE and PDE–PDE cascades [30, 38, 51].
Powerful techniques proposed to analyze the ISS property include: criteria of ISS and ISS-like properties in terms of weaker stability concepts [20, 40, 48], constructions of ISS Lyapunov functions for PDEs with in-domain and/or boundary controls [10, 44, 55, 60], efficient functional-analytic methods for the study of linear systems with unbounded input operators (including linear boundary control systems) [20, 22, 24, 29, 30, 33, 59], non-coercive ISS Lyapunov functions [19, 40], as well as small-gain stability analysis of finite [8, 26, 28, 35] and infinite networks, [9, 31, 32, 37], etc.
To make this powerful machinery work for any given system, one needs to verify its well-posedness, properties of reachability sets, and regularity of the flow induced by this system. Usually this is done for PDE systems in a case-by-case manner. In this paper, motivated by [51], we develop sufficient conditions that help to derive these crucial properties for systems (1), which cover many important PDE systems.
1.3 State of the art
The systems (1) have been studied (up to the assumptions on f, and the choice of the space of admissible inputs) in [41] under the requirement that its linearization is an exponentially stable regular linear system in the sense of [53, 56, 57]. Natarajan and Bentsman [41] ensure local well-posedness of regular nonlinear systems assuming the Lipschitz continuity of nonlinearity, and invoking regularity of the linearization. On this basis, the authors show in [41] that an error feedback controller designed for robust output regulation of a linearization of a regular nonlinear system achieves approximate local output regulation for the original regular nonlinear system.
Control of systems (1) has been studied recently in several papers. In particular, in [42], the exact controllability of a class of regular nonlinear systems was studied using back-and-forth iterations. A problem of robust observability was studied for a related class of systems in [25].
Stabilization of linear port-Hamiltonian systems by means of nonlinear boundary controllers was studied in [2, 46]. Bounded controls with saturations (a priori limitations of the input signal) have been employed for PDE control in [39, 45, 54]. Recently, several papers appeared that treat nonlinear boundary control systems within the input-to-state stability framework. Nonlinear boundary feedback was employed for the ISS stabilization of linear port-Hamiltonian systems in [50].
Several types of infinite-dimensional systems, distinct from (1), have been studied as well. One of such classes is time-variant infinite-dimensional semilinear systems that have been first studied (as far as the author is concerned) for systems without disturbances in [18]. Recently, in [49], sufficient conditions for well-posedness and uniform global stability have been obtained for scattering-passive semilinear systems (see [49, Theorem 3.8]).
Another important extension of (1) are semilinear systems with outputs. Such systems with globally Lipschitz nonlinearities have been analyzed in [57, Section 7], and it was shown that such systems are well-posed and forward complete provided that the Lipschitz constant is small enough. In [14] employing a counterexample, it was shown that a linear transport equation with a locally Lipschitz boundary feedback might fail to be well-posed. Well-posedness of incrementally scattering-passive nonlinear systems with outputs has been analyzed in [52] by applying Crandall–Pazy theorem [5] on generation of nonlinear contraction semigroups to a Lax–Phillips nonlinear semigroup representing the system together with its inputs and outputs.
1.4 Contribution
Our first main result is Theorem 3.7 guaranteeing (under proper conditions on f and the input operators) the local existence and uniqueness of solutions for the system (1) with a locally essentially bounded input u.
There are several existence and uniqueness theorems in the literature. For example, [41, Proposition 3.2] covers semilinear systems with \(L^\infty \)-inputs; [57, Theorem 7.6], [16, Lemma 2.8] treat the case of bilinear systems of various type, and [51] considers the case of systems with linearly bounded nonlinearities. In contrast to the usual formulations of such results (including a closely related result [41, Proposition 3.2]), we also provide a uniform existence time for solutions that controls the maximal deviation of the trajectory from the given set of initial conditions.
Next, we show in Theorems 3.17 and 3.18 that under natural conditions, the system (1) is a well-posed control system in the sense of [38]. Finally, we study the fundamental properties of the flow map, such as Lipschitz continuity with respect to initial states, boundedness of reachability sets, and boundedness-implies-continuation property. These properties are important in their own right. Moreover, they are key components for the robust stability analysis of systems (1) as we explained before.
The structure of semilinear evolution equations allows combining the “linear” methods of admissibility theory with “nonlinear” methods, such as fixed point theorems and Lyapunov methods. We consider the case of general \(C_0\)-semigroups and the special case of analytic semigroups, for which one can achieve stronger results. This synergy of tools is one of the novelties of this paper. For systems without inputs and without the presence of unbounded operators, the existence and uniqueness results as well as the properties of the flow are classical both for general and analytic case [15, 43]. To show the applicability of our methods, we analyze well-posedness of semilinear parabolic systems with Dirichlet boundary inputs (motivated by [15, p. 57]). Also, we reformulate semilinear boundary control systems in terms of evolution equations, which makes our results applicable to boundary control systems as well.
As argued at the previous pages, having developed conditions ensuring the well-posedness and “nice” properties of the flow map of systems (1), we can analyze the ISS of (1) via such powerful tools as coercive and non-coercive ISS Lyapunov functions [19], ISS superposition theorems [40], small-gain theorems for general systems [37], etc. We expect that this will help to prove many results available for particular PDE systems, in a more general fashion. For example, see [51] for an abstract version of the results obtained for particular classes of parabolic systems in [60]. To make the paper accessible for the researchers trained primarily in nonlinear control and nonlinear ISS theory, we spell out the proofs in great detail with a tutorial flavor.
1.5 Notation
By \(\mathbb {N}\), \(\mathbb {R}\), \(\mathbb {R}_+\), we denote the sets of natural, real, and nonnegative real numbers, respectively. \(\overline{S}\) denotes the closure of a set S (in a given topology).
By \(t \rightarrow a \pm 0\), we denote the fact that t approaches a from the right/left.
Vector spaces considered in this paper are assumed to be real.
Let S be a normed vector space. The distance from \(z\in S\) to the set \(Z \subset S\) we denote by \(\textrm{dist}\,(z,Z){:}{=}\inf \{\Vert z-y\Vert _S: y \in Z \}\). We denote an open ball of radius r around \(Z \subset S\) by \(B_{r,S}(Z){:}{=}\{y \in X:\textrm{dist}\,(y,Z)<r\}\), and we set also \(B_{r,S}(x){:}{=}B_{r,S}(\{x\})\) for \(x \in X\), and \(B_{r,S}{:}{=}B_{r,S}(0)\). If \(S =X\) (the state space of the system), we write for short \(B_{r}(Z){:}{=}B_{r,X}(Z)\), \(B_{r}(x){:}{=}B_{r,X}(x)\), etc.
Denote by \(\mathcal {K}\) the class of continuous strictly increasing functions \(\gamma :\mathbb {R}_+\rightarrow \mathbb {R}_+\), satisfying \(\gamma (0)=0\). \(\mathcal {K_\infty }\) denotes the set of unbounded functions from \(\mathcal {K}\).
For normed vector spaces X, U, denote by L(X, U) the space of bounded linear operators from X to U. We endow L(X, U) with the standard operator norm \(\Vert A\Vert {:}{=}\sup _{\Vert x\Vert _X=1}\Vert Ax\Vert _U\). We write for short \(L(X){:}{=}L(X,X)\). By C(X, U), we denote the space of continuous maps from X to U. Similarly, by \(C(\mathbb {R}_+,X)\) we understand the space of continuous maps from \(\mathbb {R}_+\) to X. The domain of definition, kernel, and image of an operator A we denote by D(A), \(\textrm{Ker}\,(A)\), and \(\textrm{Im}(A)\), respectively. By \(\sigma (A)\), we denote the spectrum of a closed operator \(A:D(A)\subset X \rightarrow X\), and by \(\rho (A)\) the resolvent set of A. We denote by \(\omega _0(T)\) the growth bound of a \(C_0\)-semigroup T.
Let X be a Banach space, and let I be a closed subset of \(\mathbb {R}\). We define for \(p\in [1,\infty )\) the following spaces of vector-valued functions
Denote also \(L^p(a,b)\) \({:}{=}L^p([a,b],\mathbb {R})\), where \(p\in [1,\infty ]\). The space \(H^{k}(a,b)\), \(k\in \mathbb {N}\), is a Sobolev space of functions \(u \in L^2(a,b)\), such that for each natural \(j\le k\), the weak derivative \(u^{(j)}\) exists and belongs to \(L^2(a,b)\). \(H^{k}(a,b)\) is endowed with the norm \(u\mapsto \Big (\sum _{j \le k}\int _a^b{\big | u^{(j)} (x)\big |^2 dx} \Big )^{\frac{1}{2}}\). \(H^{k}_0(a,b)\) denotes the closure of smooth functions with compact support in (a, b) in the norm of \(H^{k}(a,b)\), \(k\in \mathbb {N}\).
2 General class of systems
We start with a general definition of a control system that we adopt from [38].
Definition 2.1
Consider the triple \(\Sigma =(X,\mathcal {U},\phi )\) consisting of
-
(i)
A normed vector space \((X,\Vert \cdot \Vert _X)\), called the state space, endowed with the norm \(\Vert \cdot \Vert _X\).
-
(ii)
A normed vector space of inputs \(\mathcal {U}\subset \{u:\mathbb {R}_+ \rightarrow U\}\) endowed with a norm \(\Vert \cdot \Vert _{\mathcal {U}}\), where U is a normed vector space of input values. We assume that the following two axioms hold: The axiom of shift invariance: for all \(u \in \mathcal {U}\) and all \(\tau \ge 0\) the time shift \(u(\cdot + \tau )\) belongs to \(\mathcal {U}\) with \(\Vert u\Vert _\mathcal {U}\ge \Vert u(\cdot + \tau )\Vert _\mathcal {U}\). The axiom of concatenation: for all \(u_1,u_2 \in \mathcal {U}\) and for all \(t>0\) the concatenation of \(u_1\) and \(u_2\) at time t, defined by
$$\begin{aligned} {\big ({u_1\, \underset{t}{\lozenge }\,{u_2}}\big )(\tau ){:}{=}} {\left\{ \begin{array}{ll} u_1(\tau ), &{} \text { if } \tau \in [0,t], \\ u_2(\tau -t), &{} \text { otherwise}, \end{array}\right. } \end{aligned}$$(3)belongs to \(\mathcal {U}\).
-
(iii)
A map \(\phi :D_{\phi } \rightarrow X\), \(D_{\phi }\subseteq \mathbb {R}_+ \times X \times \mathcal {U}\) (called transition map), such that for all \((x,u)\in X \times \mathcal {U}\) it holds that \(D_{\phi } \cap \big (\mathbb {R}_+ \times \{(x,u)\}\big ) = [0,t_m)\times \{(x,u)\} \subset D_{\phi }\), for a certain \(t_m=t_m(x,u)\in (0,+\infty ]\). The corresponding interval \([0,t_m)\) is called the maximal domain of definition of \(t\mapsto \phi (t,x,u)\).
The triple \(\Sigma \) is called a (control) system, if the following properties hold:
- \((\sum 1)\):
-
The identity property: for every \((x,u) \in X \times \mathcal {U}\) it holds that \(\phi (0, x,u)=x\).
- \((\sum 2)\):
-
Causality: for every \((t,x,u) \in D_\phi \), for every \(\tilde{u} \in \mathcal {U}\), such that \(u(s) = \tilde{u}(s)\) for all \(s \in [0,t]\) it holds that \([0,t]\times \{(x,\tilde{u})\} \subset D_\phi \) and \(\phi (t,x,u) = \phi (t,x,\tilde{u})\).
- \((\sum 3)\):
-
Continuity: for each \((x,u) \in X \times \mathcal {U}\) the map \(t \mapsto \phi (t,x,u)\) is continuous on its maximal domain of definition.
- \((\sum 4)\):
-
The cocycle property: for all \(x \in X\), \(u \in \mathcal {U}\), for all \(t,h \ge 0\) so that \([0,t+h]\times \{(x,u)\} \subset D_{\phi }\), we have
$$\begin{aligned} \phi \big (h,\phi (t,x,u),u(t+\cdot )\big )=\phi (t+h,x,u). \end{aligned}$$
Definition 2.1 can be viewed as a direct generalization, and a unification of the concepts of strongly continuous nonlinear semigroups [4, 5] with abstract linear control systems [58].
