Well-posedness and properties of the flow for semilinear evolution equations

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.


Introduction
Semilinear evolution equations.In this work, we analyze the well-posedness and properties of the flow for semilinear evolution equations of the form ẋ(t) = Ax(t) + B 2 f (x(t), u(t)) + Bu(t), t > 0, (1a) x(0) = x 0 .(1b) 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 [43], [15], [6,Chapter 11], [3], 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 [6,23], [56,57], [11] for an overview).In particular, this class includes linear evolution PDEs with boundary inputs.• Consider a linear system ẋ = Ax + Bv, (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.
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 infinitedimensional 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].
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.
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 [56,57,53].[41] ensures 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.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.

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 ∞ -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, 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, boundedness-implies-continuation property, etc.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 [43,15].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 wellposedness 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.E.g., 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.
Notation.By N, R, R + , we denote the sets of natural, real, and nonnegative real numbers, respectively.S denotes the closure of a set S (in a given topology).
By t → a ± 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 ∈ S to the set Z ⊂ S we denote by dist (z, Z) := inf{ z − y S : y ∈ Z}.We denote an open ball of radius r around Z ⊂ S by B r,S (Z) := {y ∈ X : dist (y, Z) < r}, and we set also B r,S (x) := B r,S ({x}) for x ∈ 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 K the class of continuous strictly increasing functions γ : R + → R + , satisfying γ(0) = 0. K ∞ denotes the set of unbounded functions from 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 A := sup x X =1 Ax 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(R + , X) we understand the space of continuous maps from R + to X.The domain of definition, kernel, and image of an operator A we denote by D(A), Ker (A), and Im(A) respectively.By σ(A), we denote the spectrum of a closed operator A : D(A) ⊂ X → X, and by ρ(A) the resolvent set of A. We denote by ω 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 R. We define for p ∈ [1, ∞) the following spaces of vector-valued functions

General class of systems
We start with a general definition of a control system that we adopt from [38].
(ii) A normed vector space of inputs U ⊂ {u : R + → U } endowed with a norm • 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 ∈ U and all τ ≥ 0 the time shift The axiom of concatenation: for all u 1 , u 2 ∈ U and for all t > 0 the concatenation of u 1 and u 2 at time t, defined by The triple Σ is called a (control) system, if the following properties hold: (Σ1) The identity property: for every (x, u) ∈ X × U it holds that φ(0, x, u) = x.(Σ2) Causality: for every (t, x, u) ∈ D φ , for every ũ ∈ U, such that u(s) = ũ(s) for all s ∈ [0, t] it holds that [0, t] × {(x, ũ)} ⊂ D φ and φ(t, x, u) = φ(t, x, ũ).(Σ3) Continuity: for each (x, u) ∈ X × U the map t → φ(t, x, u) is continuous on its maximal domain of definition.(Σ4) The cocycle property: for all x ∈ X, u ∈ U, for all t, h ≥ 0 so that [0, t + h] × {(x, u)} ⊂ D φ , we have φ h, φ(t, x, u), u(t + •) = φ(t + h, x, u).
Definition 2.1 can be viewed as a direct generalization, and a unification of the concepts of strongly continuous nonlinear semigroups [5,4] 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 Σ = (X, U, φ) is forward complete (FC), if D φ = R + × X × U, that is for every (x, u) ∈ X × U and for all t ≥ 0 the value φ(t, x, u) ∈ 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 Σ = (X, U, φ) has bounded reachability sets (BRS), if for any C > 0 and any τ > 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 Σ = (X, U, φ) satisfies the boundedness-implies-continuation (BIC) property if for each (x, u) ∈ X × U with t m (x, u) < ∞ it holds that lim sup t→tm(x,u)−0 φ(t, x, u) X = ∞.

Semilinear evolution equations with unbounded input operators
Consider a Cauchy problem for infinite-dimensional evolution equations of the form where A : D(A) ⊂ X → X generates a strongly continuous semigroup T = (T (t)) t≥0 of bounded linear operators on a Banach space X; U is a Banach space of input values, and x 0 ∈ X is a given initial condition.As the input space, we take U := L ∞ (R + , U ).
