Abstract
We introduce the notions of semiuniform inputtostate stability and its subclass, polynomial inputtostate stability, for infinitedimensional systems. We establish a characterization of semiuniform inputtostate stability based on attractivity properties as in the uniform case. Sufficient conditions for linear systems to be polynomially inputtostate stable are provided, which restrict the range of the input operator depending on the rate of polynomial decay of the product of the semigroup and the resolvent of its generator. We also show that a class of bilinear systems are polynomially integral inputtostate stable under a certain smoothness assumption on nonlinear operators.
Introduction
Consider a semilinear system with state space X and input space U (both Banach spaces):
where A with domain D(A) is the generator of a \(C_0\)semigroup \((T(t))_{t\ge 0}\) on X and \(F:X\times U \rightarrow X\) is a nonlinear operator such that the solution x of (1.1) exists on \([0,\infty )\) for each essentially bounded input u. We are interested in the cases where

the \(C_0\)semigroup \((T(t))_{t\ge 0}\) is semiuniformly stable, that is, \((T(t))_{t\ge 0}\) is uniformly bounded and \(\Vert T(t)(IA)^{1} \Vert \rightarrow 0\) as \(t\rightarrow \infty \);

the \(C_0\)semigroup \((T(t))_{t\ge 0}\) is polynomially stable with parameter \(\alpha >0\), that is, \((T(t))_{t\ge 0}\) is semiuniformly stable and \(\Vert T(t)(IA)^{1} \Vert = O(t^{1/\alpha })\) as \(t\rightarrow \infty \), which means that there is a constant \(M>0\) such that for all \(t>0\),
$$\begin{aligned} \Vert T(t)(IA)^{1} \Vert \le \frac{M}{t^{1/\alpha }}. \end{aligned}$$
In this paper, we introduce and study new notions of inputtostate stability (ISS), which are closely related to semiuniformly stable semigroups and polynomially stable semigroups.
The concept of ISS has been introduced for ordinary differential equations in [31]. This concept combines asymptotic stability with respect to initial states and robustness against external inputs. Motivated by robust stability analysis of partial differential equations, ISS has been recently studied for infinitedimensional systems, e.g., in [10, 13,14,15, 17, 18, 22, 23, 30]; see also the survey [21]. Exponential stability of \(C_0\)semigroups plays an important role in the theory of uniform ISS. The concept of ISS related to strong stability of \(C_0\)semigroups has been also introduced in [22]. This stability concept is called strong ISS.
Exponential stability of \(C_0\)semigroups is a strong property in terms of quantitative asymptotic character and robustness against perturbations. However, we sometimes encounter systems that is strongly stable but not exponentially stable. Strong stability is distinctly qualitative in character unlike exponential stability and has a much weaker asymptotic property than exponential stability. Hence, it does not hold in general that the system (1.1) with generator A of a strongly stable semigroup is uniformly globally stable, that is, there exist functions \(\gamma ,\mu \in {\mathcal {K}}_{\infty }\) such that
for all \(x_0 \in X\), essentially bounded functions \(u: [0,\infty )\rightarrow U\), and \(t\ge 0\), even when \(F(\xi ,v) = Bv\) (\(\xi \in X\), \(v \in U\)) for some bounded linear operator B from U to X, as shown in Theorem 3 of [23]. Here, \({\mathcal {K}}_\infty \) is the set of the classic comparison functions from nonlinear systems theory; see the notation paragraph at the end of this section. The same is true for the sets \(\mathcal {KL}\) and \({\mathcal {K}}\) we will use below.
Semiuniform stability and its subclass, polynomial stability, lie between the two notions of semigroup stability, exponential stability and strong stability, in the sense that semiuniform stability leads to the quantified asymptotic behavior of trajectories with initial states in the domain of the generator. Semiuniformly stable semigroups have been extensively studied, and it has been shown that various partial differential equations such as weakly damped wave equations are semiuniformly stable. We refer, for example, to [3,4,5, 7, 8, 19, 25,26,27,28,29, 34] for the developments of semiuniformly stable semigroups and polynomially stable semigroups. The main motivation of introducing semiuniform and polynomial versions of ISS is to bridge the gap between uniform ISS and strong ISS as in the semigroup case.
In a manner analogous to semiuniform stability of semigroups, we define semiuniform ISS of the system (1.1) as follows. The semilinear system (1.1) is semiuniformly ISS if the system is uniformly globally stable and if there exist functions \(\kappa \in \mathcal {KL}\) and \(\mu \in {\mathcal {K}}_{\infty }\) such that
for all \(x_0 \in D(A)\), essentially bounded functions \(u: [0,\infty )\rightarrow U\), and \(t\ge 0\), where \(\Vert \cdot \Vert _A\) is the graph norm of A, i.e., \(\Vert x_0\Vert _A := \Vert x_0\Vert + \Vert Ax_0\Vert \) for \(x_0 \in D(A)\). We provide a characterization of semiuniform ISS based on attractivity properties called the limit property and the asymptotic gain property. These properties have been introduced in [33] in order to characterize ISS of ordinary differential equations. For infinitedimensional systems, such attractivitybased characterizations have been established for uniform ISS [22, Theorem 5], strong ISS [22, Theorem 12], and weak ISS [30, Theorem 3.1]. Using the attractivity properties, we show that semiuniform ISS implies strong ISS for linear systems and bilinear systems.
The semilinear system (1.1) is called polynomially ISS with parameter \(\alpha >0\) if the system is semiuniformly ISS and if a \(\mathcal {KL}\) function \(\kappa \) in (1.2) satisfies \(\kappa (r,t) = O(t^{1/\alpha })\) as \(t\rightarrow \infty \) for all \(r > 0\). We study polynomial ISS of the linear system
where A is the generator of a polynomially stable semigroup \((T(t))_{t\ge 0}\) with parameter \(\alpha >0\) on X and the input operator B is is a bounded linear operator from U to X. It is readily verified that polynomial ISS is equivalent to infinitetime admissibility of input operators (together with polynomial stability of \(C_0\)semigroups); see, e.g., [36] and [35, Chapter 4] for admissibility. Using this equivalence, we show that the linear system (1.3) is polynomial ISS if B maps into the domain of the fractional power of \(A\) with exponent \(\beta >\alpha \). Similar conditions are placed for the range of a perturbation operator in the robustness analysis of polynomial stability [25,26,27]. This sufficient condition is refined in the case where A is a diagonalizable operator (see Definition 4.5) on a Hilbert space and B is of finite rank. Moreover, when the eigenvalues \((\lambda _n)_{n \in {\mathbb {N}}}\) of the diagonalizable operator A satisfy \({\mathrm {Im}}\lambda _n  {\mathrm{Im}} \lambda _m \ge d\) for some \(d >0\) and for all distinct \(n,m \in {\mathbb {N}}\) with \(\lambda _n,\lambda _m \) near the imaginary axis, we give a necessary and sufficient condition for polynomial ISS. To this end, we utilize the relation between Laplace–Carleson embeddings and infinitetime admissibility established in Theorem 2.5 of [16].
Even in the linear case, uniform global stability and hence polynomial ISS impose a strict condition on boundedness of input operators when \(C_0\)semigroups are polynomially stable but not exponentially stable. This observation motivates us to study a variant of ISS called integral ISS [32]. We introduce a notion of polynomial integral ISS with parameter \(\alpha >0\), which means that there exist functions \(\kappa \in \mathcal {KL}\), \(\gamma , \theta \in {\mathcal {K}}_{\infty }\), and \(\mu \in {\mathcal {K}}\) such that the following three conditions hold: (i) \(\Vert x(t)\Vert \le \gamma (\Vert x_0\Vert )\) for all \(x_0 \in X\) and \(t \ge 0\) in the zeroinput case \(u(t)\equiv 0\); (ii) it holds that