This class of systems encompasses control systems generated by ordinary differential equations (ODEs), switched systems, time-delay systems, evolution partial differential equations (PDEs), abstract differential equations in Banach spaces and many others [27, Chapter 1].
Definition 2.2
We say that a control system \(\Sigma =(X,\mathcal {U},\phi )\) is forward complete (FC), if \(D_\phi = \mathbb {R}_+ \times X\times \mathcal {U}\), that is for every \((x,u) \in X \times \mathcal {U}\) and for all \(t \ge 0\) the value \(\phi (t,x,u) \in X\) is well defined.
Forward completeness alone does not imply, in general, the existence of any uniform bounds on the trajectories emanating from bounded balls that are subject to uniformly bounded inputs [40, Example 2, p. 1612]. Systems exhibiting such bounds deserve a special name.
Definition 2.3
We say that a control system \(\Sigma =(X,\mathcal {U},\phi )\) has bounded reachability sets (BRS), if for any \(C>0\) and any \(\tau >0\) it holds that
For a wide class of control systems, the boundedness of a solution implies the possibility of prolonging it to a larger interval, see [27, Chapter 1]. Next, we formulate this property for abstract systems:
Definition 2.4
We say that a control system \(\Sigma =(X,\mathcal {U},\phi )\) satisfies the boundedness-implies-continuation (BIC) property if for each \((x,u)\in X \times \mathcal {U}\) with \(t_m(x,u)<\infty \) it holds that
3 Semilinear evolution equations with unbounded input operators
Consider a Cauchy problem for infinite-dimensional evolution equations of the form
where \(A: D(A)\subset X \rightarrow X\) generates a strongly continuous semigroup \(T=(T(t))_{t\ge 0}\) of bounded linear operators on a Banach space X; U is a Banach space of input values, and \(x_0\in X\) is a given initial condition. As the input space, we take \(\mathcal {U}{:}{=}L^\infty (\mathbb {R}_+,U)\).
The map \(f: X\times U \rightarrow V\) is defined on the whole \(X \times U\) and maps to a Banach space V. Furthermore, \(B\in L(U,X_{-1})\) and \(B_2 \in L(V,X_{-1})\). Here, the extrapolation space \(X_{-1}\) is the closure of X in the norm \(x \mapsto \Vert (aI-A)^{-1}x\Vert _X\), \(x \in X\), where \(a \in \rho (A)\) (different choices of \(a \in \rho (A)\) induce equivalent norms on X). Note that the operators B and \(B_2\) are unbounded, if they are understood as operators that map to X.
3.1 Admissible input operators and mild solutions
First, consider the linear counterpart of the system (4).
for the same A, B as above. As the image of B does not necessarily lie in X, one has to be careful when defining the concept of a solution for (5). Since \(B \in L(U,X_{-1})\), it is natural to consider the system (5) on the space \(X_{-1}\). Note that the semigroup \((T(t))_{t\ge 0}\) extends uniquely to a strongly continuous semigroup \((T_{-1}(t))_{t\ge 0}\) on \(X_{-1}\) whose generator \(A_{-1}\) acting in \(X_{-1}\) is an extension of A with \(D(A_{-1}) = X\), see, e.g., [12, Section II.5]. Recall the definitions of the spaces \(L^p\), \(L^p_{{{\,\textrm{loc}\,}}}\) from Sect. 1.
The mild solution of (5) for any \(x \in X\) and \(u \in L^1_{{{\,\textrm{loc}\,}}}(\mathbb {R}_+,U)\) is given by
The integral term here, however, belongs in general to \(X_{-1}\).
Thus, the existence and uniqueness of a mild solution depend on whether \(\int _0^t T_{-1}(t-s)Bu(s){\text {d}}s \in X\). This leads to the following concept:
Definition 3.1
Let \(q\in [1,\infty ]\). The operator \(B\in L(U,X_{-1})\) is called a q-admissible control operator for \((T(t))_{t\ge 0}\), if there is \(t>0\) so that
Define for each \(t\ge 0\) an operator \(\Phi (t): L^q_{{{\,\textrm{loc}\,}}}(\mathbb {R}_+,U) \rightarrow X_{-1}\) by
Note that as \(B \in L(U,X_{-1})\), the operators \(\Phi (t)\) are well defined as maps from \(L^1_{{{\,\textrm{loc}\,}}}(\mathbb {R}_+,U)\) to \(X_{-1}\) for all t. The next result (see [58, Proposition 4.2], [56, Proposition 4.2.2]) shows that q-admissibility of B ensures that the image of \(\Phi (t)\) is in X for all \(t \ge 0\) and \(\Phi (t)\in L(L^q(\mathbb {R}_+,U),X)\) for all \(t>0\).
Proposition 3.2
Let X, U be Banach spaces, and let \(q \in [1,\infty ]\) be given. Then, \(B \in L(U,X_{-1})\) is q-admissible if and only if for all \(t>0\) there is \(h_t>0\) so that for all \(u \in L^{q}_{{{\,\textrm{loc}\,}}}(\mathbb {R}_+,U)\) it holds that \(\Phi (t)u \in X\) and
The function \(t\mapsto h_t\) we assume wlog to be non-decreasing in t.
An important consequence of Proposition 3.2 is that well-posedness (and thus forward completeness) of the system (5) already implies the boundedness of reachability sets property for (5), with a bound given by (7).
As \(t\mapsto h_t\) is non-decreasing in t, there is a limit \(h_0{:}{=}\lim _{t\rightarrow +0}h_t \ge 0\), which is not necessarily zero. Operators for which \(h_0=0\) deserve a special name.
Definition 3.3
Let \(q \in [1,\infty ]\). A q-admissible operator \(B\in L(U, X_{-1})\) is called zero-class q-admissible, if the constants \((h_t)_{t>0}\) can be chosen such that \(h_0=0\).
All \(B \in L(U,X)\) are zero-class 1-admissible. If X is reflexive, then 1-admissible operators are necessarily bounded. At the same time, there are unbounded zero-class admissible operators, see Proposition 4.2. Consider [21, Examples 3.8, 3.9] for unbounded admissible observation operators that are not zero-class admissible.
The above considerations motivate us to impose
Assumption 3.1
The operator \(B\in L(U,X_{-1})\) is \(\infty \)-admissible, and the map \((t,u) \mapsto \Phi (t)u\) is continuous on \(\mathbb {R}_+\times L^\infty (\mathbb {R}_+,U)\).
In particular, this assumption holds if B is a q-admissible operator with \(q<\infty \), see [58, Proposition 2.3].
To define the concept of a mild solution, we also require the following:
Assumption 3.2
We assume that \(B_2\) is zero-class \(\infty \)-admissible and for all \(u\in L^\infty (\mathbb {R}_+,U)\) and any \(x\in C(\mathbb {R}_+,X)\) the map \(s\mapsto f\big (x(s),u(s)\big )\) is in \(L^\infty _{{{\,\textrm{loc}\,}}}(\mathbb {R}_+,V)\).
Due to [20, Proposition 2.5], these conditions ensure that for above x, u the map
is well defined and continuous on \(\mathbb {R}_+\).
Remark 3.4
Assumption 3.2 holds, in particular, if \(B_2 \in L(V,X)\), and
-
(i)
\(f(x,u)=g(x) + Ru\), \(x \in X\), \(u\in U\), where \(R\in L(U,V)\), and g is continuous on X. Indeed, for a continuous x, the map \(s\mapsto g\big (x(s)\big )\) is continuous either, and thus Riemann integrable. The map \(s\mapsto T(t-s) B_2Ru(s)\) is Bochner integrable for any \(u \in L^1_{{{\,\textrm{loc}\,}}}(\mathbb {R}_+,U)\) by [1, Proposition 1.3.4], [23, Lemma 10.1.6]. This ensures that Assumption 3.2 holds.
-
(ii)
If f is continuous on \(X \times U\), and u is piecewise right-continuous, then the map \(s\mapsto f\big (x(s),u(s)\big )\) is also piecewise right-continuous, and thus, it is Riemann integrable.
-
(iii)
(ODE systems). Let \(X=\mathbb {R}^n\), \(U=\mathbb {R}^m\), \(A=0\) (and thus \(T(t)={\text {id}}\) for all t), \(B_2={\text {id}}\), \(B=0\), and f be continuous on \(X \times U\). With these assumptions, the equations (4) take the form
$$\begin{aligned} \dot{x} = f(x,u). \end{aligned}$$(9)Then for each \(u\in L^\infty (\mathbb {R}_+,U)\) and each \(x \in C(\mathbb {R}_+,X)\), the map \(s\mapsto f(x(s),u(s))\) is Lebesgue integrable, and thus, Assumption 3.2 holds. Indeed, as x is a solution of (9) on \([0,\tau )\), x is continuous on \([0,\tau )\). By assumptions, u is measurable on \([0,\tau )\), and f is continuous on \(\mathbb {R}^n \times \mathbb {R}^m\). Arguing similarly to [47, Proposition 7] (where it was shown that a composition of a continuous and measurable function defined on a measurable set E is measurable on E), we see that the map \(q: [0,\tau )\rightarrow \mathbb {R}^n\), \(q(s){:}{=} f(x(s),u(s))\), is a measurable map. As u is essentially bounded, and x and f map bounded sets into bounded sets, q is essentially bounded on \([0,\tau )\). Thus, \(q \in L^{\infty }(\mathbb {R}_+,\mathbb {R}^n)\), and thus, q is integrable on \([0,\tau )\).
Next we define mild solutions of (4).
Definition 3.5
(Mild solutions) Let Assumptions 3.1 and 3.2 hold and \(\tau >0\) be given. A function \(x \in C([0,\tau ], X)\) is called a mild solution of (4) on \([0,\tau ]\) corresponding to certain \(x_0\in X\) and \(u \in L^\infty _{{{\,\textrm{loc}\,}}}(\mathbb {R}_+,U)\), if x solves the integral equation
Here, the integrals are Bochner integrals of \(X_{-1}\)-valued maps.
We say that \(x:\mathbb {R}_+\rightarrow X\) is a mild solution of (4) on \(\mathbb {R}_+\) corresponding to certain \(x_0\in X\) and \(u \in L^\infty _{{{\,\textrm{loc}\,}}}(\mathbb {R}_+,U)\), if \(x|_{[0,\tau ]}\) is a mild solution of (4) (with \(x_0, u\)) on \([0,\tau ]\) for all \(\tau >0\).
3.2 Local existence and uniqueness
Assumptions 3.1, 3.2 guarantee that the integral terms in (10) are well defined. To ensure the existence and uniqueness of mild solutions, we impose further restrictions on f.
Recall the notation \(B_{C,U} = \{v \in U:\Vert v\Vert _U < C\}\) and \(B_{C} = \{v \in X:\Vert v\Vert _X < C\}\).
Definition 3.6
We call \(f:X \times U \rightarrow V\)
-
(i)
Lipschitz continuous (with respect to the first argument) on bounded subsets of X if for any \(C>0\) there is \(L(C)>0\), such that \(\forall x,y \in B_C\), \(\forall v \in B_{C,U}\) it holds that
$$\begin{aligned} \Vert f(y,v)-f(x,v)\Vert _V \le L(C) \Vert y-x\Vert _X. \end{aligned}$$(11) -
(ii)
uniformly globally Lipschitz continuous (with respect to the first argument) if (11) holds for all \(x,y \in X\), and all \(v \in U\) with a constant L that does not depend on x, y, v.
We omit the indication “with respect to the first argument” wherever this is clear from the context.
For the well-posedness analysis, we rely on the following assumption on the nonlinearity f in (4).
Assumption 3.3
The nonlinearity f satisfies the following properties:
-
(i)
\(f: X \times U \rightarrow V\) is Lipschitz continuous on bounded subsets of X.
-
(ii)
\(f(x,\cdot )\) is continuous for all \(x \in X\).