The map f : X × U → V is defined on the whole X × U and maps to a Banach space V .Furthermore, B ∈ L(U, X −1 ) and B 2 ∈ L(V, X −1 ).Here the extrapolation space X −1 is the closure of X in the norm x → (aI − A) −1 x X , x ∈ X, where a ∈ ρ(A) (different choices of a ∈ ρ(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).ẋ(t) = Ax(t) + Bu(t), t > 0, (5a) x(0) = x 0 , (5b) 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 ∈ L(U, X −1 ), it is natural to consider the system (5) on the space X −1 .Note that the semigroup (T (t)) t≥0 extends uniquely to a strongly continuous semigroup (T −1 (t)) t≥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 loc from Section 1.The mild solution of (5) for any x ∈ X and u ∈ L 1 loc (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 t 0 T −1 (t − s)Bu(s)ds ∈ X.This leads to the following concept: ) is called a qadmissible control operator for (T (t)) t≥0 , if there is t > 0 so that Define for each t ≥ 0 an operator Φ(t) : Note that as B ∈ L(U, X −1 ), the operators Φ(t) are well-defined as maps from L 1 loc (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 Φ(t) is in X for all t ≥ 0 and Φ(t) ∈ L(L q (R + , U ), X) for all t > 0. Proposition 3.2.Let X, U be Banach spaces and let q ∈ [1, ∞] be given.Then B ∈ 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 ∈ L q loc (R + , U ) it holds that Φ(t)u ∈ X and (7) The function t → h t we assume wlog to be nondecreasing 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 → h t is nondecreasing in t, there is a limit h 0 := lim t→+0 h t ≥ 0, which is not necessarily zero.Operators for which h 0 = 0 deserve a special name.
In particular, this assumption holds if B is a q-admissible operator with q < ∞, see [58,Proposition 2.3].
To define the concept of a mild solution, we also require the following: Due to [20, Proposition 2.5], these conditions ensure that for above x, u the map is well-defined and continuous on R + .Remark 3.4.Assumption 3.2 holds, in particular, if B 2 ∈ L(V, X), and ), and g is continuous on X.Indeed, for a continuous x, the map s → g x(s) is continuous either, and thus Riemann integrable.The map s → T [23,Lemma 10.1.6].This ensures that Assumption 3.2 holds.(ii) If f is continuous on X × U , and u is piecewise right-continuous, then the map s → f x(s), u(s) is also piecewise right-continuous, and thus it is Riemann integrable.
(iii) (ODE systems).Let X = R n , U = R m , A = 0 (and thus T (t) = id for all t), B 2 = id, B = 0, and f be continuous on X × U .With these assumptions the equations ( 4) take the form Then for each u ∈ L ∞ (R + , U ) and each x ∈ C(R + , X) the map s → f (x(s), u(s)) is Lebesgue integrable, and thus Assumption 3.2 holds.
Indeed, as x is a solution of ( 9) on [0, τ ), x is continuous on [0, τ ).By assumptions, u is measurable on [0, τ ), and f is continuous on R n × 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, τ ) → 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, τ ).Thus, q ∈ L ∞ (R + , R n ), and thus q is integrable on [0, τ ).
Here the integrals are Bochner integrals of X −1 -valued maps.
We say that x : R + → X is a mild solution of (4) on R + corresponding to certain is a mild solution of (4) (with x 0 , u) on [0, τ ] for all τ > 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
(ii) uniformly globally Lipschitz continuous (with respect to the first argument) if (11) holds for all x, y ∈ X, and all v ∈ 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: (iii) There exist σ ∈ K ∞ and c > 0 so that for all u ∈ U the following holds: Recall the notation for the distances and balls in normed vector spaces, introduced in the end of Section 1. Finally, for a set S ⊂ U , denote the set of inputs with essential image in S as U S : We start with the following sufficient condition for the existence and uniqueness of solutions of a system (4) with inputs in L ∞ (R + , U ).
Recall the notation h 0 := lim t→+0 h t , where h t is defined as in (7).