for all \(x_0 \in D(A)\), essentially bounded functions \(u: [0,\infty )\rightarrow U\), and \(t\ge 0\); (iii) \(\kappa (r,t) = O(t^{1/\alpha })\) as \(t\rightarrow \infty \) for all \(r >0\). By definition, we immediately see that the linear system (1.3) is polynomially integral ISS for all generators of polynomially stable semigroups and all bounded input operators. Moreover, we prove that bilinear systems are also polynomially integral ISS provided that the product of the \(C_0\)semigroup and the nonlinear operator has the same polynomial decay rate as \(\Vert T(t)(IA)^{1}\Vert \). This result is a polynomial analogue of Theorem 4.2 in [20] on uniform integral ISS.
This paper is organized as follows. In Sect. 2, we review some basic facts on semiuniform stability and polynomial stability of \(C_0\)semigroups. In Sect. 3, we provide a characterization of semiuniform ISS and investigate the relation between semiuniform ISS and strong ISS. Polynomial ISS of linear systems and polynomial integral ISS of bilinear systems are studied in Sects. 4 and 5, respectively.
Notation: Let \({\mathbb {N}}_0\) and \({\mathbb {R}}_+\) denote the set of nonnegative integers and the set of nonnegative real numbers, respectively. Define \(i{\mathbb {R}} := \{i s : s \in {\mathbb {R}}\}\). For realvalued functions f, g on \({\mathbb {R}}\), we write
if there exist \(M>0\) and \(t_0 \in {\mathbb {R}}\) such that \(f(t) \le Mg(t)\) for all \(t \ge t_0\). Let X and Y be Banach spaces. The space of all bounded linear operators from X to Y is denoted by \({\mathcal {L}}(X,Y)\). We write \({\mathcal {L}}(X) := {\mathcal {L}}(X,X)\). The domain and the range of a linear operator \(A:X \rightarrow Y\) are denoted by D(A) and \({\mathrm{ran}}(A)\), respectively. We denote by \(\sigma (A)\) and \(\varrho (A)\) the spectrum and the resolvent set of a linear operator \(A:D(A) \subset X \rightarrow X\), respectively. We write \(R(\lambda ,A) := (\lambda I  A)^{1}\) for \(\lambda \in \varrho (A)\). The graph norm \(\Vert \cdot \Vert _A\) of a linear operator \(A:D(A) \subset X \rightarrow X\) is defined by \(\Vert x\Vert _A := \Vert x\Vert +\Vert Ax\Vert \) for \(x \in D(A)\). We denote by \(L^{\infty }({\mathbb {R}}_+,X)\) the space of all measurable functions \(f : {\mathbb {R}}_+ \rightarrow X\) such that \(\Vert f\Vert _{\infty } := \mathop {\mathrm{ess~sup}}_{t \in {\mathbb {R}}_+ }\Vert f(t)\Vert < \infty \). We denote by \(C(\varOmega ,X)\) the space of all continuous functions from a topological space \(\varOmega \) to X. Let Z and W be Hilbert spaces. The Hilbert space adjoint of \(T \in {\mathcal {L}}(Z,W)\) is denoted by \(T^*\).
Classes of comparison functions for ISS are defined as follows:
Basic facts on semiuniform stability and polynomial stability of semigroups
We start by recalling the notion of semiuniform stability of \(C_0\)semigroups introduced in Definition 1.2 of [4].
Definition 2.1
A \(C_0\)semigroup \((T(t))_{t\ge 0}\) on a Banach space with generator A is semiuniformly stable if \((T(t))_{t\ge 0}\) is uniformly bounded and satisfies
The following characterization of (2.1) in terms of the intersection of \(\sigma (A)\) with \(i {\mathbb {R}}\) has been established in Theorem 1.1 of [4].
Theorem 2.2
Let A be the generator of a uniformly bounded semigroup \((T(t))_{t\ge 0}\) on a Banach space. Then, (2.1) holds if and only if \(\sigma (A) \cap i {\mathbb {R}}\) is empty.
Quantitative statements on the decay rate, as \(t\rightarrow \infty \), of \(\Vert T(t)R(1,A)\Vert \) and the blowup rate, as \(s \rightarrow \infty \), of R(is, A) have been also given, e.g., in [3,4,5, 7, 19, 29]. In particular, we are interested in semiuniform stability with polynomial decay rates studied in [3, 5, 19]. Note that \(\Vert T(t)R(1,A)\Vert \) and \(\Vert T(t)A^{1}\Vert \) are asymptotically of the same order, which easily follows from the resolvent equation.
Definition 2.3
A \(C_0\)semigroup \((T(t))_{t\ge 0}\) on a Banach space with generator A is polynomially stable with parameter \(\alpha >0\) if \((T(t))_{t\ge 0}\) is semiuniformly stable and satisfies
For the generator A of a uniformly bounded semigroup, \(A\) is sectorial in the sense of [12, Chapter 2]. Therefore, if A is injective, then the fractional power \((A)^{\beta }\) is well defined for \(\beta \in {\mathbb {R}}\). The following result gives the rate of polynomial decay of \(\Vert T(t)(A)^{\beta }\Vert \) for \(\beta >0\); see Proposition 3.1 of [3] for the proof.
Proposition 2.4
Let \((T(t))_{t\ge 0}\) be a uniformly bounded semigroup on a Banach space with generator A such that \(0 \in \varrho (A)\). For fixed \(\alpha ,\beta >0\),
if and only if
For \(C_0\)semigroups generated by normal operators on Hilbert spaces, a spectral condition equivalent to polynomial decay is known. The proof can be found in Proposition 4.1 of [3].
Proposition 2.5
Let \((T(t))_{t\ge 0}\) be the \(C_0\)semigroup on a Hilbert space generated by a normal operator A whose spectrum \(\sigma (A)\) is contained in the open left halfplane \(\{\lambda \in {\mathbb {C}}:{\mathrm{Re}}\lambda <0 \}\). For a fixed \(\alpha >0\),
if and only if there exist \(C,p>0\) such that
for all \(\lambda \in \sigma (A)\) with \({\mathrm{Re}} \lambda > p\).
Characterization of semiuniform inputtostate stability
In this section, we first present the nonlinear system we consider and introduce the notion of semiuniform inputtostate stability. Next, we develop a characterization of this stability. Finally, the relation between semiuniform inputtostate stability and strong inputtostate stability is investigated.
System class
Let X and U be Banach spaces with norm \(\Vert \cdot \Vert \) and \(\Vert \cdot \Vert _U\), respectively. Let \({\mathcal {U}}\) be a normed vector space contained in the space \(L^1_{\text {loc}}({\mathbb {R}}_+,U)\) of all locally integrable functions from \({\mathbb {R}}_+\) to U. We denote by \(\Vert \cdot \Vert _{{\mathcal {U}}}\) the norm on \({\mathcal {U}}\). Assume that \(u(\cdot + \tau ) \in {\mathcal {U}}\) and \(\Vert u\Vert _{{\mathcal {U}}} \ge \Vert u(\cdot + \tau )\Vert _{{\mathcal {U}}} \) for all \(u \in {\mathcal {U}}\) and \(\tau \ge 0\). We are interested in the case \({\mathcal {U}} = L^{\infty }({\mathbb {R}}_+,U)\), but a general space \({\mathcal {U}}\) is used for the definition of semiuniform inputtostate stability and its characterization.
Consider a semilinear system with state space X and input space U:
where A is the generator of a \(C_0\)semigroup \((T(t))_{t\ge 0}\) on X, \(F:X\times U \rightarrow X\) is a nonlinear operator, \(x_0 \in X\) is an initial state, and \(u \in {\mathcal {U}}\) is an input.
Definition 3.1
Suppose that for every \(\tau >0\), \(f \in C([0,\tau ],X)\), and \(g\in {\mathcal {U}}\), the map \(t \mapsto F(f(t),g(t))\) is integrable on \([0,\tau ]\). For \(\tau >0\), a function \(x \in C([0,\tau ],X)\) is called a mild solution of \(\varSigma (A,F)\) on \([0,\tau ]\) if x satisfies the integral equation
Moreover, we say that \(x \in C({\mathbb {R}}_+,X)\) is a mild solution of \(\varSigma (A,F)\) on \({\mathbb {R}}_+\) if \(x_{[0,\tau ]}\) is a mild solution of \(\varSigma (A,F)\) on \([0,\tau ]\) for all \(\tau >0\).
By Proposition 1.3.4 of [2], the integrability of the map \(t \mapsto F(x(t),u(t))\) guarantees that the integral in (3.1) exists as a Bochner integral and is continuous with respect to t. If the nonlinear operator F satisfies certain Lipschitz conditions, then \(\varSigma (A,F)\) has a unique mild solution on \([0,\tau ]\) for some \(\tau >0\). We refer, e.g., to [14, 22] for the mild solution and uniform ISS properties of \(\varSigma (A,F)\) in the case where \({\mathcal {U}}\) is the space of piecewise continuous functions that are bounded and rightcontinuous. In this paper, we sometimes consider a class of bilinear systems whose nonlinear operators F satisfy the next assumption.
Assumption 3.2
The nonlinear operator \(F:X\times U \rightarrow X\) of \(\varSigma (A,F)\) is decomposed into \(F(\xi ,v) = B\xi + G(\xi ,v)\) for all \(\xi \in X\) and \(v \in U\), where \(B \in {\mathcal {L}}(X,U)\) and \(G:X \times U \rightarrow X\) is a nonlinear operator satisfying the following conditions:

1.
\(G(0,v) = 0\) for all \(v \in U\).

2.
For all \(r>0\), there exist \(K_r>0\) and \(\chi _r \in {\mathcal {K}}\) such that for all \(\xi ,\zeta \in X\) with \(\Vert \xi \Vert ,\Vert \zeta \Vert \le r\) and all \(v \in U\),
$$\begin{aligned} \Vert G(\xi ,v)  G(\zeta ,v) \Vert \le K_r \Vert \xi  \zeta \Vert \chi _r (\Vert v\Vert _U). \end{aligned}$$ 
3.
For all \(\tau >0\) and all functions \(f \in C ([0,\tau ], X)\), \(g \in L^{\infty }([0,\tau ],U)\), the map \(t \mapsto G(f(t),g(t))\) is measurable on \([0,\tau ]\).
Note that the nonlinear operator G has the following property under Assumption 3.2: For all \(r>0\), there exist \(K_r>0\) and \(\chi _r \in {\mathcal {K}}\) such that for all \(\xi \in X\) with \(\Vert \xi \Vert \le r\) and all \(v \in U\),
A standard argument using Gronwall’s inequality and Banach’s fixed point theorem shows that there exists a unique mild solution of \(\varSigma (A,F)\) for \({\mathcal {U}} = L^{\infty }({\mathbb {R}}_+,U)\) under Assumption 3.2; see, e.g., Section 4.3.1 of [6] or Lemma 2.8 of [13]. More precisely, if Assumption 3.2 holds, then one of the following statements is true for all initial states \(x_0 \in X\) and all inputs \(u \in L^{\infty }({\mathbb {R}}_+,U)\):

1.
There exists a unique mild solution of \(\varSigma (A,F)\) on \({\mathbb {R}}_+\).

2.
There exist \(t_{\max }\in (0,\infty )\) and \(x \in C([0,t_{\max }),X)\) such that \(x_{[0,\tau ]}\) is a unique mild solution of \(\varSigma (A,F)\) on \([0,\tau ]\) for all \(\tau \in (0, t_{\max })\) and
$$\begin{aligned} \lim _{t \uparrow t_{\max }}\Vert x(t)\Vert = \infty . \end{aligned}$$
Throughout this section, we consider only forward complete systems; see also [33, p. 1284] and [1] for forward completeness.
Definition 3.3
The semilinear system \(\varSigma (A,F)\) is forward complete if there exists a unique mild solution of \(\varSigma (A,F)\) on \({\mathbb {R}}_+\) for all \(x_0 \in X\) and \(u \in {\mathcal {U}}\).
We denote by \(\phi (t,x_0,u)\) the unique mild solution of the forward complete semilinear system \(\varSigma (A,F)\) with initial state \(x_0 \in X\) and input \(u \in {\mathcal {U}}\), i.e.,
The mild solution satisfies the cocycle property
for all \(x_0 \in X\), \(u \in {\mathcal {U}}\), and \(t,\tau \ge 0\).
Definition of semiuniform inputtostate stability
For the forward complete semilinear system \(\varSigma (A,F)\), we introduce the notion of semiuniform inputtostate stability. Before doing so, we recall the definition of uniform global stability; see [33, p. 1285] and [22, Definition 6].
Definition 3.4
The semilinear system \(\varSigma (A,F)\) is called uniformly globally stable (UGS) if the following two conditions hold:

1.
\(\varSigma (A,F)\) is forward complete.

2.
There exist \(\gamma , \mu \in {\mathcal {K}}_{\infty }\) such that
$$\begin{aligned} \Vert \phi (t,x_0,u)\Vert \le \gamma (\Vert x_0\Vert ) + \mu (\Vert u\Vert _{{\mathcal {U}}}) \end{aligned}$$(3.3)for all \(x_0 \in X\), \(u \in {\mathcal {U}}\), and \(t \ge 0\).
Definition 3.5
The semilinear system \(\varSigma (A,F)\) is called semiuniformly inputtostate stable (semiuniformly ISS) if the following two conditions hold:

1.
\(\varSigma (A,F)\) is UGS.