-
(iii)
There exist \(\sigma \in \mathcal {K_\infty }\) and \(c>0\) so that for all \(u \in U\) the following holds:
$$\begin{aligned} \Vert f(0,u)\Vert _V \le \sigma (\Vert u\Vert _U) + c. \end{aligned}$$(12)
Recall the notation for the distances and balls in normed vector spaces, introduced in the end of Sect. 1. Finally, for a set \(\mathcal {S}\subset U\), denote the set of inputs with essential image in \(\mathcal {S}\) as \(\mathcal {U}_\mathcal {S}\):
We start with the following sufficient condition for the existence and uniqueness of solutions of a system (4) with inputs in \(L^\infty (\mathbb {R}_+,U)\).
Recall the notation \(h_0{:}{=}\lim _{t\rightarrow +0}h_t\), where \(h_t\) is defined as in (7).
Theorem 3.7
(Picard–Lindelöf theorem) Let Assumptions 3.1, 3.2, and 3.3 hold.
Assume that \((T(t))_{t\ge 0}\) satisfies for certain \(M \ge 1\), \(\lambda >0\) the estimate
For any compact set \(Q \subset X\), any \(r>0\), any bounded set \(\mathcal {S}\subset U\), and any \(\delta >0\), there is a time \(t_1 = t_1(Q,r,\mathcal {S},\delta )>0\), such that for any \(w\in Q\), for any \(x_0 \in W{:}{=}B_r(w)\), and for any \(u\in \mathcal {U}_{\mathcal {S}}\) there is a unique mild solution of (4) on \([0,t_1]\), and \(\phi ([0,t_1],x_0,u) \subset B_{Mr+h_0\Vert u\Vert _{L^\infty ([0,t_1],U)}+\delta }(w)\).
Proof
First, we show the claim for the case if Q is a single point in X, that is, \(Q=\{w\}\), for some \(\omega \in X\). Pick any \(C>0\) such that \(W {:}{=} B_r(w) \subset B_C\), and \(\mathcal {U}_\mathcal {S}\subset B_{C,\mathcal {U}}\). Pick any \(u \in \mathcal {U}_\mathcal {S}\). Also, take any \(\delta >0\), and consider the following sets (depending on the parameter \(t>0\)):
endowed with the metric \(\rho _{t}(x,y){:}{=}\sup _{s \in [0,t]} \Vert x(s)-y(s)\Vert _X\). As the sets \(Y_t\) are closed subsets of the Banach spaces C([0, t], X), for all \(t>0\), the space \(Y_t\) is a complete metric space.
Pick any \(x_0 \in W\). We are going to prove that for small enough t, the spaces \(Y_{t}\) are invariant under the operator \(\Phi _u\), defined for any \(x \in Y_{t}\) and all \(\tau \in [0,t]\) by
By Assumptions 3.1, 3.2, the function \(\Phi _u(x)\) is continuous for any \(x \in Y_t\).
Fix any \(t>0\) and pick any \(x \in Y_{t}\). As \(x_0 \in W=B_r(w)\), there is \(a \in B_r\) such that \(x_0=w+a\).
Then for any \(\tau <t\), it holds that
Now, for all \(s\in [0,t]\)
In view of Assumption 3.3(iii), it holds that
As \(M\ge 1\), it holds that \(K>C\), and the Lipschitz continuity of f on bounded balls ensures that there is \(L(K)>0\), such that for all \(\tau \in [0,t]\)
Since T is a strongly continuous semigroup, as \(h_t\rightarrow h_0\) whenever \(t\rightarrow +0\), and since \(c_t\rightarrow 0\) as \(t\rightarrow +0\), there exists \(t_1\), such that
This means that \(Y_{t}\) is invariant with respect to \(\Phi _u\) for all \(t \in (0,t_1]\), and \(t_1\) does not depend on the choice of \(x_0 \in W\).
Now, pick any \(t>0\), \(\tau \in [0, t]\), and any \(x, y \in Y_{t}\). It holds that
for \(t \le t_2\), where \(t_2>0\) is a small enough real number, that does not depend on the choice of \(x_0 \in W\).
According to the Banach fixed point theorem, there exists a unique solution of \(x(t)=\Phi _u(x)(t)\) on \([0,\min \{t_1,t_2\}]\), which is a mild solution of (4).
General compact Q Till now, we have shown that for any \(w\in Q\), any \(r>0\), any bounded set \(\mathcal {S}\subset U\), and any \(\delta >0\), there is a time \(t_1 = t_1(w,r,\mathcal {S},\delta )>0\) (that we always take the maximal possible), such that for any \(x_0 \in W{:}{=}B_r(w)\), and for any \(u\in \mathcal {U}_{\mathcal {S}}\) there is a unique solution of (4) on \([0,t_1]\), and it lies in the ball \(B_{Mr+h_0\Vert u\Vert _{L^\infty ([0,t_1],U)}+\delta }(w)\).
It remains to show that \(t_1\) can be chosen uniformly in \(w \in Q\), that is \(\inf _{w \in Q} t_1(w,r,\mathcal {S},\delta )>0\). Let this not be so, that is, \(\inf _{w \in Q} t_1(w,r,\mathcal {S},\delta )=0\). Then, there is a sequence \((w_k) \subset Q\), such that the corresponding times \(\big (t_1(w_k,r,\mathcal {S},\delta )\big )_{k\in \mathbb {N}}\) monotonically decay to zero. As Q is compact, there is a converging subsequence of \((w_k)\), converging to some \(w^* \in Q\). However, \(t_1(w^*,r,\mathcal {S},\delta )>0\), which easily leads to a contradiction. \(\square \)
Remark 3.8
The technique of proving the Picard–Lindelöf theorem is quite classical. Note, however, that here we need to tackle the influence of unbounded input operators, and also we provide a uniform existence time for solutions that controls the maximal deviation of the trajectory from the given set of initial conditions, which is realized by the choice of the spaces \(Y_t\) in (15). This leads to several changes in the proof of the invariance of \(Y_t\) with respect to the operator \(\Phi _u(x)\).
Corollary 3.9
(Picard–Lindelöf theorem for zero-class admissible B and quasi-contractive semigroups) Let Assumptions 3.1, 3.2, 3.3 hold. Let also B be zero-class admissible, and T be a quasi-contractive strongly continuous semigroup; that is, there is \(\lambda >0\) such that
For any bounded ball \(W \subset X\) (with corresponding \(w\in X\) and \(r>0\): \(W= B_r(w)\)), any bounded set \(\mathcal {S}\subset U\), and any \(\delta >0\), there is a time \(t_1 = t_1(W,\mathcal {S},\delta )>0\), such that for any \(x_0 \in W\) and any \(u\in \mathcal {U}_{\mathcal {S}}\) there is a unique solution of (4) on \([0,t_1]\), and it lies in the ball \(B_{r+\delta }(w)\).
Proof
The claim follows directly from Theorem 3.7. \(\square \)
Remark 3.10
Without an assumption of quasi-contractivity, Corollary 3.9 does not hold. Consider the special case \(f \equiv 0\) and \(B\equiv 0\). Then, the system (4) is linear, and for a given \(x_0\in X\) the solution of (4) exists globally and equals \(t\mapsto T(t)x_0\). Now, take \(w{:}{=}0\) and pick any \(r>0\) and \(t_1>0\). Then,
Since T is merely strongly continuous, the map \(t\mapsto \Vert T(t)\Vert \) does not have to be continuous at \(t=0\), and it may happen that \(\lim _{t_1\rightarrow 0}\sup _{\tau \in [0,t_1]}\Vert T(\tau )\Vert >1\).
Hence, in general, it is not possible to prove that the solution starting at arbitrary \(x_0 \in B_r(w)\) will stay in \(B_{r+\delta }(w)\) during a sufficiently small and uniform in \(x_0 \in B_r(w)\) time.
The following example shows that Theorem 3.7 does not hold in general if W is a bounded set (and not only a bounded ball over a compact set), even for linear systems governed by contraction semigroups on a Hilbert space.
Example 3.11
Let \(X=\ell _2\), and consider a diagonal semigroup, defined by \(T(t)x{:}{=}(e^{-kt}x_k)_k\), for all \(x=(x_k)_k\in X\) and all \(t\ge 0\). This semigroup is strongly continuous and contractive. Consider a bounded and closed set \(W{:}{=}\{x \in \ell _2:\ \Vert x\Vert _X =1\}\). Yet \(\Vert T(t)e_k\Vert _X = e^{-kt}\), and thus for each \(\delta \in (0,1)\) and for each time \(t_1>0\), we can find \(k\in \mathbb {N}\), such that \(\Vert T(t_1)e_k\Vert _X<1-\delta \), which means that \(T(t_1)e_k \notin B_\delta (W)\).
At the same time, a stronger Picard–Lindelöf-type theorem can be shown for uniformly continuous semigroups (this encompasses, in particular, the case of infinite ODE systems, also called “ensembles”), which fully extends the corresponding result for ODE systems, see [36, Chapter 1].
Theorem 3.12
(Picard–Lindelöf theorem for uniformly continuous semigroups) Let Assumptions 3.1, 3.2, 3.3 hold. Let further T be a uniformly continuous semigroup (not necessarily quasi-contractive). For any bounded set \(W \subset X\), any bounded set \(\mathcal {S}\subset U\) and any \(\delta >0\), there is a time \(\tau = \tau (W,\mathcal {S},\delta )>0\), such that for any \(x_0 \in W\), and \(u\in \mathcal {U}_{\mathcal {S}}\) there is a unique solution of (4) on \([0,\tau ]\), and it lies in \(B_\delta (W)\).
Proof
First note that since \(A \in L(X)\), for any \(a \in \rho (A)\) the norm \(x \mapsto \Vert (aI-A)^{-1}x\Vert _X\), \(x \in X\), is equivalent to the original norm on X. Thus, \(X = X_{-1}\) up to the equivalence of norms. Hence, as \(B \in L(U,X_{-1})\), then also \(B \in L(U,X)\), and thus, in particular, B is zero-class \(\infty \)-admissible operator.
Pick any \(C>0\) such that \(W \subset B_C\), and \(\mathcal {U}_\mathcal {S}\subset B_{C,\mathcal {U}}\). Take also any \(\delta >0\), and consider the following sets (depending on a parameter \(t>0\)):
endowed with the metric \(\rho _{t}(x,y){:}{=}\sup _{s \in [0,t]} \Vert x(s)-y(s)\Vert _X\), making them complete metric spaces.
Pick any \(x_0 \in W\) and any \(u \in \mathcal {U}_\mathcal {S}\). We are going to prove that for small enough t, the spaces \(Y_{t}\) are invariant under the operator \(\Phi _u\), defined for any \(x \in Y_{t}\) and all \(\tau \in [0,t]\) by (16). By Assumptions 3.1, 3.2, the function \(\Phi _u(x)\) is continuous.
Fix any \(t>0\) and pick any \(x \in Y_{t}\). Then for any \(\tau <t\), it holds that
Now for all \(s\in [0,t]\)
In view of Assumption 3.3(iii), it holds that
Now, Lipschitz continuity of f on bounded balls ensures that there is \(L(K)>0\), such that for all \(\tau \in [0,t]\)
Since T is a uniformly continuous semigroup, \(h_t\rightarrow 0\) as \(t\rightarrow +0\), and \(c_t\rightarrow 0\) as \(t\rightarrow +0\), from this estimate it is clear that there exists \(t_1>0\), depending solely on C and \(\delta \), such that
This means that \(Y_{t}\) is invariant with respect to \(\Phi _u\) for all \(t \in (0,t_1]\), and \(t_1\) does not depend on the choice of \(x_0 \in W\). The rest of the proof is analogous to the proof of Theorem 3.7. \(\square \)
3.3 Well-posedness
Our next aim is to study the prolongations of solutions and their asymptotic properties.
Definition 3.13
Let \(x_1(\cdot )\), \(x_2(\cdot )\) be mild solutions of (4) defined on the intervals \([0,t_1)\) and \([0,t_2)\), respectively, \(t_1,t_2>0\). We call \(x_2\) an extension of \(x_1\) if \(t_2>t_1\), and \(x_2(t)=x_1(t)\) for all \(t \in [0,t_1)\).