Assume that (T (t)) t≥0 satisfies for certain M ≥ 1, λ > 0 the estimate For any compact set Q ⊂ X, any r > 0, any bounded set S ⊂ U , and any δ > 0, there is a time t 1 = t 1 (Q, r, S, δ) > 0, such that for any w ∈ Q, for any x 0 ∈ W := B r (w), and for any u ∈ U S there is a unique mild solution of (4) on [0, t 1 ], and Proof.First, we show the claim for the case if Q is a single point in X, that is, Q = {w}, for some ω ∈ X. Pick any C > 0 such that W := B r (w) ⊂ B C , and U S ⊂ B C,U .Pick any u ∈ U S .Also take any δ > 0, and consider the following sets (depending on the parameter t > 0): endowed with the metric ρ t (x, y) := sup s∈[0,t] x(s) − y(s) 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 ∈ W .We are going to prove that for small enough t, the spaces Y t are invariant under the operator Φ u , defined for any x ∈ Y t and all τ ∈ [0, t] by Fix any t > 0 and pick any x ∈ Y t .As Then for any τ < t, it holds that In view of Assumption 3.3(iii), it holds that As M ≥ 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 τ ∈ [0, t] Since T is a strongly continuous semigroup, as h t → h 0 whenever t → +0, and since c t → 0 as t → +0, there exists t 1 , such that This means, that Y t is invariant with respect to Φ u for all t ∈ (0, t 1 ], and t 1 does not depend on the choice of x 0 ∈ W . Now pick any t > 0, τ ∈ [0, t], and any x, y ∈ Y t .It holds that for t ≤ t 2 , where t 2 > 0 is a small enough real number, that does not depend on the choice of x 0 ∈ W .
According to the Banach fixed point theorem, there exists a unique solution of which is a mild solution of (4).
General compact Q. Till now, we have shown that for any w ∈ Q, any r > 0, any bounded set S ⊂ U , and any δ > 0, there is a time t 1 = t 1 (w, r, S, δ) > 0 (that we always take the maximal possible), such that for any x 0 ∈ W := B r (w), and for any u ∈ U S there is a unique solution of (4) on [0, t 1 ], and it lies in the ball It remains to show that t 1 can be chosen uniformly in w ∈ Q, that is inf w∈Q t 1 (w, r, S, δ) > 0. Let this not be so, that is, inf w∈Q t 1 (w, r, S, δ) = 0. Then there is a sequence (w k ) ⊂ Q, such that the corresponding times t 1 (w k , r, S, δ) k∈N monotonically decay to zero.As Q is compact, there is a converging subsequence of (w k ), converging to some w * ∈ Q.However, t 1 (w * , r, S, δ) > 0, which easily leads to a contradiction.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 Φ 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 λ > 0 such that For any bounded ball W ⊂ X (with corresponding w ∈ X and r > 0: W = B r (w)), any bounded set S ⊂ U , and any δ > 0, there is a time t 1 = t 1 (W, S, δ) > 0, such that for any x 0 ∈ W and any u ∈ U S there is a unique solution of (4) on [0, t 1 ], and it lies in the ball B r+δ (w).
Proof.The claim follows directly from Theorem 3.7.
Remark 3.10.Without an assumption of quasicontractivity, Corollary 3.9 does not hold.Consider the special case f ≡ 0 and B ≡ 0. Then the system (4) is linear, and for a given x 0 ∈ X the solution of (4) exists globally and equals t → 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 → T (t) does not have to be continuous at t = 0, and it may happen that lim t1→0 sup τ ∈[0,t1] T (τ ) > 1.
Hence, in general, it is not possible to prove that the solution starting at arbitrary x 0 ∈ B r (w), will stay in B r+δ (w) during a sufficiently small and uniform in x 0 ∈ 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 = ℓ 2 , and consider a diagonal semigroup, defined by T (t)x := (e −kt x k ) k , for all x = (x k ) k ∈ X and all t ≥ 0. This semigroup is strongly continuous and contractive.Consider a bounded and closed set W := {x ∈ ℓ 2 : x X = 1}.Yet T (t)e k X = e −kt , and thus for each δ ∈ (0, 1) and for each time t 1 > 0, we can find k ∈ N, such that T (t 1 )e k X < 1 − δ, which means that T (t 1 )e k / ∈ B δ (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 quasicontractive).For any bounded set W ⊂ X, any bounded set S ⊂ U and any δ > 0, there is a time τ = τ (W, S, δ) > 0, such that for any x 0 ∈ W , and u ∈ U S there is a unique solution of (4) on [0, τ ], and it lies in B δ (W ).
Proof.First note that since A ∈ L(X), for any a ∈ ρ(A) the norm x → (aI − A) −1 x X , x ∈ X, is equivalent to the original norm on X.Thus X = X −1 up to the equivalence of norms.Hence, as B ∈ L(U, X −1 ), then also B ∈ L(U, X), and thus, in particular, B is zero-class ∞-admissible operator.