2.
There exist \(\kappa \in \mathcal {KL}\) and \(\mu \in {\mathcal {K}}_{\infty }\) such that
$$\begin{aligned} \Vert \phi (t,x_0,u)\Vert \le \kappa (\Vert x_0\Vert _A, t) + \mu (\Vert u\Vert _{{\mathcal {U}}}) \end{aligned}$$(3.4)for all \(x_0 \in D(A)\), \(u \in {\mathcal {U}}\), and \(t \ge 0\).
In particular, if there exists \(\alpha >0\) such that for all \(r > 0\), \(\kappa (r,t) = O(t^{1/\alpha })\) as \(t \rightarrow \infty \), then \(\varSigma (A,F)\) is called polynomially inputtostate stable (polynomially ISS) with parameter \(\alpha >0\).
Assume that the nonlinear operator \(F:X\times U \rightarrow X\) satisfies \(F(\xi ,0) = 0\) for all \(\xi \in X\). Then, one can easily see that if the semilinear system \(\varSigma (A,F)\) is semiuniformly (resp. polynomially) ISS, then A generates a semiuniformly (resp. polynomially) stable semigroup \((T(t))_{t\ge 0}\) on X. In fact, \(\phi (t,x_0,0) = T(t)x_0\) for all \(x_0 \in X\) and \(t \ge 0\) by assumption. Therefore, \((T(t))_{t\ge 0}\) is uniformly bounded by UGS with \(u(t) \equiv 0\). Take \(\xi \in X\) with \(\Vert \xi \Vert = 1\). Then,
Since the inequality (3.4) with \(u(t) \equiv 0\) yields
for all \(t \ge 0\), it follows that
Hence, \((T(t))_{t\ge 0}\) is semiuniformly stable. Note that semiuniform stability of \((T(t))_{t\ge 0}\) generated by A is equivalent to \(i {\mathbb {R}} \subset \varrho (A)\) by Theorem 2.2. A similar calculation shows that \( \Vert T(t)A^{1}\Vert \le \kappa (1+\Vert A^{1}\Vert ,t) \) for all \(t \ge 0\). Thus, polynomial ISS of \(\varSigma (A,F)\) implies polynomial stability of \((T(t))_{t\ge 0}\).
We conclude this subsection by giving an example of polynomially ISS (but not necessarily uniform ISS) nonlinear systems.
Example 3.6
Let X and U be Banach spaces. Let A be the generator of a polynomially stable semigroup \((T(t))_{t \ge 0}\) with parameter \(\alpha >0\) on X and let \(H \in {\mathcal {L}}(U,X)\) satisfy \({\mathrm{ran}}(H) \subset D((A)^\beta )\) for some \(\beta > \alpha \). Assume that \(q : {\mathbb {R}}_+ \rightarrow {\mathbb {R}}\) satisfies the following conditions:

1.
\(q(0) = 0\).

2.
For all \(r >0\), there exists \(K_r >0\) such that
$$\begin{aligned} q(z)  q(w) \le K_r zw\qquad \forall z,w \in [0,r]. \end{aligned}$$ 
3.
\(\sup _{z \ge 0} q(z) < \infty \).
Define a nonlinear operator \(F:X \times U \rightarrow X\) by
A routine calculation shows that Assumption 3.2 holds for the nonlinear operator F.
We show that \(\varSigma (A,F)\) is polynomially ISS with parameter \(\alpha \) for \({\mathcal {U}}= L^{\infty }({\mathbb {R}}_+,U)\). By Proposition 2.4, there is a constant \(M>0\) such that
Since \((A)^{\beta }\) is closed and since \({\mathrm{ran}}(H) \subset D((A)^\beta )\), it follows that \((A)^{\beta } H \in {\mathcal {L}}(U,X)\). Let \(c :=\sup _{z \ge 0} q(z) < \infty \) and \(t >0\). We obtain
for all \(x\in C([0,t],X)\) and \(u \in L^{\infty }({\mathbb {R}}_+,U)\). From this estimate, we see that \(\varSigma (A,F)\) is polynomially ISS with parameter \(\alpha \) for \({\mathcal {U}}= L^{\infty }({\mathbb {R}}_+,U)\).
Characterization of semiuniform inputtostate stability
We define a semiuniform version of the properties of uniform attractivity and strong attractivity studied in [22]. The attractivity properties has been originally introduced in [33, pp. 1284–1285] in order to characterize ISS of ordinary differential equations.
Definition 3.7
The forward complete semilinear system \(\varSigma (A,F)\) has the semiuniform limit property if there exists \(\mu \in {\mathcal {K}}_{\infty }\) such that the following statement holds: For all \(\varepsilon , r >0\), there is \(\tau = \tau (\varepsilon ,r) < \infty \) such that for all \(x_0 \in D(A)\),
Definition 3.8
The forward complete semilinear system \(\varSigma (A,F)\) has the semiuniform asymptotic gain property if there exists \(\mu \in {\mathcal {K}}_{\infty }\) such that the following statement holds: For all \(\varepsilon ,r >0\), there is \(\tau = \tau (\varepsilon ,r) < \infty \) such that for all \(x_0 \in D(A)\) with \(\Vert x_0\Vert _A \le r\) and all \(u \in {\mathcal {U}}\),
By definition, the asymptotic gain property is stronger than the limit property. We will see that both properties are equivalent if the system is UGS. Moreover, based on these attractivity properties, a characterization of semiuniform ISS is established. The attractivitybased characterization of ISS is useful when the construction of a \(\mathcal {KL}\) function \(\kappa \) is involved. The proof for the semiuniform case is obtained by a slight modification of the proof of Theorem 5 in [22] for the uniform case. We sketch it for the sake of completeness.
Theorem 3.9
The following statements on the semilinear system \(\varSigma (A,F)\) are equivalent:

1.
\(\varSigma (A,F)\) is semiuniformly ISS.

2.
\(\varSigma (A,F)\) is UGS and has the semiuniform limit property.

3.
\(\varSigma (A,F)\) is UGS and has the semiuniform asymptotic gain property.
Proof
[1. \(\Rightarrow \) 2.] Suppose that \(\varSigma (A,F)\) is semiuniformly ISS. By definition, \(\varSigma (A,F)\) is UGS. There exist \(\kappa \in \mathcal {KL}\) and \(\mu \in {\mathcal {K}}_{\infty }\) such that
for all \(x_0 \in D(A)\), \(u \in {\mathcal {U}}\), and \(t \ge 0\). Take \(\varepsilon ,r >0\). We obtain \(\kappa (r,\tau ) \le \varepsilon \) for some \(\tau = \tau (\varepsilon ,r) < \infty \). Therefore, if \(x_0 \in D(A)\) satisfies \(\Vert x_0\Vert _A \le r\), then
Thus, \(\varSigma (A,F)\) has the semiuniform limit property.
[2. \(\Rightarrow \) 3.] Suppose that \(\varSigma (A,F)\) is UGS and has the semiuniform limit property. By assumption, there exist \(\gamma , \mu \in {\mathcal {K}}_{\infty }\) such that the following statement holds: For every \(\varepsilon ,r >0\), there is \(\tau = \tau (\varepsilon ,r) < \infty \) such that for all \(x_0 \in D(A)\),
and for all \(s \ge 0\),
Using the cocycle property (3.2), we obtain
Since \(\gamma (a+b) \le \gamma (2a) + \gamma (2b)\) for all \(a,b \ge 0\), it follows that
where \({{\widetilde{\mu }}} := \gamma \circ (2\mu ) +\mu \in {\mathcal {K}}_{\infty }\).
Choose \({{\widetilde{\varepsilon }}}, {{\widetilde{r}}}>0\) arbitrarily and set
We have shown that there is
such that for all \(x_0 \in D(A)\) with \(\Vert x_0\Vert _A \le {{\widetilde{r}}}\) and all \(u \in {\mathcal {U}}\),
Hence, \(\varSigma (A,F)\) has the semiuniform asymptotic gain property.
[3. \(\Rightarrow \) 1.] Suppose that \(\varSigma (A,F)\) is UGS and has the semiuniform asymptotic gain property. There exist \(\gamma ,\mu \in {\mathcal {K}}_{\infty }\) such that the following two properties hold:

(a)
For all \(x_0 \in X\), \(u \in {\mathcal {U}}\), and \(t \ge 0\),
$$\begin{aligned} \Vert \phi (t,x_0,u)\Vert \le \gamma (\Vert x_0\Vert ) + \mu (\Vert u\Vert _{\mathcal {U}}). \end{aligned}$$(3.5) 
(b)
For all \(\varepsilon ,r >0\), there is \(\tau = \tau (\varepsilon ,r) < \infty \) such that for all \(x_0 \in D(A)\) with \(\Vert x_0\Vert _A \le r\) and all \(u \in {\mathcal {U}}\),
$$\begin{aligned} t \ge \tau \quad \Rightarrow \quad \Vert \phi (t,x_0,u)\Vert \le \varepsilon + \mu (\Vert u\Vert _{\mathcal {U}}). \end{aligned}$$(3.6)
Let \(r >0\). Set \(\varepsilon _n := 2^{n}\gamma (r)\) for \(n \in {\mathbb {N}}_0\) and \(\tau _0 := 0\). By the property (3.6), there exist \(\tau _n = \tau _n(\varepsilon _n,r)\), \(n \in {\mathbb {N}}\), such that for all \(x_0 \in D(A)\) with \(\Vert x_0\Vert _A \le r\) and all \(u \in {\mathcal {U}}\),
For \(n=0\), we also obtain (3.7) by the property (3.5) and the inequality
We may assume without loss of generality that \( \inf _{n \in {\mathbb {N}}} (\tau _n  \tau _{n1}) >0. \) For these sequences \((\varepsilon _n)_{n \in {\mathbb {N}}_0}\) and \((\tau _n)_{n \in {\mathbb {N}}_0}\), one can construct a function \(\kappa \in \mathcal {KL}\) satisfying
see the proof of Lemma 7 of [22] for the detailed construction. From (3.7) and (3.8), we have that for all \(x_0 \in D(A)\) with \(\Vert x_0\Vert _A \le r\) and all \(u \in {\mathcal {U}}\),
Take \(x_0 \in D(A)\) and \(u \in {\mathcal {U}}\) arbitrarily. If \(\Vert x_0\Vert _A = 0\), then the property (3.5) yields
If \(\Vert x_0\Vert _A >0\), then it follows from (3.9) with \(r:= \Vert x_0\Vert _A\) that
Thus, \(\varSigma (A,F)\) is semiuniformly ISS. \(\square \)
Relation between semiuniform inputtostate stability and strong inputtostability
After recalling the notion of strong inputtostate stability introduced in [22, Definition 13], we study its relation to semiuniform ISS with the help of the characterization in Theorem 3.9.
Definition 3.10
The semilinear system \(\varSigma (A,F)\) is strongly inputtostate stable (strongly ISS) if \(\varSigma (A,F)\) is forward complete and if there exist \(\gamma , \mu \in {\mathcal {K}}_{\infty }\) and \(\kappa : X \times {\mathbb {R}}_+ \rightarrow {\mathbb {R}}_+\) such that the following three conditions hold:

1.
\(\kappa (x_0,\cdot ) \in {\mathcal {L}}\) for all \(x_0 \in X\) with \(x_0\not =0\).

2.
\(\kappa (x_0,t) \le \gamma (\Vert x_0\Vert )\) for all \(x_0 \in X\) and \(t \ge 0\).

3.
\(\Vert \phi (t,x_0,u)\Vert \le \kappa (x_0,t) + \mu (\Vert u\Vert _{{\mathcal {U}}})\) for all \(x_0 \in X\), \(u \in {\mathcal {U}}\), and \(t \ge 0\).
For some special classes of semilinear systems, semiuniform ISS implies strong ISS.
Theorem 3.11
Assume that the operator F of \(\varSigma (A,F)\) satisfies one of the following conditions:

1.
There exists \(B \in {\mathcal {L}}(U,X)\) such that \(F(\xi ,v) = Bv\) for all \(\xi \in X\) and \(v \in U\).

2.
\(F(\xi \zeta ,v) = F(\xi ,v)  F(\zeta ,v) \) for all \(\xi ,\zeta \in X\) and \(v \in U\).
Then, semiuniform ISS implies strong ISS for \(\varSigma (A,F)\).
Proof
By Theorem 12 of [22], the semilinear system \(\varSigma (A,F)\) is strongly ISS if and only if \(\varSigma (A,F)\) is UGS and has the strong asymptotic gain property, which means that there exists \(\mu \in {\mathcal {K}}_{\infty }\) such that the following statement holds: For all \(\varepsilon >0\) and \(x_0 \in X\), there exists \(\tau = \tau (\varepsilon ,x_0)<\infty \) such that for all \(u \in {\mathcal {U}}\),
It suffices to show that semiuniform ISS implies the strong asymptotic gain property.
1. Assume that there exists \(B \in {\mathcal {L}}(U,X)\) such that \(F(\xi ,v) = Bv\) for all \(\xi \in X\) and \(v \in U\). By linearity, \( \phi (t,x_0,u) = \phi (t,x_0,0) + \phi (t,0,u) \) for all \(x_0 \in X\), \(u \in {\mathcal {U}}\), and \(t \ge 0\). Since \(\varSigma (A,F)\) is semiuniformly ISS, \((T(t))_{t\ge 0}\) is uniformly bounded and \(\lim _{t\rightarrow \infty }T(t)x_0 = 0\) as \(t \rightarrow \infty \) for all \(x_0 \in D(A)\). Hence, strong stability of \((T(t))_{t\ge 0}\) follows by the density of D(A); see also Proposition A.3 of [11]. For all \(\varepsilon >0\) and \(x_0 \in X\), there exists \(\tau = \tau (\varepsilon ,x_0)<\infty \) such that
Since \(\varSigma (A,F)\) is UGS, there exists \(\mu \in {\mathcal {K}}_{\infty }\) such that
Thus, \(\varSigma (A,F)\) has the strong asymptotic gain property.
2. Assume that \(F(\xi \zeta ,v) = F(\xi ,v)  F(\zeta ,v) \) for all \(\xi ,\zeta \in X\) and \(v \in U\). Since \(\varSigma (A,F)\) is UGS and has the semiuniform asymptotic gain property by Theorem 3.9, there exist \(\gamma ,\mu \in {\mathcal {K}}_{\infty }\) such that the following two properties hold:

(a)
For all \(x_0 \in X\), \(u \in {\mathcal {U}}\), and \(t \ge 0\),
$$\begin{aligned} \Vert \phi (t,x_0,u)\Vert \le \gamma (\Vert x_0\Vert ) + \mu (\Vert u\Vert _{\mathcal {U}}). \end{aligned}$$(3.11) 
(b)
For all \(\varepsilon , r >0\), there exists \(\tau = \tau (\varepsilon ,r) < \infty \) such that for all \(x_0 \in D(A)\) with \(\Vert x_0\Vert _A \le r\) and all \(u \in {\mathcal {U}}\),
$$\begin{aligned} t \ge \tau \quad \Rightarrow \quad \Vert \phi (t,x_0,u)\Vert \le \varepsilon + \mu (\Vert u\Vert _{{\mathcal {U}}}). \end{aligned}$$(3.12)
Take \(\varepsilon >0\) and \(x_0 \in X\). There exists \(y_0 \in D(A)\) such that \(\Vert x_0  y_0\Vert \le \gamma ^{1}(\varepsilon /2)\). By assumption, for all \(u \in {\mathcal {U}}\), \(\phi (t,x_0,u)\phi (t,y_0,u)\) is the mild solution of \(\varSigma (A,F)\) with initial state \(x_0 y_0\) and input u. Therefore, the property (3.11) implies that
for all \(u \in {\mathcal {U}}\) and \(t \ge 0\).
Since \(y_0 \in D(A)\), it follows from the property (3.12) that in the case \(y_0 \not =0\), there exists \(\tau = \tau (\varepsilon ,\Vert y_0\Vert _A) <\infty \) such that for all \(u \in {\mathcal {U}}\),
In the case \(y_0 = 0\), the property (3.11) yields that (3.14) holds with \(\tau = 0\). Combining the estimates (3.13) and (3.14), we obtain
for all \(u \in {\mathcal {U}}\), where \({{\widetilde{\mu }}} := 2\mu \in {\mathcal {K}}_{\infty }\). Since \(y_0\) depends only on \(\varepsilon \) and \(x_0\), it follows that \(\varepsilon \) and \(x_0\) determine \(\tau = \tau (\varepsilon ,\Vert y_0\Vert _A) \). Thus, \(\varSigma (A,F)\) has the strong asymptotic gain property. \(\square \)
Suppose that \(\varSigma (A,F)\) is strong ISS. If the input \(u \in {\mathcal {U}}\) satisfies
then \(\Vert \phi (t,x_0,u)\Vert \rightarrow 0\) as \(t \rightarrow \infty \) for all \(x_0 \in X\); see Lemma 2.5 of [30], where this convergence result has been proved under a weaker assumption. We obtain a convergence property of semiuniform ISS as a corollary of Theorem 3.11.
Corollary 3.12
Under the same assumption on the operator F as in Theorem 3.11, if \(\varSigma (A,F)\) is semiuniformly ISS, then \(\Vert \phi (t,x_0,u)\Vert \rightarrow 0\) as \(t \rightarrow \infty \) for all \(x_0 \in X\) and all \(u \in {\mathcal {U}}\) satisfying \(\Vert u(\cdot +\tau )\Vert _{{\mathcal {U}}} \rightarrow 0\) as \(\tau \rightarrow \infty \).
Polynomial inputtostate stability of linear systems
In this section, we focus on polynomial ISS of linear systems for \({\mathcal {U}} = L^{\infty }({\mathbb {R}}_+,U)\). First, we give a sufficient condition for general linear systems to be polynomially ISS. Next, we consider linear systems with diagonalizable generators and finiterank input operators and refine the sufficient condition. Finally, a necessary and sufficient condition for polynomial ISS is presented in the case where the eigenvalues of the diagonalizable generator near the imaginary axis have uniformly separated imaginary parts.
Polynomial inputtostate stability for general linear systems
Let X and U be Banach spaces. Consider a linear system with state space X and input space U:
where A is the generator of a \(C_0\)semigroup \((T(t))_{t\ge 0}\) on X, \(B \in {\mathcal {L}}(U,X)\) is an input operator, \(x_0 \in X\) is an initial state, and \(u \in L^{\infty }({\mathbb {R}}_+,U)\) is an input.
To study ISS of linear systems, we employ the notion of admissibility studied in the seminal work [36].
Definition 4.1
We call the operator \(B \in {\mathcal {L}}(U,X)\) infinitetime \(L^{\infty }\)admissible for a \(C_0\)semigroup \((T(t))_{t\ge 0}\) on X if there exists a constant \(c >0\) such that
for all \(u \in L^{\infty }({\mathbb {R}}_+,U)\) and \(t \ge 0\).
Let \((T(t))_{t\ge 0}\) be a \(C_0\)semigroup on X and let \(B \in {\mathcal {L}}(U,X)\). If there exists \(\mu \in {\mathcal {K}}_{\infty }\) such that
then
for all \(u \in L^{\infty }({\mathbb {R}}_+,U) \setminus \{0\}\) and \(t \ge 0\). Hence, B is infinitetime \(L^{\infty }\)admissible for \((T(t))_{t\ge 0}\).
As in the case of strong ISS [23, Proposition 1], polynomial ISS for \({\mathcal {U}} = L^{\infty }({\mathbb {R}}_+,U)\) is equivalent to the combination of polynomial stability of \(C_0\)semigroups and infinitetime \(L^{\infty }\)admissibility of input operators.
Lemma 4.2
Let X and U be Banach spaces. The linear system \(\varSigma _{\mathrm {lin}}(A,B)\) is polynomially ISS with parameter \(\alpha >0\) for \({\mathcal {U}}= L^{\infty }({\mathbb {R}}_+,U)\) if and only if the \(C_0\)semigroup \((T(t))_{t\ge 0}\) on X generated by A is polynomially stable with parameter \(\alpha \) and the input operator \(B\in {\mathcal {L}}(U,X)\) is infinitetime \(L^{\infty }\)admissible for \((T(t))_{t\ge 0}\).
Proof
By the remarks following Definitions 3.5 and 4.1, polynomial ISS of \(\varSigma _{\mathrm {lin}}(A,B)\) for \({\mathcal {U}} = L^{\infty }({\mathbb {R}}_+,U)\) implies polynomial stability of \((T(t))_{t\ge 0}\) and infinitetime \(L^{\infty }\)admissibility of B. The converse implication immediately follows, since there exist constants \(M,c>0\) such that
for all \(x_0 \in D(A)\), \(u \in L^{\infty }({\mathbb {R}}_+,U)\), and \(t \ge 0\). \(\square \)
We provide a simple sufficient condition for \(\varSigma _{\mathrm{lin}}(A,B)\) to be polynomially ISS, by restricting the range of the input operator B.
Proposition 4.3
Let X and U be Banach spaces. Suppose that A is the generator of a polynomially stable semigroup with parameter \(\alpha >0\) on X. If \(B \in {\mathcal {L}}(U,X)\) satisfies \({\mathrm{ran}}(B) \subset D((A)^{\beta })\) for some \(\beta > \alpha \), then \(\varSigma _{\mathrm {lin}}(A,B)\) is polynomially ISS with parameter \(\alpha \) for \({\mathcal {U}}= L^{\infty }({\mathbb {R}}_+,U)\).
Proof
Let \((T(t))_{t\ge 0}\) be the polynomially stable semigroup on X generated by A. By Proposition 2.4, there exists \(M>0\) such that
Since \((A)^{\beta }\) is closed, we have that \((A)^{\beta } B \in {\mathcal {L}}(U,X)\) by assumption. For all \(u \in L^{\infty }({\mathbb {R}}_+,U)\) and \(t \ge 0\), we obtain
Hence, B is infinitetime \(L^{\infty }\)admissible for \((T(t))_{t \ge 0}\). Thus, \(\varSigma _{\mathrm {lin}}(A,B)\) is polynomially ISS by Lemma 4.2. \(\square \)
From an argument similar to that in Example 18 of [28], we see that if \(\beta < \alpha \), then the condition \({\mathrm{ran}}(B) \subset D((A)^{\beta })\) may not lead to UGS.
Example 4.4
Let A be the generator of a polynomially stable semigroup \((T(t))_{t \ge 0}\) with parameter \(\alpha >0\) on a Banach space X. Set \(U := X\) and \(B := (A)^{\beta }\) with \(0< \beta < \alpha \). Taking the input \(u(t) := T(t)y_0\) with \(y_0\in X\), we obtain
for all \(t \ge 0\). If the linear system \(\varSigma _{\mathrm {lin}}(A,B)\) is UGS for \({\mathcal {U}} = L^{\infty }({\mathbb {R}}_+,U)\), then the uniform boundedness principle implies that
However, one can easily find polynomially stable semigroups with parameter \(\alpha \) for which the condition (4.1) does not hold. Hence, the condition \({\mathrm{ran}}(B) \subset D((A)^{\beta })\) with \(\beta <\alpha \) does not imply UGS in general. The case \(\alpha = \beta \) remains open except in the diagonalizable case studied in the next subsection.
Polynomial inputtostate stability for diagonalizable linear systems
In this subsection, we consider linear systems with diagonalizable generators and finiterank input operators. We aim to refine the condition on the range of the input operator obtained in Proposition 4.3. To this end, we first review the definition and basic properties of diagonalizable operators; see Section 2.6 of [35] for details.
Definition 4.5
Let X be a Hilbert space. The linear operator \(A :D(A) \subset X \rightarrow X\) is diagonalizable if \(\varrho (A) \not = \emptyset \) and there exists a Riesz basis \((\varphi _n)_{n \in {\mathbb {N}}}\) in X consisting of eigenvectors of A.
Throughout this subsection, we place the following assumption.
Assumption 4.6
Let X be a Hilbert space with inner product \(\langle \cdot , \cdot \rangle \). The operator \(A:D(A) \subset X \rightarrow X \) is diagonalizable, and \((\varphi _n)_{n \in {\mathbb {N}}}\) is a Riesz basis in X consisting of eigenvectors of A. The biorthogonal sequence for \((\varphi _n)_{n \in {\mathbb {N}}}\) and the eigenvalue corresponding to the eigenvector \(\varphi _n\) are given by \((\psi _n)_{n \in {\mathbb {N}}}\) and \(\lambda _n\), respectively.
Proposition 4.7
Suppose that Assumption 4.6 is satisfied. Then, the following statements hold:

1.
The operator A may be written as
$$\begin{aligned} Ax = \sum _{n=1}^{\infty } \lambda _n \langle x , \psi _n \rangle \varphi _n\qquad \forall x \in D(A) \end{aligned}$$and
$$\begin{aligned} D(A) = \left\{ x \in X : \sum _{n=1}^{\infty } \lambda _n^2 ~\!\langle x , \psi _n \rangle ^2 < \infty \right\} . \end{aligned}$$ 
2.
The operator A is the generator of a \(C_0\)semigroup \((T(t))_{t\ge 0}\) on X if and only if
$$\begin{aligned} \sup _{n \in {\mathbb {N}}} {\mathrm{Re}} \lambda _n < \infty . \end{aligned}$$In this case, the exponential growth bound of \((T(t))_{t\ge 0}\) is given by \(\sup _{n \in {\mathbb {N}}} {\mathrm{Re}} \lambda _n\), and for all \(x \in X\) and \(t \ge 0\),
$$\begin{aligned} T(t)x = \sum _{n=1}^{\infty } e^{t \lambda _n } \langle x , \psi _n \rangle \varphi _n. \end{aligned}$$
Suppose that the eigenvalues \((\lambda _n)_{n \in {\mathbb {N}}}\) of a diagonalizable operator A satisfy \({\mathrm {Re}}\lambda _n \le 0\) for all \(n \in {\mathbb {N}}\). Then, A generates a uniformly bounded semigroup. Moreover, \(A\) is sectorial in the sense of [12, Chapter 2], and hence, the fractional power \((A)^{\alpha }\) is well defined for every \(\alpha > 0\). The domain of the fractional power \((A)^{\alpha }\) is given by
for all \(\alpha > 0\), where \((\psi _n)_{n \in {\mathbb {N}}}\) is as in Assumption 4.6.
A diagonalizable operator is similar to a normal operator. Hence, by Proposition 2.5, a diagonalizable operator with eigenvalues \((\lambda _n)_{n \in {\mathbb {N}}}\) generates a polynomially stable semigroup with parameter \(\alpha >0\) if and only if \({\mathrm {Re}}\lambda _n <0\) for all \(n \in {\mathbb {N}}\) and there exist \(C,p>0\) such that
We obtain a refined sufficient condition for linear systems with diagonalizable generators and finiterank input operators to be polynomially ISS.
Theorem 4.8
Let Assumption 4.6 be satisfied, and let U be a Banach space. Suppose that the diagonalizable operator A generates a polynomially stable semigroup with parameter \(\alpha >0\) on X. If \(B \in {\mathcal {L}}(U,X)\) is a finiterank operator and satisfies \( {\mathrm{ran}}(B) \subset D((A)^{\alpha }) \), then \(\varSigma _{\mathrm {lin}}(A,B)\) is polynomially ISS with parameter \(\alpha \) for \({\mathcal {U}}= L^{\infty }({\mathbb {R}}_+,U)\).
Proof
By Lemma 4.2, it suffices to show that B is infinitetime \(L^{\infty }\)admissible for \((T(t))_{t\ge 0}\) in the case \(B \not = 0\).
By a property of a Riesz basis (see, e.g., Proposition 2.5.2 of [35]), there exists a constant \(M_1>0\) such that
for all \(u \in L^{\infty }({\mathbb {R}}_+,U)\) and \(t\ge 0\). Since B is a finiterank operator, there is an orthonormal basis \((\xi _k)_{k=1}^m\) of the finitedimensional space \({\mathrm{ran}}(B)\), where \(m\in {\mathbb {N}}\) is the dimension of \({\mathrm{ran}}(B)\). Then, we obtain
Since
for all \(s \ge 0\), it follows that
for all \(u \in L^{\infty }({\mathbb {R}}_+,U)\) and \(t\ge 0\). Therefore,
Combining \(\xi _k \in D((A)^{\alpha }) \) with the geometric condition (4.2) on \((\lambda _n)_{n \in {\mathbb {N}}}\), we obtain
for every \(k=1,\dots ,m\). Therefore,
for all \(u \in L^{\infty }({\mathbb {R}}_+,U)\) and \(t\ge 0\). From the estimates (4.3) and (4.4), we obtain
for all \(u \in L^{\infty }({\mathbb {R}}_+,U)\) and \(t\ge 0\). Thus, B is infinitetime \(L^{\infty }\)admissible for \((T(t))_{t\ge 0}\). \(\square \)
We apply Theorem 4.8 to an Euler–Bernoulli beam with weak damping.
Example 4.9
Consider a simply supported Euler–Bernoulli beam with weak damping, which is described by the following partial differential equation on (0, 1):
where \(b \in L^2(0,1)\) is a “shaping function” for the external input u and h is the damping coefficient. Here, we set \(h(\zeta ) := 1\zeta \) for \(\zeta \in (0,1)\).
It is well known that the partial differential Equ. (4.5) can be written as a firstorder linear system in the following way; see, e.g., Exercise 3.18 of [9]. Define \(X_0 := L^2(0,1)\) and
with domain
The operator \(A_0\) has a positive selfadjoint square root \(A_0^{1/2}= \frac{\mathrm{d}^2}{\mathrm{d}\zeta ^2}\) with domain
The space \(X := D(A_0^{1/2}) \times L^2(0,1)\) equipped with an inner product
is a Hilbert space. Define the operators \(A_1:D(A_1) \subset X \rightarrow X\) and \(B,H \in {\mathcal {L}}({\mathbb {C}},X)\) by
with domain \(D(A_1) = D(A_0) \times D(A_0^{1/2})\) and
For
the partial differential Equ. (4.5) can be written as
The operator \(A_1\) is diagonalizable with simple eigenvalues
Since \(A_1\) has compact resolvents by Lemma 3.2.12 of [9], it follows from Theorem 1 of [37] that \(A := A_1HH^*\) is also diagonalizable. Moreover, A generates a polynomially stable semigroup with parameter \(\alpha = 1\) by Corollary 6.6 of [7]. Thus, Theorem 4.8 shows that \(\varSigma _{\text {lin}}(A, B)\) is polynomially ISS with parameter \(\alpha = 1\) if \({\mathrm{ran}}(B) \subset D(A) = D(A_1)\), i.e., \(b \in D(A_0^{1/2})\).
Case where eigenvalues near the imaginary axis have uniformly separated imaginary parts
We investigate how sharp the condition \({\mathrm{ran}}(B) \subset D((A)^{\alpha })\) is. To this end, we employ the relation between Laplace–Carleson embeddings and infinitetime \(L^{\infty }\)admissibility established in [16].
Let \(A:D(A) \subset X \rightarrow X\) be diagonalizable and generate a strongly stable semigroup \((T(t))_{t\ge 0}\) on X. Let \(B \in {\mathcal {L}}({\mathbb {C}},X)\) be represented as \(Bv = bv\) for some \(b \in X\) and all \(v \in {\mathbb {C}}\). Define the Borel measure \(\nu \) on the open right halfplane \(\{ \lambda \in {\mathbb {C}}: {\mathrm{Re}} \lambda >0 \}\) by