Lemma 3.14
Let Assumptions 3.1, 3.2, 3.3 hold. Take any \(x_0 \in X\) and \(u\in \mathcal {U}\). Any two solutions of (4) coincide in their common domain of existence.
The proof is similar to the ODE case [36, Lemma 1.13] as is omitted.
Definition 3.15
A solution \(x(\cdot )\) of (4) is called
-
(i)
maximal if there is no solution of (4) that extends \(x(\cdot )\),
-
(ii)
global if \(x(\cdot )\) is defined on \(\mathbb {R}_+\).
A central property of the system (4) is
Definition 3.16
We say that the system (4) is well-posed if for every initial value \(x_0 \in X\) and every external input \(u \in \mathcal {U}\), there exists a unique maximal solution \(\phi (\cdot ,x_0,u):[0,t_m(x_0,u)) \rightarrow X\), where \(0 < t_m(x_0,u) \le \infty \).
We call \(t_m(x_0,u)\) the maximal existence time of a solution corresponding to \((x_0,u)\).
The map \(\phi \), defined in Definition 3.16, and describing the evolution of the system (4), is called the flow map, or just flow. The domain of definition of the flow \(\phi \) is
In the following pages, we will always deal with maximal solutions. We will usually denote the initial condition by \(x \in X\).
Theorem 3.17
(Well-posedness) Let Assumptions 3.1, 3.2, 3.3 hold. Then, (4) is well-posed.
The proof is similar to the ODE case [36, Theorem 1.16] as is omitted.
Now, we show that well-posed systems (4) are a special case of general control systems, introduced in Definition 2.1.
Theorem 3.18
Let (4) be well-posed. Then, the triple \((X,\mathcal {U},\phi )\), where \(\phi \) is a flow map of (4), constitutes a control system in the sense of Definition 2.1.
Proof
The continuity axiom holds by the definition of a mild solution. Let us check the cocycle property.
Take any initial condition \(x \in X\), any input \(u \in \mathcal {U}\), and any \(t,\tau \ge 0\), such that \([0,t+\tau ]\times \{(x,u)\} \subset D_{\phi }\). Define an input v by \(v(r)=u(r+\tau )\), \(r \ge 0\).
Due to (10), we have:
As \(T_{-1}(t)\) is a bounded operator, it can be taken out of the Bochner integral:
As B is \(\infty \)-admissible, we have that \(\int _0^{\tau } T_{-1}(\tau -s) Bu(s){\text {d}}s \in X\). Since \(T_{-1}(\cdot )\) coincides with \(T(\cdot )\) on X, we infer
Finally,
and the cocycle property holds. The rest of the properties of control systems are fulfilled by construction. \(\square \)
We proceed with a proof of a boundedness-implies-continuation property.
Proposition 3.19
Let Assumptions 3.1, 3.2, 3.3 hold. Then, (4) has the BIC property.
Proof
Pick any \(x \in X\), any \(u \in \mathcal {U}\), and consider the corresponding maximal solution \(\phi (\cdot ,x,u)\), defined on \([0,t_m(x,u))\). Assume that \(t_m(x,u)<+\infty \), but at the same time \(\mathop {\underline{\lim }}_{t\rightarrow t_m(x,u)-0}\Vert \phi (t,x,u)\Vert _X <\infty \). Then, there is a sequence \((t_k)\), such that \(t_k \rightarrow t_m(x,u)\) as \(k\rightarrow \infty \) and \(\lim _{k\rightarrow \infty }\Vert \phi (t_k,x,u)\Vert _X <\infty \). Hence, also \(\sup _{k\in \mathbb {N}}\Vert \phi (t_k,x,u)\Vert _X =:C<\infty \).
Let \(\tau (C)>0\) be a uniform existence time for the solutions starting in the ball \(\overline{B_C}\) subject to inputs of a magnitude not exceeding \(\Vert u\Vert \), which exists and is positive in view of Theorem 3.7. Then, the solution of (4) starting in \(\phi (t_k,x,u)\), corresponding to the input \(u(\cdot + t_k)\), exists and is unique on \([0,\tau (C)]\) by Theorem 3.7, and by the cocycle property, \(\phi (\cdot ,x,u)\) can be prolonged to \([0,t_k +\tau (C))\), which (since \(t_k\rightarrow t_m(x,u)\) as \(k\rightarrow \infty \)) contradicts to the maximality of the solution corresponding to (x, u).
Hence, \(\mathop {\underline{\lim }}_{t\rightarrow t_m(x,u)-0}\Vert \phi (t,x,u)\Vert _X =\infty \), which implies the claim. \(\square \)
3.4 Forward completeness and boundedness of reachability sets
Local Lipschitz continuity guarantees the local existence of solutions. To ensure the global existence of solutions, stronger requirements on nonlinearity are needed.
Proposition 3.20
Let Assumptions 3.1, 3.2, 3.3 hold. Let further f be uniformly globally Lipschitz. Then, (4) is forward complete and has BRS.
Proof
By Theorem 3.7, for any \(x_0 \in X\) and any \(u \in \mathcal {U}\) there exists a mild solution of (4), with a maximal existence time \(t_m (x_0, u)\), which may be finite or infinite. Let \(t_m (x_0, u)\) be finite.
Let \(L>0\) be a uniform global Lipschitz constant for f. As \(\Vert T(t)\Vert \le Me^{\lambda t}\) for some \(M\ge 1\), \(\lambda \ge 0\) and all \(t\ge 0\), for any \(t<t_m (x_0, u)\) we have according to the formula (10) the following estimates
Since \(c_t \rightarrow 0\) whenever \(t \rightarrow +0\), there is some \(t_1 \in (0,t_m (x_0, u))\) such that \(c_{t_1}L\le \frac{1}{2}\). Then, it holds that
Note that \(t_1\) does not depend on \(x_0\) and u. Hence, using cocycle property and with \(\phi (t_1,x_0,u)\) instead of \(x_0\), we obtain a uniform bound for \(\phi (\cdot ,x_0,u)\) on \(2t_1\), \(3t_1\), and so on. Thus, \(\phi (\cdot ,x_0,u)\) is uniformly bounded on \([0,t_m (x_0, u))\), and hence can be prolonged to a larger interval by the BIC property ensured by Proposition 3.19, a contradiction to the definition of \(t_m (x_0, u)\). Overall, \(\Sigma \) is forward complete, and the estimate (19) iterated as above to larger intervals shows that (4) has BRS. \(\square \)
3.5 Regularity of the flow map
We start this section with a basic result describing the exponential deviation between two trajectories.
Theorem 3.21
Let Assumptions 3.1, 3.2, 3.3 hold. Take \(M \ge 1\), \(\lambda \ge 0\) such that \(\Vert T(t)\Vert \le Me^{\lambda t}\) for all \(t\ge 0\). Pick any \(x_1,x_2 \in X\), any \(u\in \mathcal {U}\), and let \(\phi (\cdot ,x_1,u)\) and \(\phi (\cdot ,x_2,u)\) be defined on a certain common interval \([0,\tau ]\).
Then, there exists \(R=R(x_1,x_2,\tau ,u)>0\), such that
Proof
Pick any \(x_1,x_2 \in X\), any \(u \in \mathcal {U}\), and let \(\phi _i(t){:}{=}\phi (t,x_i,u)\), \(i=1,2\) be the corresponding (unique) maximal solutions of (4) (guaranteed by Theorem 3.17), defined on \([0,\tau ]\), for a certain \(\tau >0\).
Set
where K is finite due to the continuity of trajectories.
Due to (10), and using Lipschitz continuity of f (see (11)), we have for any \(t \in [0,\tau ]\):
As \(c_t \rightarrow 0\) as \(t \rightarrow +0\), there is some \(t_1 \in (0,\tau )\) such that \(c_{t_1}L(K)\le \frac{1}{2}\). Note that \(t_1\) depends on \(\tau \) only (as K does).
Then, taking the supremum of the previous expression over [0, t], with \(t<t_1\), we obtain that
Take \(k\in \mathbb {N}\) such that \(kt_1 <\tau \) and \((k+1)t_1>\tau \). Then, using the cocycle property, for any \(l\in \mathbb {N}\), \(l\le k\) and all \(t\in [0,t_1]\) s.t. \(lt_1+t<\tau \) we have
This shows (20). \(\square \)
Definition 3.22
The flow of a forward complete control system \(\Sigma =(X,\mathcal {U},\phi )\), is called Lipschitz continuous on compact intervals (for uniformly bounded inputs), if for any \(\tau >0\) and any \(C>0\) there exists \(L>0\) so that for any \(x_1,x_2 \in \overline{B_C}\), for all \(u \in B_{C,\mathcal {U}}\), it holds that
Theorem 3.21 estimates the deviation between two trajectories. To have a stronger result, showing the Lipschitz continuity of the flow map \(\phi \), we additionally assume the BRS property of (4).
Theorem 3.23
Suppose that Assumptions 3.1, 3.2, 3.3 hold and (4) has BRS. Then, the flow of (4) is Lipschitz continuous on compact intervals for uniformly bounded inputs.
Proof
Take any \(C>0\) and pick any \(x_1,x_2 \in B_C\), and any \(u \in \mathcal {U}\) with \(\Vert u\Vert _{\mathcal {U}} \le C\). Let \(\phi _i(\cdot ){:}{=}\phi (\cdot ,x_i,u)\), \(i=1,2\) be the corresponding maximal solutions of (4). These solutions are global since we assume that (4) is forward complete.
As (4) is BRS, the following quantity is finite for any \(\tau >0\):
Following the lines of the proof of Theorem 3.21, we obtain the claim. \(\square \)
Definition 3.24
Let \(\Sigma =(X,\mathcal {U},\phi )\) be a forward complete control system. We say that the flow \(\phi \) depends continuously on inputs and on initial states, if for all \(x \in X\), \(u \in \mathcal {U}\), \(\tau >0\), and all \(\varepsilon >0\) there exists \(\delta >0\), such that \(\forall x' \in X: \Vert x-x'\Vert _X< \delta \) and \(\forall u' \in \mathcal {U}: \Vert u-u'\Vert _{\mathcal {U}}< \delta \) it holds that
To obtain the continuity of the flow map with respect to both states and inputs, which is important for the application of the density argument, we impose additional conditions on the nonlinearity f.
Theorem 3.25
Let Assumptions 3.1, 3.2, 3.3 hold. Let further there exists \(q \in \mathcal {K_\infty }\) such that for all \(C>0\) there is \(L(C)>0\): for all \(x_1,x_2 \in \overline{B_C}\) and all \(v_1,v_2 \in \overline{B_{C,U}}\) it holds that
If (4) has the BRS property, then the flow of (4) depends continuously on initial states and inputs.
Proof
Pick any time \(\tau >0\). Take any \(C>0\), any \(x_1,x_2 \in \overline{B_C}\), and any \(u_1,u_2 \in \overline{B_{C,\mathcal {U}}}\). Let \(\phi _i(\cdot )=\phi (\cdot ,x_i,u_i)\), \(i=1,2\) be the corresponding global solutions.
Due to (10), we have:
In view of the boundedness of reachability sets for the system (4), we have
As \(\Vert T(t)\Vert \le Me^{\lambda t}\) for some \(M,\lambda \ge 0\) and all \(t\ge 0\), and due to the property (22) with \(L{:}{=}L(K)\) (note that \(K\ge C\)), we can continue above estimates to obtain
Since B is zero-class admissible, there is \(t_1>0\) such that \(c_{t_1} L(K) = \frac{1}{2}\), and thus taking the supremum of both sides over \(t\in [0,t_1]\), we have for all \(t\in [0,t_1]\) that
Thus, for each \(\varepsilon >0\) there is \(\delta >0\) so that for all \(x_2 \in B_\delta (x_1)\) and for all \(u_2 \in B_{\delta ,\mathcal {U}}(u_1)\) it holds that
This establishes the continuity over the interval \([0,t_1]\). To obtain continuity over the interval \([0,\tau ]\), one can follow the strategy in the second part of the proof of Lemma 3.27 (and noting that at all the steps the parameter K does not change). \(\square \)
3.6 Continuity at trivial equilibrium
Without loss of generality, we restrict our analysis to fixed points of the form \((0,0) \in X \times \mathcal {U}\). Note that (0, 0) is in \(X \times \mathcal {U}\) since both X and \(\mathcal {U}\) are linear spaces.