Pick any C > 0 such that W ⊂ B C , and U S ⊂ B C,U .Take also any δ > 0, and consider the following sets (depending on a parameter t > 0): endowed with the metric ρ t (x, y) := sup s∈[0,t] x(s)−y(s) X , making them complete metric spaces.
Pick any x 0 ∈ W and any u ∈ U S .We are going to prove that for small enough t, the spaces Y t are invariant under the operator Φ u , defined for any x ∈ Y t and all τ ∈ [0, t] by (16).By Assumptions 3.1, 3.2, the function Φ u (x) is continuous.
Fix any t > 0 and pick any x ∈ Y t .Then for any τ < t, it holds that Now Lipschitz continuity of f on bounded balls ensures that there is L(K) > 0, such that for all τ ∈ [0, t] Since T is a uniformly continuous semigroup, h t → 0 as t → +0, and c t → 0 as t → +0, from this estimate it is clear that there exists t 1 > 0, depending solely on C and δ, such that This means, that Y t is invariant with respect to Φ u for all t ∈ (0, t 1 ], and t 1 does not depend on the choice of x 0 ∈ W .The rest of the proof is analogous to the proof of Theorem 3.7.
The proof is similar to the ODE case [36,Lemma 1.13] as is omitted.Definition 3.15.A solution x(•) of (4) is called (i) maximal if there is no solution of (4) that extends x(•), 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 ∈ X and every external input u ∈ U, there exists a unique maximal solution φ(•, x 0 , u) : [0, t m (x 0 , u)) → X, where 0 < t m (x 0 , u) ≤ ∞.
We call t m (x 0 , u) the maximal existence time of a solution corresponding to (x 0 , u).
The map φ, 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 φ is In the following pages, we will always deal with maximal solutions.We will usually denote the initial condition by x ∈ 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, U, φ), where φ 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 ∈ X, any input u ∈ U, and any t, τ ≥ 0, such that [0, t + τ ] × {(x, u)} ⊂ D φ .Define an input v by v(r) = u(r + τ ), r ≥ 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 ∞-admissible, we have that Finally, and the cocycle property holds.The rest of the properties of control systems are fulfilled by construction.
We proceed with a proof of a boundedness-implies-continuation property.Proof.Pick any x ∈ X, any u ∈ U, and consider the corresponding maximal solution φ(•, x, u), defined on [0, t m (x, u)).Assume that t m (x, u) < +∞, but at the same time lim t→tm(x,u)−0 φ(t, x, u) X < ∞.Then there is a sequence (t k ), such that t k → t m (x, u) as k → ∞ and lim k→∞ φ(t k , x, u) X < ∞.Hence, also sup k∈N φ(t k , x, u) X =: C < ∞.
Let τ (C) > 0 be a uniform existence time for the solutions starting in the ball B C subject to inputs of a magnitude not exceeding u , which exists and is positive in view of Theorem 3.7.Then the solution of (4) starting in φ(t k , x, u), corresponding to the input u(•+t k ), exists and is unique on [0, τ (C)] by Theorem 3.7, and by the cocycle property, φ(•, x, u) can be prolonged to [0, t k + τ (C)), which (since t k → t m (x, u) as k → ∞) contradicts to the maximality of the solution corresponding to (x, u).
Hence lim t→tm(x,u)−0 φ(t, x, u) X = ∞, which implies the claim.Proof.By Theorem 3.7, for any x 0 ∈ X and any u ∈ 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 T (t) ≤ M e λt for some M ≥ 1, λ ≥ 0 and all t ≥ 0, for any t < t m (x 0 , u) we have according to the formula (10) the following estimates Since c t → 0 whenever t → +0, there is some t 1 ∈ (0, t m (x 0 , u)) such that c t1 L ≤ 1 2 .Then it holds that sup Note that t 1 does not depend on x 0 and u.Hence, using cocycle property and with φ(t 1 , x 0 , u) instead of x 0 , we obtain a uniform bound for φ(•, x 0 , u) on 2t 1 , 3t 1 , and so on.Thus, φ(•, 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, Σ is forward complete, and the estimate (19) iterated as above to larger intervals shows that (4) has BRS.
where K is finite due to the continuity of trajectories.