To describe the behavior of solutions near the equilibrium, the following notion is of importance:
Definition 3.26
Consider a system \(\Sigma =(X,\mathcal {U},\phi )\) with equilibrium point \(0\in X\). We say that \(\phi \) is continuous at the equilibrium if for every \(\varepsilon >0\) and for any \(h>0\) there exists a \(\delta = \delta (\varepsilon ,h)>0\), so that \([0,h] \times B_\delta \times B_{\delta ,\mathcal {U}} \subset D_\phi \), and
In this case, we will also say that \(\Sigma \) has the CEP property.
CEP property is a “local in time version” of Lyapunov stability and is important, in particular, for the ISS superposition theorems [40] and for the applications of the non-coercive ISS Lyapunov theory [19].
Lemma 3.27
(Continuity at equilibrium for (4)) Let Assumptions 3.1, 3.2, 3.3 hold, and let \(f(0,0)=0\). Then, \(\phi \) is continuous at the equilibrium.
Proof
Consider the following auxiliary system
where
and the saturation function is given for the vectors z in X and in U, respectively, by
As f satisfies Assumption 3.3, one can show that \(\tilde{f}\) is uniformly globally Lipschitz continuous. Hence, (24) is forward complete and has BRS property by Proposition 3.20.
We denote the flow of (24) by \(\tilde{\phi } = \tilde{\phi }(t,x,u)\). As \(f(x,u)=\tilde{f}(x,u)\) whenever \(\Vert x\Vert _X\le 1\) and \(\Vert u\Vert _U\le 1\), it holds also
provided that \(\Vert u\Vert _{\mathcal {U}}\le 1\), \(\phi (\cdot ,x,u)\) exists on [0, t], and \(\Vert \phi (s,x,u)\Vert _X\le 1\) for all \(s\in [0,t]\).
Pick any \(\varepsilon \in (0,1)\), \(\tau \ge 0\), \(\delta \in (0,\varepsilon )\), \(x \in B_\delta \), and any \(u\in B_{\delta ,\mathcal {U}}\). It holds that
Since (24) has BRS property, by Theorem 3.23, the flow of (24) is Lipschitz continuous on compact time intervals. Hence, there exists a \(L(\tau ,\delta )>0\) so that for all \(t\in [0,\tau ]\)
Let us estimate \(\Vert \tilde{\phi }(t,0,u)\Vert _X\). We have:
Since \(\tilde{f}(0,\cdot )\) is continuous, for any \(\varepsilon _2>0\) there exists \(\delta _2<\delta \) so that \(u(s) \in B_{\delta _2}\) implies that \(\Vert \tilde{f}(0,u(s))-\tilde{f}(0,0)\Vert _X \le \varepsilon _2\). Since \(\tilde{f}(0,0)=0\), for the above u we have \(\Vert \tilde{f}(0,u(s))\Vert _X \le \varepsilon _2\).
As \(\tilde{f}\) is uniformly globally Lipschitz, there is \(L>0\) such that for the inputs satisfying \(\Vert u\Vert _\mathcal {U}\le \delta _2\) we have
As \(c_t\rightarrow 0\) for \(t\rightarrow +0\), there is \(t_1>0\), such that \(c_{t_1}L\le \frac{1}{2}\).
Then, we have that
Combining (25) with (26), we see that whenever \(\Vert x\Vert _X\le \delta _2\) and \(\Vert u\Vert _\mathcal {U}\le \delta _2\), it holds that
Now, for any \(\varepsilon <1\) we can find \(\delta _2<\varepsilon \), such that
As \(\tilde{\phi }(t,x,u) = \phi (t,x,u)\) whenever \(\Vert \tilde{\phi }(t,x,u)\Vert _X<1\), we obtain that
Note that \(t_1\) depends on L only and does not depend on \(\delta _2\). Thus, one can find \(\delta _3<\delta _2\), such that
By the cocycle property, we obtain that
Iterating this procedure, we obtain that there is some \(\omega >0\), such that
This shows the CEP property. \(\square \)
4 Semilinear analytic systems
4.1 Preliminaries for analytic semigroups and admissibility
Recall that \(\omega _0(T)\) denotes the growth bound of the semigroup T. Pick any \(\omega >\omega _0(T)\) and define the space \(X_\alpha \) and the norm in it as
Furthermore, define the spaces \(X_{-\alpha }\) as the completion of X with respect to the norm \(x \mapsto \Vert (\omega I-A)^{-\alpha }x\Vert _X\). For the theory of fractional powers of operators and fractional spaces, see [13] and [15, Section 1.4] and, for a very brief description of the essentials required here, [51].
The following well-known property holds:
Proposition 4.1
Let T be an analytic semigroup on a Banach space X with the generator A. Then for each \(\omega ,\kappa >\omega _0(T)\), each \(\alpha \in [0,1)\), and each \(t>0\) we have \(\textrm{Im}(T(t)) \subset X_\alpha \), and there is \(C_\alpha >0\) such that
Furthermore, the map \(t\mapsto (\omega I-A)^\alpha T(t)\) is continuous on \((0,+\infty )\) in the uniform operator topology.
Next, we formulate a sufficient condition for the zero-class admissibility of input operators for analytic systems. Part (ii) of the following proposition is (up to the zero-class statement) contained in [51, Proposition 2.13]. We, however, provide a short proof based on the statement (i) to be self-contained.
Proposition 4.2
Assume that A generates an analytic semigroup T and \(B\in L(U,X_{-1+\alpha })\) for some \(\alpha \in (0,1)\). Then:
- (i):
-
For any \(\omega >\omega _0(T)\), any \(d \in [0,1)\) the operator \((\omega I-A_{-1})^{d}\) is zero-class p-admissible for any \(p\in (\frac{1}{1-d},+\infty ]\). In particular, for any \(g \in L^p_{{{\,\textrm{loc}\,}}}(\mathbb {R}_+,X)\), the following map
$$\begin{aligned} {\xi :t \mapsto \int _0^t (\omega I-A)^d T(t-s)g(s){\text {d}}s = \int _0^t T_{-1}(t-s)(\omega I-A)^d g(s){\text {d}}s} \nonumber \\ \end{aligned}$$(29)is well defined and continuous on \(\mathbb {R}_+\). Furthermore, for any \(\kappa >\omega _0(T)\) there is \(R=R(\kappa ,d)\) such that for any \(g \in L^\infty _{{{\,\textrm{loc}\,}}}(\mathbb {R}_+,X)\) the following holds:
$$\begin{aligned} \int _0^t \big \Vert (\omega I-A)^d T(t-s) g(s)\big \Vert _X {\text {d}}s \le R t^{1-d} e^{\kappa t} \Vert g\Vert _{L^\infty ([0,t],X)}. \end{aligned}$$(30) - (ii):
-
B is zero-class q-admissible for \(q\in (\frac{1}{\alpha },+\infty ]\).
- (iii):
-
For any \(\omega >\omega _0(T)\), any \(d \in [0,\alpha )\) the operator \((\omega I-A)^{d}B\) is zero-class \(\infty \)-admissible.
Proof
(i). Since T is an analytic semigroup, T(t) maps X to D(A) for any \(t>0\). As \(D(A)\subset X_d\) for all \(d\in [0,1]\), the integrand in (29) is in X for a.e. \(s\in [0,t)\). Let us show the Bochner integrability of X-valued map \(s \mapsto (\omega I-A)^d T(t-s)g(s)\) on [0, t].
As \(g \in L^{1}_{{{\,\textrm{loc}\,}}}(\mathbb {R}_+,X)\), by the criterion of Bochner integrability, g is strongly measurable and \(\int _I \Vert g(s)\Vert _X {\text {d}}s < \infty \) for any bounded interval \(I \subset \mathbb {R}_+\).
Denote by \(\chi _{\Omega }\) the characteristic function of the set \(\Omega \subset \mathbb {R}_+\). Recall that the map \(t\mapsto (\omega I-A)^d T(t)\) is continuous outside of \(t=0\) in view of Proposition 4.1.
If \(g(s) = \chi _{\Omega }(s)x\) for some measurable \(\Omega \subset \mathbb {R}_+\) and \(x \in X\), then the function
is measurable as a product of a measurable scalar function and a continuous (and thus measurable) vector-valued function. By linearity, \(s\mapsto (\omega I-A)^d T(t-s)g(s)\) is strongly measurable if g is a simple function (see [1, Section 1.1] for definitions).
As g is strongly measurable, there is a sequence of simple functions \((g_n)_{n\in \mathbb {N}}\), converging pointwise to g almost everywhere. Consider a sequence
and take any \(s \in [0,t)\) such that \(g_n(s) \rightarrow g(s)\) as \(n\rightarrow \infty \). We have that
Hence, a sequence of strongly measurable functions (31) converges a.e. to \(s\mapsto (\omega I-A)^d T(t-s)g(s)\), and thus, \(s\mapsto (\omega I-A)^d T(t-s)g(s)\) is strongly measurable by [1, Corollary 1.1.2].
Furthermore, for any \(t >0\), using Proposition 4.1, we have that for any \(\kappa >\omega _0(T)\) there is \(C_d>0\) such that
Using Hölder’s inequality with a finite \(p>\frac{1}{1-d}\), we obtain
where \(\frac{1}{b} + \frac{1}{p} = 1\), and thus, b satisfies \(b<\frac{1}{d}\).
Finally, by [1, Theorem 1.1.4], the map \(s \mapsto (\omega I-A)^d T(t-s)g(s)\) is Bochner integrable on each \([0,t]\subset \mathbb {R}_+\). This shows p-admissibility of \((\omega I-A)^d\), and if \(p<+\infty \), (33) implies zero-class p-admissibility of \((\omega I-A)^d\). Continuity of the map \(\xi \) follows from [58, Proposition 2.3].
For the last claim of item (i), we take \(g\in L^\infty _{{{\,\textrm{loc}\,}}}(\mathbb {R}_+,X)\) and continue the estimates in (32) as follows:
and (30) holds with \(R=\frac{C_d}{1-d}\). This implies zero-class \(\infty \)-admissibility of \((\omega I-A)^d\).
(ii). Take any \(\omega >\omega _0(T)\), and consider the corresponding norm on \(X_{-1+\alpha }\):
Thus, the condition \(B\in L(U,X_{-1+\alpha })\) is equivalent to \((\omega I-A)^{-1+\alpha }B \in L(U,X)\). With this in mind, we have
Due to [43, Theorem 2.6.13, p. 74], on \(X_{1-\alpha } = D((\omega I-A)^{1-\alpha })\) it holds that
Now, take any \(f \in L^q_{{{\,\textrm{loc}\,}}}(\mathbb {R}_+,U)\) with \(q>\frac{1}{\alpha }\). Representing
and applying item (i) of this proposition and in particular the estimate (33) with \(d{:}{=}1-\alpha \), \(p{:}{=}q\), and with \(g{:}{=}(\omega I-A)^{-1+\alpha }B f\) we see that B is zero-class q-admissible for \(q \in (\frac{1}{\alpha },+\infty )\).
(iii). It holds that \(\Vert (\omega I-A)^{d} B\Vert _{L(U,X_{-1+\alpha -d})} = \Vert (\omega I-A)^{-1+\alpha }B\Vert _{L(U,X)}\), and item (i) implies the claim. \(\square \)
Proposition 4.3
Assume that A generates an analytic semigroup T and \(B\in L(U,X_{-1+\alpha })\) for some \(\alpha \in (0,1)\).
For any \(\omega >\omega _0(T)\), any \(d \in [0,\alpha )\), and any \(\kappa >\omega _0(T)\), there is \(R>0\) such that for any \(g \in L^\infty _{{{\,\textrm{loc}\,}}}(\mathbb {R}_+,X)\) the map
is continuous in X-norm, and the following holds:
Proof
For \(d<\alpha \) and \(g \in L^\infty _{{{\,\textrm{loc}\,}}}(\mathbb {R}_+,U)\), consider the map
By item (i) of Proposition 4.2, this map is Bochner integrable and in view of (30) with \(1-\alpha +d\) instead of d and \((\omega I-A)^{-1+\alpha }Bg\) instead of g, we see that the map (35) is continuous and (36) holds. \(\square \)
4.2 Semilinear analytic systems and their mild solutions
Consider again the system (4) with \(B_2={\text {id}}\) that we restate next:
In Sect. 3, we have assumed that f is a well-defined map from \(X \times U\) to X. Although it sounds natural, it is, in fact, a quite restrictive assumption, as already basic nonlinearities, such as pointwise polynomial maps, do not satisfy it. Indeed, if \(f(x) = x^2\), where \(x \in X{:}{=}L^2(0,1)\), then f maps X to the space \(L^1(0,1)\). However, as A generates an analytic semigroup, the requirements on f can be considerably relaxed. Namely, we assume in this section that there is \(\alpha \in [0,1]\) such that f is a well-defined map from \(X_\alpha \times U \) to X.
We note that systems (37) without inputs (\(u=0\)) have been analyzed several decades ago, see the classical monographs [15, 43]. The main difference to these works is the presence of unbounded input operators.
Next, we define mild solutions of (37). Note that the nonlinearity f is defined on \(X_\alpha \times U\), and thus, we must require that the mild solution lies in \(X_\alpha \) for all positive times. We cannot expect such a nice behavior for general semigroups, but thanks to the smoothing effect of analytic semigroups, this is what we can expect in the analytic case.
Definition 4.4
Let \(\tau >0\) and \(\alpha \in [0,1]\) be given. A function \(x \in C([0,\tau ], X)\) is called a mild solution of (37) on \([0,\tau ]\) corresponding to certain \(x_0\in X\) and \(u \in L^1_{{{\,\textrm{loc}\,}}}(\mathbb {R}_+,U)\), if \(x(s) \in X_\alpha \) for \(s\in (0,\tau ]\), and x solves the integral equation
We say that \(x:\mathbb {R}_+\rightarrow X\) is a mild solution of (37) on \(\mathbb {R}_+\) corresponding to certain \(x_0\in X\) and \(u \in L^1_{{{\,\textrm{loc}\,}}}(\mathbb {R}_+,U)\), if its restriction to \([0,\tau ]\) is a mild solution of (37) (with \(x_0, u\)) on \([0,\tau ]\) for all \(\tau >0\).
Remark 4.5
Note that if \(\alpha = 0\), then \(X_\alpha = X_0 = X\), and the concept of a mild solution introduced for general and analytic semigroups coincide.
Assumption 4.1
Let the following hold:
-
(i)
\(\alpha \in (0,1)\).
-
(ii)
\(B\in L(U,X_{-1+\alpha +\varepsilon })\) for sufficiently small \(\varepsilon >0\).
-
(iii)
\(f \in C(X_\alpha \times U, X)\), and f is Lipschitz continuous in the first argument in the following sense: For each \(r>0\), there is \(L=L(r)>0\) such that for each \(x_1,x_2 \in B_{r,X_\alpha }\) and all \(u \in B_{r,U}\) it holds that
$$\begin{aligned} \Vert f(x_1,u) - f(x_2,u)\Vert _X \le L\Vert x_1-x_2\Vert _{X_\alpha }. \end{aligned}$$(39) -
(iv)
For all \(u\in L^\infty (\mathbb {R}_+,U)\) and any \(x\in C(\mathbb {R}_+,X)\) with \(x((0,+\infty )) \subset X_\alpha \), the map \(s\mapsto f\big (x(s),u(s)\big )\) is in \(L^p_{{{\,\textrm{loc}\,}}}(\mathbb {R}_+,X)\) with a certain \(p>\frac{1}{1-\alpha }\).
-
(v)
There is \(\sigma \in \mathcal {K_\infty }\) such that
$$\begin{aligned} \Vert f(0,u)\Vert _X \le \sigma (\Vert u\Vert _U) + c,\quad u\in U. \end{aligned}$$
4.3 Local existence and uniqueness
By Proposition 4.2, the condition \(B\in L(U,X_{-1+\alpha +\varepsilon })\) with \(\alpha ,\varepsilon >0\) implies that B is zero-class q-admissible for any \(q\in (\frac{1}{\alpha +\varepsilon },+\infty ]\). This in turn implies that for such q the map \(t\mapsto \int _0^t T_{-1}(t-s) Bu(s){\text {d}}s\) is continuous for any \(u \in L^q(\mathbb {R}_+,U)\), by [58, Proposition 2.3].
By Assumption 4.1(iv), we see that for any \(u\in L^\infty (\mathbb {R}_+,U)\) the map
is well defined and continuous.
Hence, if \(x\in C(\mathbb {R}_+,X)\) with \(x((0,+\infty )) \subset X_\alpha \), then for any \(u\in L^\infty (\mathbb {R}_+,U)\) the right-hand side of (38) is a continuous function of time.
Our next result is the local existence and uniqueness theorem for analytic systems with initial states in \(X_\alpha \) and the inputs in \(\mathcal {U}{:}{=}L^\infty _{{{\,\textrm{loc}\,}}}(\mathbb {R}_+,U)\). Recall the notation \(\mathcal {U}_\mathcal {S}\) from (13).
Theorem 4.6
(Picard–Lindelöf theorem for analytic systems) Let Assumption 4.1 hold. Assume that T is an analytic semigroup, satisfying for certain \(M \ge 1\), \(\lambda >0\) the estimate
For any compact set \(Q \subset X_\alpha \), any \(r>0\), any bounded set \(\mathcal {S}\subset U\), and any \(\delta >0\), there is a time \(t_1 = t_1(Q,r,\mathcal {S},\delta )>0\), such that for any \(w \in Q\), any \(x_0 \in W{:}{=}B_{r,X_\alpha }(w)\), and any \(u\in \mathcal {U}_{\mathcal {S}}\) there is a unique mild solution of (4) on \([0,t_1]\), and it lies in the ball \(B_{Mr+\delta ,X_\alpha }(w)\).
Proof
First, we show the claim for the case if \(Q=\{w\}\) is a single point in \(X_\alpha \).
(i) Take any \(\omega >\omega _0(T)\), and consider the corresponding space \(X_\alpha \). Pick any \(r>0\) and any \(C>0\) such that \(W {:}{=} B_{r,X_\alpha }(w) \subset B_{C,X_\alpha }\), and \(\mathcal {U}_\mathcal {S}\subset B_{C,\mathcal {U}}\). Pick any \(u \in \mathcal {U}_\mathcal {S}\). Take also any \(\delta >0\), and consider the following sets (depending on a parameter \(t>0\)):
endowed with the metric \(\rho _{t}(y_1,y_2){:}{=}\sup _{s \in [0,t]} \Vert y_1(s)-y_2(s)\Vert _X\), which makes \(Y_t\) complete metric spaces for all \(t>0\).
(ii) Pick any \(x_0 \in W\). We are going to prove that C([0, t], X) is invariant under the operator \(\Phi _u\), defined for any \(y \in Y_{t}\) and all \(\tau \in [0,t]\) by
Since \(y\in C([0,t],X)\), the map \(s \mapsto (\omega I-A)^{-\alpha } y(s)\) is in \(C([0,t],X_\alpha )\), as for any \(s_1,s_2 \in [0,\tau ]\) we have that
By Assumption 4.1, the map \(s\mapsto f\big ((\omega I-A)^{-\alpha } y(s),u(s)\big )\) is in \(L^p_{{{\,\textrm{loc}\,}}}(\mathbb {R}_+,X)\), with a certain \(p>\frac{1}{1-\alpha }\). Proposition 4.2 ensures that the map
is continuous.
Since \(B\in L(U,X_{-1+\alpha +\varepsilon })\), Proposition 4.2(ii) implies that
belongs to \(C([0,\tau ],X)\).
Overall, the function \(\Phi _u(y)\) is continuous, and thus, \(\Phi _u\) maps C([0, t], X) to C([0, t], X).
(iii) Now, we prove that for small enough t the spaces \(Y_{t}\) are invariant under the operator \(\Phi _u\).
Fix any \(t>0\) and pick any \(y \in Y_{t}\). As \(x_0 \in W=B_{r,X_\alpha }(w)\), there is \(a \in X_\alpha \): \(\Vert a\Vert _{X_\alpha } <r\) such that \(x_0=w+a\).
Then for any \(\tau <t\), we obtain that
We substitute \(x_0{:}{=}w+a\) into the first term on the right-hand side of the above inequality. The last term we estimate using (28). To estimate the second term, we use that \((\omega I-A)^{\alpha } B \in L(U,X_{-1+\varepsilon })\). By Proposition 4.2, \((\omega I-A)^{\alpha } B\) is zero-class \(\infty \)-admissible, and thus, there is an increasing continuous function \(t \mapsto h_t\) satisfying \(h_0=0\), such that:
To estimate the latter expression, note that
-
\(\Vert (\omega I-A)^\alpha a\Vert _X = \Vert a\Vert _{X_\alpha } < r\).
-
For all \(s\in [0,t]\), we have
$$\begin{aligned} \Vert (\omega I-A)^{-\alpha }y(s)-0\Vert _{X_\alpha } = \Vert y(s)\Vert _X&\le \Vert (\omega I-A)^\alpha w\Vert _X+Mr+\delta \\&\le M(\Vert w\Vert _{X_\alpha }+r) +\delta \le MC+\delta {:}{=}K. \end{aligned}$$ -
In view of Assumption 4.1, it holds that
$$\begin{aligned} \Vert f(0,u(s))\Vert _X \le \sigma (\Vert u(s)\Vert _U) + c,\quad \text { for a.e. } s \in [0,t]. \end{aligned}$$ -
h is a monotonically increasing continuous function.
As \(M\ge 1\), it holds that \(K>C\), and Lipschitz continuity of f on bounded balls ensures that there is \(L(K)>0\), such that for all \(\tau \in [0,t]\)
Since T is a strongly continuous semigroup, and \(h_t\rightarrow 0\) as \(t\rightarrow +0\), from this estimate, it is clear that there exists \(t_1\), such that
This means, that \(Y_{t}\) is invariant with respect to \(\Phi _u\) for all \(t \in (0,t_1]\), and \(t_1\) does not depend on the choice of \(x_0 \in W\).
(iv) Now, pick any \(t>0\), \(\tau \in [0, t]\), and any \(y_1, y_2 \in Y_{t}\). Then, it holds that
for \(t \le t_2\), where \(t_2>0\) is a small enough real number that does not depend on the choice of \(x_0 \in W\).
According to Banach fixed point theorem, there exists a unique \(y \in Y_t\) that is a fixed point of \(\Phi _u\), that is,
on \([0,\min \{t_1,t_2\}]\).
As \((\omega I-A)^{\alpha }\) is invertible with a bounded inverse, y solves (42) if and only if y solves
As \(y \in C([0, \min \{t_1,t_2\}],X)\), the map \(x{:}{=}(\omega I-A)^{-\alpha } y\) is in \(C([0, \min \{t_1,t_2\}],X_\alpha )\), and is the unique mild solution of (37).
(v). General compact Q. Similar to the corresponding part of the proof of Theorem 3.7. \(\square \)
Remark 4.7
For systems without inputs, Theorem 4.6 was shown (in a somewhat different formulation without bounds on the growth of the solution) in [43, Theorem 3.1]. We have proved our local existence result for initial conditions that are in \(X_\alpha \). To ensure local existence and uniqueness for the initial states outside of \(X_\alpha \), stronger requirements on f have to be imposed, see [34, Theorems 7.1.5, 7.1.6].
Introducing the concepts of maximal solutions and of well-posedness and arguing similar to Sects. 3.2, 3.3, we obtain the following well-posedness theorem.