Then, taking the supremum of the previous expression over [0, t], with t < t 1 , we obtain that Take k ∈ N such that kt 1 < τ and (k + 1)t 1 > τ .Then, using the cocycle property, for any l ∈ N, l ≤ k and all t ∈ [0, t 1 ] s.t.lt 1 + t < τ we have This shows (20).
Definition 3.22.The flow of a forward complete control system Σ = (X, U, φ), is called Lipschitz continuous on compact intervals (for uniformly bounded inputs), if for any τ > 0 and any C > 0 there exists L > 0 so that for any x 1 , x 2 ∈ B C , for all u ∈ B C,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 φ, 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 ∈ B C , and any u ∈ U with u U ≤ C. Let φ i (•) := φ(•, 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 τ > 0: Following the lines of the proof of Theorem 3.21, we obtain the claim.
Definition 3.24.Let Σ = (X, U, φ) be a forward complete control system.We say that the flow φ depends continuously on inputs and on initial states, if for all x ∈ X, u ∈ U, τ > 0, and all ε > 0 there exists δ > 0, such that ∀x ′ ∈ X : x − x ′ X < δ and ∀u ′ ∈ U : u − u ′ U < δ 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 ∈ K ∞ such that for all C > 0 there is L(C) > 0: for all If (4) has the BRS property, then the flow of (4) depends continuously on initial states and inputs.
Proof.Pick any time τ > 0. Take any C > 0, any x 1 , x 2 ∈ B C , and any , 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 T (t) ≤ M e λt for some M, λ ≥ 0 and all t ≥ 0, and due to the property (22) with L := L(K) (note that K ≥ C), we can continue above estimates to obtain Since B is zero-class admissible, there is t 1 > 0 such that c t1 L(K) = 1 2 , and thus taking the supremum of both sides over t ∈ [0, t 1 ], we have for all t ∈ [0, Thus, for each ε > 0 there is δ > 0 so that for all x 2 ∈ B δ (x 1 ) and for all u 2 ∈ B δ,U (u 1 ) it holds that This establishes the continuity over the interval [0, t 1 ].To obtain continuity over the interval [0, τ ], 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).
3.6.Continuity at trivial equilibrium.Without loss of generality, we restrict our analysis to fixed points of the form (0, 0) ∈ X × U. Note that (0, 0) is in X × U since both X and 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 Σ = (X, U, φ) with equilibrium point 0 ∈ X.We say that φ is continuous at the equilibrium if for every ε > 0 and for any h > 0 there exists a δ = δ(ε, h) > 0, so that [0, h] × B δ × B δ,U ⊂ D φ , and In this case, we will also say that Σ 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 φ is continuous at the equilibrium.
Proof.Consider the following auxiliary system 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 f is uniformly globally Lipschitz continuous.Hence, ( 24) is forward complete and has BRS property by Proposition 3.20.
As f is uniformly globally Lipschitz, there is L > 0 such that for the inputs satisfying u U ≤ δ 2 we have As c t → 0 for t → +0, there is t 1 > 0, such that c t1 L ≤ 1  2 .Then we have that φ(t, 0, u) Combining (25) with (26), we see that whenever

Now for any
As φ(t, x, u) = φ(t, x, u) whenever φ(t, x, u) X < 1, we obtain that Note that t 1 depends on L only, and does not depend on δ 2 .Thus, one can find By the cocycle property, we obtain that Iterating this procedure, we obtain that there is some ω > 0, such that This shows the CEP property.
4. Semilinear analytic systems 4.1.Preliminaries for analytic semigroups and admissibility.Recall that ω 0 (T ) denotes the growth bound of the semigroup T .Pick any ω > ω 0 (T ) and define the space X α and the norm in it as Furthermore, define the spaces X −α as the completion of X with respect to the norm x → (ωI − A) −α x 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 ω, κ > ω 0 (T ), each α ∈ [0, 1), and each t > 0 we have Im(T (t)) ⊂ X α , and there is C α > 0 such that Furthermore, the map t → (ωI − A) α T (t) is continuous on (0, +∞) 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 ∈ L(U, X −1+α ) for some α ∈ (0, 1).Then: In particular, for any g ∈ L p loc (R + , X), the following map is well-defined and continuous on R + .