Theorem 4.8
Let A generate an analytic semigroup, Assumption 4.1 hold, and let \(\mathcal {U}{:}{=}L^\infty (\mathbb {R}_+,U)\). Then:
-
(i)
For each \(x\in X_\alpha \) and each \(u \in \mathcal {U}\), there is a unique maximal solution of (37), defined over the certain maximal time-interval \([0,t_m(x,u))\). We denote this solution as \(\phi (\cdot ,x,u)\).
-
(ii)
The triple \(\Sigma {:}{=}(X_\alpha ,\mathcal {U},\phi )\) is a well-defined control system in the sense of Definition 2.1.
-
(iii)
\(\Sigma \) satisfies the BIC property; that is, if for a certain \(x \in X_\alpha \) and \(u\in \mathcal {U}\) we have \(t_m(x,u)<\infty \), then \(\Vert \phi (t,x,u)\Vert _{X_\alpha } \rightarrow \infty \) as \(t \rightarrow t_m(x,u)-0\).
4.4 Global existence
Motivated by [43, Section 6.3, Theorem 3.3], we have the following result guaranteeing the forward completeness and BRS property for semilinear analytic systems.
Theorem 4.9
Let A generate an analytic semigroup, Assumption 4.1 hold, and let \(\mathcal {U}{:}{=}L^\infty (\mathbb {R}_+,U)\). Assume further that there are \(L,c>0\) and \(\sigma \in \mathcal {K_\infty }\) such that
Then, \(\Sigma {:}{=}(X_\alpha ,\mathcal {U},\phi )\) is a forward complete control system.
Proof
Take any positive \(\omega >\omega _0(T)\) and define \(X_\alpha \) as in (27).
We argue by a contradiction. Let \(\Sigma \) be not forward complete. Then there are \((x_0,u)\in X_\alpha \times \mathcal {U}\) such that \(t_m(x_0,u)<\infty \). By Theorem 4.8, we have that \(\Vert \phi (t,x_0,u)\Vert _{X_\alpha } \rightarrow \infty \) as \(t \rightarrow t_m(x_0,u)-0\).
For \(t<t_m(x_0,u)\) denote \(x(t){:}{=}\phi (t,x_0,u)\). As \(x(\cdot ) \subset X_\alpha \), we can apply \((\omega I-A)^\alpha \) along the trajectory \(x(\cdot )\) to obtain
We obtain
We now estimate the second term as in (41), where h is a continuous increasing function with \(h_0=0\). The last term we estimate using (28). Overall:
Defining \(z(t){:}{=}x(t)e^{-\omega t}\), we obtain from the previous estimate that
An analytic version of Gronwall inequality [15, p. 6] shows that z, and hence, x is uniformly bounded on \([0,t_m(x_0,u))\), and BIC property (Theorem 4.8(iii)) shows that \(t_m(x_0,u)\) is not the finite maximal existence time. A contradiction. \(\square \)
4.5 Example: well-posedness of a Burgers’ equation with a distributed input
We consider the following semilinear reaction–diffusion equation of Burgers’ type on a domain \([0,\pi ]\), with distributed input u, boundary input d at \(z=0\), and homogeneous Dirichlet boundary condition at \(\pi \).
Here, \(f: [0,\pi ]\times \mathbb {R}\rightarrow \mathbb {R}\) is measurable in z, locally Lipschitz continuous in x uniformly in z, and
where \(h\in L^2(0,\pi )\), and g is continuous, increasing, and both h and g are positive.
This system with \(u=0\) and \(d=0\) was investigated in [15, p. 57]. Here, we give a detailed analysis of this system with distributed and boundary inputs.
We denote \(X{:}{=}L^2(0,\pi )\). The operator \(A{:}{=} \frac{d^2}{{\text {d}}z^2}\) with the domain \(D(A) = H^2(0,\pi )\cap H^1_0(0,\pi )\) generates an analytic semigroup on X.
We assume that the distributed input u belongs to the space \(\mathcal {U}=L^\infty (\mathbb {R}_+,U)\), with \(U{:}{=}L^2(0,\pi )\), and the boundary input d belongs to \(\mathcal {D}{:}{=}L^\infty (\mathbb {R}_+,\mathbb {R})\).
The system (45) can be reformulated as a semilinear evolution equation
where we slightly abuse the notation and use x as an argument of the evolution equation.
The condition (ii) in Assumption 4.1 characterizing the admissibility properties of the boundary input operator B holds in view of [51, Example 2.16].
The space \(X_{\frac{1}{2}}\) corresponding to the operator A is given by (see [56, Proposition 3.6.1])
which is a Banach space with the norm
The nonlinearity \(F: X_{\frac{1}{2}} \rightarrow X\) in (47) is given by
Proposition 4.10
For each \(x_0 \in X_{\frac{1}{2}}\), each \(u \in \mathcal {U}=L^\infty _{{{\,\textrm{loc}\,}}}(\mathbb {R}_+,U)\), and each boundary input \(d \in \mathcal {D}=L^\infty _{{{\,\textrm{loc}\,}}}(\mathbb {R}_+,\mathbb {R})\), the system (45) possesses a unique maximal mild solution \(\phi (\cdot ,x_0,(u,d))\). The system \(\Sigma = (X_{\frac{1}{2}},\mathcal {U}\times \mathcal {D},\phi )\) is a control system satisfying the BIC property.
Proof
We proceed in 3 steps:
Step 1: F maps bounded sets of \( X_{\frac{1}{2}}\) to bounded sets of X. Since the elements of \(X_{\frac{1}{2}}=H^1_0(0,\pi )\) are absolutely continuous functions, using the Cauchy–Schwarz inequality, we obtain that for any \(x \in X_{\frac{1}{2}}\) it holds that
For any \(x \in X_{\frac{1}{2}}\), consider
Using (48) and (46), we continue the estimates as follows:
Taking the square root and using that \(\sqrt{a+b}\le \sqrt{a} + \sqrt{b}\), for all \(a,b\ge 0\), we finally obtain
This shows that F is well defined as a map from \(X_{\frac{1}{2}}\) to X and F maps bounded sets of \(X_{\frac{1}{2}}\) to bounded sets of X.
Step 2: F is Lipschitz continuous in x. For \(x_0 \in X_{\frac{1}{2}}\), there is a neighborhood V of a compact set \( \{(z,x_0(z)):z\in [0,\pi ]\} \) in \([0,\pi ]\times \mathbb {R}\) and positive constants \(L,\theta \) so that for \((z,x_1) \in V\), \((z,x_2) \in V\) it holds that
Thus, there is a neighborhood U of \(x_0\) in \(X_{\frac{1}{2}}\) such that \(x\in U\) implies that \((z,x(z))\in V\) for a.e. \(z\in [0,\pi ]\) and for \(x_1,x_2 \in U\) it holds that
and using (48) we have that
Taking the square root, we have that
Finally, for any \(x_1,x_2\in X_{\frac{1}{2}}\) it holds that
and again using (48), we proceed to
Combining (50) and (51), we obtain the required Lipschitz property for the function F.
Step 3: Application of general well-posedness theorems. Finally, Theorem 4.6 shows that the system (45) possesses a unique mild solution for each \(x_0 \in X_{\frac{1}{2}}\), each \(u \in \mathcal {U}=L^\infty _{{{\,\textrm{loc}\,}}}(\mathbb {R}_+,U)\), and each boundary input \(d \in \mathcal {D}=L^\infty _{{{\,\textrm{loc}\,}}}(\mathbb {R}_+,\mathbb {R})\). Theorem 4.8 shows that \(\Sigma \) is a control system satisfying the BIC property. \(\square \)
5 Boundary control systems
Control systems governed by partial differential equations are defined by PDEs describing the dynamics inside of the spatial domain and boundary conditions, describing the dynamics of the system at the boundary of the domain. Such systems look (at first glance) quite differently from the evolution equations in Banach spaces, studied in Sect. 3. This motivated the development of a theory of abstract boundary control systems that allows for a more straightforward interpretation of PDEs in the language of semigroup theory.
5.1 Linear boundary control systems
Let X and U be Banach spaces. Consider a system
where the formal system operator \(\hat{A}: D( \hat{A}) \subset X \rightarrow X\) is a linear operator, the control function u takes values in U, and the boundary operator \(\hat{R}: D( \hat{R}) \subset X \rightarrow U\) is linear and satisfies \(D(\hat{A}) \subset D(\hat{R})\).
Definition 5.1
The system (52) is called a linear boundary control system (linear BCS) if the following conditions hold:
-
(i)
The operator \(A: D(A) \rightarrow X\) with \(D(A) = D({\hat{A}} ) \cap \textrm{Ker}\,({\hat{R}})\) defined by
$$\begin{aligned} Ax = {\hat{A}}x \qquad \text {for} \quad x\in D(A) \end{aligned}$$(53)is the infinitesimal generator of a \(C_0\)-semigroup \((T(t))_{t\ge 0}\) on X.
-
(ii)
There is an operator \(R \in \mathcal {L}(U,X)\) such that for all \(u \in U\) we have \(Ru \in D({\hat{A}})\), \({\hat{A}}R \in \mathcal {L}(U,X)\) and
$$\begin{aligned} {\hat{R}}Ru = u, \qquad u\in U. \end{aligned}$$(54)
The operator R in this definition is sometimes called a lifting operator. Note that R is not uniquely defined by the properties in the item (ii).
Item (i) of Definition 5.1 shows that for \(u\equiv 0\) the equations (52) are well-posed. In particular, as A is the generator of a certain strongly continuous semigroup \(T(\cdot )\), for any \(x\in D(A)\), it holds that \(T(t)x \in D(A)\) and thus \(T(t)x \in \textrm{Ker}\,({\hat{R}})\) for all \(t\ge 0\), which means that (52b) is satisfied.
Item (ii) of the definition implies, in particular, that the range of the operator \({\hat{R}}\) equals U, and thus, the values of inputs are not restricted.
5.2 Semilinear boundary control systems
Let \((\hat{A},\hat{R})\) be a linear BCS. We consider \(D(\hat{A}) \subset X\) as a linear space equipped with the graph norm
Motivated by [51], we consider the following class of semilinear boundary control systems.
Definition 5.2
Consider a linear BCS \((\hat{A},\hat{R})\). Consider the following system
with a nonlinearity \(f:X\times W\rightarrow X\), where W is a Banach space.
The system (55) we call a semilinear boundary control system (semilinear BCS).
Following [51], we define classical solutions to the semilinear BCS (55).
Definition 5.3
Let \(x_{0}\in D(\hat{A})\), \(\tau >0\) and \(u\in C([0,\tau ],U)\). A function
is called a classical solution to the semilinear BCS (55) on \([0,\tau ]\) if \(x(t)\in X\) for all \(t>0\) and the equations (55) are satisfied pointwise for \(t\in (0,\tau ]\).
A function \(x:[0,\infty )\rightarrow X\) is called (global) classical solution to the semilinear BCS (55), if \(x|_{[0,\tau ]}\) is a classical solution on \([0,\tau ]\) for every \(\tau >0\).
If \(x\in C([0,\tau ],D(\hat{A}))\cap C^{1}((0,\tau ],X)\) and \(x(t)\in X\) for all \(t>0\) and the equations (55) are satisfied pointwise for \(t\in (0,\tau ]\), then we say that x is a classical solution on \((0,\tau ]\).
The next theorem gives a representation for the (unique) solutions of (55) for smooth enough inputs.
Theorem 5.4
Consider the boundary control system (52) with \(f \in C(X\times W,X)\). Let \(u \in C^2([0,\tau ],U)\), and \(w\in C([0,\tau ],W)\) for some \(\tau >0\), and let \(x_0\in X\) be such that \(x_0 - Ru(0) \in D(A)\). Assume that the classical solution of semilinear BCS \(\phi (\cdot ,x_0,u)\) exists on \([0,\tau ]\). Then, it can be represented as
where \(A_{-1}\) and \(T_{-1}\) are the extensions of the infinitesimal generator A and of the semigroup T to the extrapolation space \(X_{-1}\). Furthermore, \(A_{-1}R \in L(U,X_{-1})\) (and thus \({\hat{A}}R - A_{-1}R \in L(U,X_{-1})\)).
The proof of the linear case (with \(f=0\)) should be well known; see, e.g., [38, Theorem 4.4], [51, pp. 93–94] for the proofs of this fact, and [23, Theorem 11.1.2] for a partial result. The nonlinear result can be obtained in a similar manner. Hence we omit the proof.
An advantage of the representation formula (56a) is in the boundedness of the operators R and \(\hat{A}R\) involved in the expression. Its disadvantage is that the derivative of u is employed. Still, the expression in the right-hand side of (56a) makes sense for any \(x \in X\) and for any \(u \in H^1([0,\tau ],U)\) and can be called a mild solution of BCS (52), as is done, e.g., in [23, p. 146].
The formula (56c) does not involve any derivatives of inputs and again is given in terms of a bounded operator \({\hat{A}}R - A_{-1}R \in L(U,X_{-1})\). Moreover, if we consider the expression in the right-hand side of (56c) in the extrapolation spaces \(X_{-1}\), then it makes sense for all \(x \in X\) and all \(u\in L^{1}_{{{\,\textrm{loc}\,}}}(\mathbb {R}_+,U)\) and constitutes a mild solution of
with
This motivates us to define the mild solutions of semilinear BCS by means of the formula (56c), as was proposed in [51].
Definition 5.5
Let \((\hat{A},\hat{R},f)\) be a semilinear boundary control system with corresponding A, R. Let \(x_{0}\in X\), \(\tau >0\), \(w\in L_{{{\,\textrm{loc}\,}}}^{1}([0,\tau ],W)\), and \(u\in L_{{{\,\textrm{loc}\,}}}^{1}([0,\tau ],U)\). A continuous function \(x:[0,\tau ]\rightarrow X\) is called mild solution to the semilinear BCS (55) on \([0,\tau ]\) if \(x(t)\in X\) for all \(t>0\) and x solves
for all \(t\in [0,\tau ]\) and where \(B=\hat{A}B_{0}-A_{-1}B_{0}\). A function \(x:\mathbb {R}_+ \rightarrow X\) is called a global mild solution if \(x|_{[0,\tau ]}\) is a mild solution on \([0,\tau ]\) for all \(\tau >0\).
In other words, x is a mild solution of a semilinear BCS (5.2), if x is a mild solution of (57) with \(B={\hat{A}}R - A_{-1}R\).
Thus, semilinear boundary control systems are a special case of semilinear evolution equations studied in Sect. 3, and we can use our well-posedness theory for semilinear evolution equations to analyze semilinear BCS.
Availability of data and materials
Not applicable.
References
Arendt W, Batty CJ, Hieber M, Neubrander F (2011) Vector-valued Laplace transforms and Cauchy problems. Springer
Augner B (2016) Stabilisation of infinite-dimensional port-Hamiltonian systems via dissipative boundary feedback. PhD thesis, Bergische Universität Wuppertal
Cazenave T, Haraux A (1998) An introduction to semilinear evolution equations. Oxford University Press, New York
Crandall MG, Liggett TM (1971) Generation of semi-groups of nonlinear transformations on general Banach spaces. Am J Math 93(2):265–298
Crandall MG, Pazy A (1969) Semi-groups of nonlinear contractions and dissipative sets. J Funct Anal 3(3):376–418
Curtain R, Zwart H (2020) Introduction to infinite-dimensional systems theory: a state-space approach. Springer
Daleckii J, Krein M (1974) Stability of solutions of differential equations in Banach space. American Mathematical Society
Dashkovskiy S, Mironchenko A (2013) Input-to-state stability of infinite-dimensional control systems. Math Control Signals Syst 25(1):1–35
Dashkovskiy S, Pavlichkov S (2020) Stability conditions for infinite networks of nonlinear systems and their application for stabilization. Automatica 112:108643
Edalatzadeh MS, Morris KA (2019) Stability and well-posedness of a nonlinear railway track model. IEEE Control Syst Lett 3(1):162–167
Emirsajlow Z, Townley S (2000) From PDEs with boundary control to the abstract state equation with an unbounded input operator: a tutorial. Eur J Control 6(1):27–49
Engel K-J, Nagel R (2000) One-parameter semigroups for linear evolution equations. Springer, New York
Haase M (2006) The functional calculus for sectorial operators. Birkhäuser Verlag, Basel
Hastir A, Califano F, Zwart H (2019) Well-posedness of infinite-dimensional linear systems with nonlinear feedback. Syst Control Lett 128:19–25
Henry D (1981) Geometric theory of semilinear parabolic equations. Springer, Berlin
Hosfeld R, Jacob B, Schwenninger FL (2022) Integral input-to-state stability of unbounded bilinear control systems. Math Control Signals Syst 34:1–23
Huhtala K, Paunonen L (2021) Approximate local output regulation for a class of nonlinear fluid flows. Eur J Control 62:136–142
Jacob B, Dragan V, Pritchard AJ (1995) Infinite dimensional time varying systems with nonlinear output feedback. Integral Equ Oper Theory 22(4):440–462
Jacob B, Mironchenko A, Partington JR, Wirth F (2020) Noncoercive Lyapunov functions for input-to-state stability of infinite-dimensional systems. SIAM J Control Optim 58(5):2952–2978
Jacob B, Nabiullin R, Partington JR, Schwenninger FL (2018) Infinite-dimensional input-to-state stability and Orlicz spaces. SIAM J Control Optim 56(2):868–889
Jacob B, Partington JR, Pott S (2009) Zero-class admissibility of observation operators. Syst Control Lett 58(6):406–412
Jacob B, Schwenninger FL, Zwart H (2019) On continuity of solutions for parabolic control systems and input-to-state stability. J Differ Equ 266:6284–6306
Jacob B, Zwart HJ (2012) Linear port-Hamiltonian systems on infinite-dimensional spaces. Springer, Basel
Jayawardhana B, Logemann H, Ryan EP (2008) Infinite-dimensional feedback systems: the circle criterion and input-to-state stability. Commun Inf Syst 8(4):413–444
Jiang W, Liu B, Zhang Z (2015) Robust observability for regular linear systems under nonlinear perturbation. Electron J Differ Equ 218:1–14
Karafyllis I, Jiang Z-P (2007) A small-gain theorem for a wide class of feedback systems with control applications. SIAM J Control Optim 46(4):1483–1517
Karafyllis I, Jiang Z-P (2011) Stability and stabilization of nonlinear systems. Springer, London
Karafyllis I, Jiang Z-P (2011) A vector small-gain theorem for general non-linear control systems. IMA J Math Control Inf 28:309–344
Karafyllis I, Krstic M (2016) ISS with respect to boundary disturbances for 1-D parabolic PDEs. IEEE Trans Autom Control 61(12):3712–3724
Karafyllis I, Krstic M (2019) Input-to-state stability for PDEs. Springer, Cham
Kawan C, Mironchenko A, Swikir A, Noroozi N, Zamani M (2021) A Lyapunov-based small-gain theorem for infinite networks. IEEE Trans Autom Control 66(12):5830–5844
Kawan C, Mironchenko A, Zamani M (2023) A Lyapunov-based ISS small-gain theorem for infinite networks of nonlinear systems. IEEE Trans Autom Control 68(3):1447–1462
Lhachemi H, Shorten R (2019) ISS property with respect to boundary disturbances for a class of Riesz-spectral boundary control systems. Automatica 109:108504
Lunardi A (2012) Analytic semigroups and optimal regularity in parabolic problems. Springer
Mironchenko A (2021) Non-uniform ISS small-gain theorem for infinite networks. IMA J Math Control Inf 38(4):1029–1045
Mironchenko A (2023) Input-to-state stability: theory and applications. Springer Nature
Mironchenko A, Kawan C, Glück J (2021) Nonlinear small-gain theorems for input-to-state stability of infinite interconnections. Math Control Signals Syst 33:573–615
Mironchenko A, Prieur C (2020) Input-to-state stability of infinite-dimensional systems: recent results and open questions. SIAM Rev 62(3):529–614
Mironchenko A, Prieur C, Wirth F (2021) Local stabilization of an unstable parabolic equation via saturated controls. IEEE Trans Autom Control 66(5):2162–2176
Mironchenko A, Wirth F (2018) Characterizations of input-to-state stability for infinite-dimensional systems. IEEE Trans Autom Control 63(6):1602–1617
Natarajan V, Bentsman J (2016) Approximate local output regulation for nonlinear distributed parameter systems. Math Control Signals Syst 28(3):1–44
Natarajan V, Zhou H-C, Weiss G, Fridman E (2019) Exact controllability of a class of nonlinear distributed parameter systems using back-and-forth iterations. Int J Control 92(1):145–162
Pazy A (1983) Semigroups of linear operators and applications to partial differential equations. Springer, New York
Prieur C, Mazenc F (2012) ISS-Lyapunov functions for time-varying hyperbolic systems of balance laws. Math Control Signals Syst 24(1–2):111–134
Prieur C, Tarbouriech S, da Silva JMG (2016) Wave equation with cone-bounded control laws. IEEE Trans Autom Control 61(11):3452–3463
Ramirez H, Zwart H, Le Gorrec Y (2017) Stabilization of infinite dimensional port-Hamiltonian systems by nonlinear dynamic boundary control. Automatica 85:61–69
Royden H, Fitzpatrick P (2010) Real analysis. Prentice Hall
Schmid J (2019) Weak input-to-state stability: characterizations and counterexamples. Math Control Signals Syst 31(4):433–454
Schmid J (2022) Well-posedness and stability of non-autonomous semilinear input-output systems. Evolution equations and control theory
Schmid J, Zwart H (2021) Stabilization of port-Hamiltonian systems by nonlinear boundary control in the presence of disturbances. ESAIM Control Optim Calc Var 27:53
Schwenninger FL (2020) Input-to-state stability for parabolic boundary control: Linear and semilinear systems. In: Control theory of infinite-dimensional systems. Springer, pp 83–116
Singh S, Weiss G, Tucsnak M (2022) A class of incrementally scattering-passive nonlinear systems. Automatica 142:110369
Staffans O (2005) Well-posed linear systems. Cambridge University Press
Tanwani A, Marx S, Prieur C (2018) Local input-to-state stabilization of 1-D linear reaction-diffusion equation with bounded feedback. In: Proceedings of 23rd international symposium on mathematical theory of networks and systems, pp 576–581
Tanwani A, Prieur C, Tarbouriech S (2018) Stabilization of linear hyperbolic systems of balance laws with measurement errors. In: Control subject to computational and communication constraints. Springer, pp 357–374
Tucsnak M, Weiss G (2009) Observation and control for operator semigroups. Basel, Birkhäuser
Tucsnak M, Weiss G (2014) Well-posed systems—the LTI case and beyond. Automatica 50(7):1757–1779
Weiss G (1989) Admissibility of unbounded control operators. SIAM J Control Optim 27(3):527–545
Zheng J, Zhu G (2018) A De Giorgi iteration-based approach for the establishment of ISS properties for Burgers’ equation with boundary and in-domain disturbances. IEEE Trans Autom Control 64(8):3476–3483
Zheng J, Zhu G (2018) Input-to-state stability with respect to boundary disturbances for a class of semi-linear parabolic equations. Automatica 97:271–277
Acknowledgements
The author thanks the Associate Editor and anonymous Reviewers for useful and constructive comments concerning the first version of this manuscript.
Funding
Open Access funding enabled and organized by Projekt DEAL. This research has been supported by the German Research Foundation (DFG) via the grant MI 1886/2-2.
Author information
Authors and Affiliations
Contributions
Not applicable.
Corresponding author
Ethics declarations
Conflict of interest
There is no conflict of interests.
Ethical approval
Not applicable.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Mironchenko, A. Well-posedness and properties of the flow for semilinear evolution equations. Math. Control Signals Syst. 36, 483–523 (2024). https://doi.org/10.1007/s00498-023-00378-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00498-023-00378-x
Keywords
- Well-posedness
- Evolution equations
- Boundary control systems
- Infinite-dimensional systems
- Analytic systems