Furthermore, for any κ > ω 0 (T ) there is R = R(κ, d) such that for any g ∈ L ∞ loc (R + , X) the following holds: Proof.(i).Since T is an analytic semigroup, T (t) maps X to D(A) for any t > 0. As D(A) ⊂ X d for all d ∈ [0, 1], the integrand in ( 29) is in X for a.e.s ∈ [0, t).Let us show the Bochner integrability of X-valued map s → (ωI As g ∈ L 1 loc (R + , X), by the criterion of Bochner integrability, g is strongly measurable and I g(s) X ds < ∞ for any bounded interval I ⊂ R + .
Denote by χ Ω the characteristic function of the set Ω ⊂ R + .Recall that the map t → (ωI − A) d T (t) is continuous outside of t = 0 in view of Proposition 4.1.
If g(s) = χ Ω (s)x for some measurable Ω ⊂ R + and x ∈ 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 → (ωI As g is strongly measurable, there is a sequence of simple functions (g n ) n∈N , converging pointwise to g almost everywhere.Consider a sequence and take any s ∈ [0, t) such that g n (s) → g(s) as n → ∞.We have that Hence a sequence of strongly measurable functions (31) converges a.e. to s → (ωI − A) d T (t − s)g(s), and thus s → (ωI Furthermore, for any t > 0, using Proposition 4.1, we have that for any κ > ω 0 (T ) there is Using Hölder's inequality with a finite p > 1  1−d , we obtain For the last claim of item (i), we take g ∈ L ∞ loc (R + , X) and continue the estimates in (32) as follows: (ii).Take any ω > ω 0 (T ), and consider the corresponding norm on X −1+α : Thus, the condition With this in mind, we have Due to [43, Theorem 2.6.13,p. 74], on and applying item (i) of this proposition and in particular the estimate (33) with d := 1 − α, p := q, and with g := (ωI − A) −1+α Bf we see that B is zero-class q-admissible for q ∈ ( 1 α , +∞).
For any ω > ω 0 (T ), any d ∈ [0, α), and any κ > ω 0 (T ) there is R > 0 such that for any g ∈ L ∞ loc (R + , X) the map is continuous in X-norm, and the following holds: By item (i) of Proposition 4.2, this map is Bochner integrable and in view of (30) with 1 − α + d instead of d and (ωI − A) −1+α Bg instead of g, we see that the map (35) is continuous and (36) holds.

4.2.
Semilinear analytic systems and their mild solutions.Consider again the system (4) with B 2 = id that we restate next: In Section 3, we have assumed that f is a well-defined map from X × 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 ∈ 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 α ∈ [0, 1] such that f is a well-defined map from X α × 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 α × U , and thus we must require that the mild solution lies in X α 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 τ > 0 and α ∈ [0, 1] be given.A function x ∈ C([0, τ ], X) is called a mild solution of (37) on [0, τ ] corresponding to certain x 0 ∈ X and u ∈ L 1 loc (R + , U ), if x(s) ∈ X α for s ∈ (0, τ ], and x solves the integral equation We say that x : R + → X is a mild solution of (37) on R + corresponding to certain x 0 ∈ X and u ∈ L 1 loc (R + , U ), if its restriction to [0, τ ] is a mild solution of (37) (with x 0 , u) on [0, τ ] for all τ > 0.
Remark 4.5.Note that if α = 0, then X α = X 0 = X, and the concept of a mild solution introduced for general and analytic semigroups coincide.
), 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 ∈ B r,Xα and all u ∈ B r,U it holds that (iv) For all u ∈ L ∞ (R + , U ) and any x ∈ C(R + , X) with x((0, +∞)) ⊂ X α , the map s → f x(s), u(s) is in
Our next result is the local existence and uniqueness theorem for analytic systems with initial states in X α and the inputs in U := L ∞ loc (R + , U ). Recall the notation U S from (13).For any compact set Q ⊂ X α , any r > 0, any bounded set S ⊂ U , and any δ > 0, there is a time t 1 = t 1 (Q, r, S, δ) > 0, such that for any w ∈ Q, any x 0 ∈ W := B r,Xα (w), and any u ∈ U S there is a unique mild solution of (4) on [0, t 1 ], and it lies in the ball B Mr+δ,Xα (w).
Proof.First, we show the claim for the case if Q = {w} is a single point in X α .
(i).Take any ω > ω 0 (T ), and consider the corresponding space X α .Pick any r > 0 and any C > 0 such that W := B r,Xα (w) ⊂ B C,Xα , and U S ⊂ B C,U .Pick any u ∈ U S .Take also any δ > 0, and consider the following sets (depending on a parameter t > 0): endowed with the metric ρ t (y 1 , y 2 ) := sup s∈[0,t] y 1 (s) − y 2 (s) X , which makes Y t complete metric spaces for all t > 0.
(iii).Now we prove that for small enough t the spaces Y t are invariant under the operator Φ u .Fix any t > 0 and pick any y ∈ Y t .As x 0 ∈ W = B r,Xα (w), there is a ∈ X α : a Xα < r such that x 0 = w + a.
Then for any τ < 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 (ωI − A) α B ∈ L(U, X −1+ε ).By Proposition 4.2, (ωI − A) α B is zero-class ∞-admissible, and thus there is an increasing continuous function t → h t satisfying h 0 = 0, such that: To estimate the latter expression, note that • In view of Assumption 4.1, it holds that f (0, u(s)) X ≤ σ( u(s) U ) + c, for a.e.s ∈ [0, t].
• h is a monotonically increasing continuous function.
As M ≥ 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 τ ∈ [0, t] Since T is a strongly continuous semigroup, and h t → 0 as t → +0, from this estimate, it is clear that there exists t 1 , such that This means, that Y t is invariant with respect to Φ u for all t ∈ (0, t 1 ], and t 1 does not depend on the choice of x 0 ∈ W . (iv).Now pick any t > 0, τ ∈ [0, t], and any y 1 , y 2 ∈ Y t .Then it holds that for t ≤ t 2 , where t 2 > 0 is a small enough real number that does not depend on the choice of x 0 ∈ W .
According to Banach fixed point theorem, there exists a unique y ∈ Y t that is a fixed point of Φ u , that is As (ωI − A) α is invertible with a bounded inverse, y solves (42) if and only if y solves As y ∈ C([0, min{t 1 , t 2 }], X), the map x := (ωI−A) −α y is in C([0, min{t 1 , t 2 }], X α ), and is the unique mild solution of (37).
(v).General compact Q.Similar to the corresponding part of the proof of Theorem 3.7.
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 α .To ensure local existence and uniqueness for the initial states outside of X α , 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 Sections 3.2, 3.3, we obtain the following well-posedness theorem.(i) For each x ∈ X α and each u ∈ 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 φ(•, x, u).
(ii) The triple Σ := (X α , U, φ) is a well-defined control system in the sense of Definition 2.1.(iii) Σ satisfies the BIC property, that is if for a certain x ∈ X α and u ∈ U we have t m (x, u) < ∞, then φ(t, x, u) Xα → ∞ as t → t m (x, u) − 0.
For t < t m (x 0 , u) denote x(t) := φ(t, x 0 , u).As x(•) ⊂ X α , we can apply (ωI − A) α along the trajectory x(•) to 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: 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. 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, π], with distributed input u, boundary input d at z = 0, and homogeneous Dirichlet boundary condition at π.
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.
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].
Proof.We proceed in 3 steps: Step 1: F maps bounded sets of X 1 2 to bounded sets of X.Since the elements of X 1 2 = H 1 0 (0, π) are absolutely continuous functions, using the Cauchy-Schwarz inequality, we obtain that for any x ∈ X Using (48) and (46), we continue the estimates as follows: This shows that F is well-defined as a map from X 1 2 to X and F maps bounded sets of X 1 2 to bounded sets of X.
Taking the square root, we have that Finally, for any x 1 , x 2 ∈ X 1 2 it holds that x ′ 2 X , 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 ∈ X 1 2 , each u ∈ U = L ∞ loc (R + , U ), and each boundary input d ∈ D = L ∞ loc (R + , R). Theorem 4.8 shows that Σ is a control system satisfying the BIC property.
then 1-admissible operators are necessarily bounded.At the same time, there are unbounded zeroclass 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 ∈ L(U, X −1 ) is ∞-admissible, and the map

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.

4. 4 .Theorem 4 . 9 .
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.Let A generate an analytic semigroup, Assumption 4.1 hold, and let U := L ∞ (R + , U ). Assume further that there are L, c > 0 and σ