where \((\lambda _n)_{n \in {\mathbb {N}}}\) and \((\psi _n)_{n \in {\mathbb {N}}}\) are as in Assumption 4.6 and \(\delta _{\lambda _n}\) is the Dirac measure at the point \(\lambda _n\) for \(n \in {\mathbb {N}}\). Define the Carleson square \(Q_I\) associated with an interval \(I \subset i {\mathbb {R}}\) and the dyadic stripe \(S_k\) for \(k \in {\mathbb {Z}}\) by
Then, Theorem 2.5 of [16] shows that B is infinitetime \(L^{\infty }\)admissible for \((T(t))_{t\ge 0}\) if and only if
Using this equivalence of admissibility, we obtain a necessary and sufficient condition for polynomial ISS. We write
for \(k \in {\mathbb {Z}}\).
Theorem 4.10
Let Assumption 4.6 hold and let \(B \in {\mathcal {L}}({\mathbb {C}},X)\) be represented as \(Bv = bv\) for some \(b \in X\) and all \(v \in {\mathbb {C}}\). Assume that there exists \(p >0\) such that
Then, \(\varSigma _{\mathrm {lin}}(A,B)\) is polynomially ISS with parameter \(\alpha >0\) for \({\mathcal {U}} = L^{\infty }({\mathbb {R}}_+,U)\) if and only if the diagonalizable operator A generates a polynomially stable semigroup with parameter \(\alpha \) and
where \(\varPhi _k\) is defined as in (4.9) for \(k \in {\mathbb {Z}}\).
Proof
We first note that a polynomially stable semigroup is strongly stable. By Lemma 4.2, it suffices to show that the conditions (4.8) and (4.11) are equivalent under the assumption (4.10).
Let \(p>0\) satisfy (4.10). There exists \(d>0\) such that \({\mathrm {Im}}\lambda _n  {\mathrm{Im}}\lambda _{m} \ge d\) for all \(n,m \in {\mathbb {N}}\) satisfying \(n\not =m\) and \({\mathrm {Re}}\lambda _n, {\mathrm{Re}}\lambda _m < p\). If an interval \(I \subset i {\mathbb {R}}\) satisfies \(I \ge d\), then for all \(k \in {\mathbb {Z}}\),
Suppose next that an interval \(I \subset i {\mathbb {R}}\) satisfies \(I < d\). If \(k \in {\mathbb {Z}}\) satisfies \(p \le 2^{k+1}\), then \(\nu (Q_I \cap S_k) >0\) implies \(I \ge p/2\), and therefore
Let \(k \in {\mathbb {Z}}\) satisfy \(p> 2^{k+1}\). For \(\lambda _n,\lambda _m \in S_k\) with \(n\not =m\), we obtain \({\mathrm{Im}} \lambda _n  {\mathrm{Im}} \lambda _{m} \ge d\) by assumption. Recalling that the interval I is chosen so that \(I < d\), we have that \(Q_I \cap S_k\) contains at most one element of \((\lambda _n)_{n \in {\mathbb {N}}}\). Since \(\nu (Q_I \cap S_k) = \langle b,\psi _n\rangle ^2\) for some \(n \in {\mathbb {N}}\) with \(\lambda _n\in S_k\) or \(\nu (Q_I \cap S_k) = 0\), it follows that
We have shown that for every interval \(I \subset i {\mathbb {R}}\) and \(k \in {\mathbb {Z}}\),
Hence,
Since \(b \in X\), it follows that \(\sum _{n \in {\mathbb {N}}} \langle b,\psi _n\rangle ^2 < \infty \). Therefore, (4.11) implies (4.8).
Conversely, for all \(k \in {\mathbb {Z}}\), if \(\lambda _n \in S_k\), then
and hence
This yields
Thus, (4.8) implies (4.11).\(\square \)
For \(k \in {\mathbb {Z}}\), define
A routine calculation shows that if \(\lim _{n \rightarrow \infty }{\mathrm{Re}} \lambda _n = 0\) and if there exists \(C>0\) such that
then the condition (4.11) is equivalent to
From this, we observe that the condition (4.11) is milder than \(b \subset D((A)^{\alpha })\). However, the following example shows that if the assumption (4.10) is not satisfied, then \(b \in D((A)^{\alpha })\) may be necessary and sufficient for infinitetime \(L^{\infty }\)admissibility.
Example 4.11
Consider a diagonalizable operator A whose eigenvalues \((\lambda _n)_{n \in {\mathbb {N}}}\) are given by
Since \((\lambda _n)_{n \in {\mathbb {N}}}\) satisfies the geometric condition (4.2) with \(\alpha = 1\), it follows that A generates a polynomially stable semigroups with parameter \(\alpha = 1\). For all \(k \in {\mathbb {N}}_0\), taking intervals \(I \subset i {\mathbb {R}}\) with center \(i2^k\), we obtain
This yields
Thus, infinitetime \(L^{\infty }\)admissibility implies \(b \in D(A)\).
Polynomial integral inputtostate stability of bilinear systems
In the previous section, we saw that polynomial ISS is restrictive even for linear systems with bounded input operators. This is because infinitetime \(L^{\infty }\)admissibility cannot be achieved for all bounded input operators due to the weak asymptotic property of polynomially stable semigroups. This motivates us to study a semiuniform version of integral inputtostate stability, which provides norm estimates of trajectories with respect to a kind of energy fed into systems.
We recall a stability notion for systems without inputs; see [22, Definition 5].
Definition 5.1
The semilinear system \(\varSigma (A,F)\) is called uniformly globally stable at zero if the following two conditions hold:

1.
\(\varSigma (A,F)\) is forward complete.

2.
There exists \(\gamma \in {\mathcal {K}}_{\infty }\) such that
$$\begin{aligned} \Vert \phi (t,x_0,0)\Vert \le \gamma (\Vert x_0\Vert ) \end{aligned}$$for all \(x_0 \in X\) and \(t \ge 0\).
We define the concept of semiuniform integral inputtostate stability.
Definition 5.2
The semilinear system \(\varSigma (A,F)\) is called semiuniformly integral inputtostate stable (semiuniformly iISS) if the following two conditions hold:

1.
\(\varSigma (A,F)\) is uniformly globally stable at zero.

2.
There exist \(\kappa \in \mathcal {KL}\), \(\theta \in {\mathcal {K}}_{\infty }\), and \(\mu \in {\mathcal {K}}\) such that
$$\begin{aligned} \Vert \phi (t,x_0,u)\Vert \le \kappa (\Vert x_0\Vert _A, t) + \theta \left( \int ^t_0 \mu (\Vert u(s)\Vert _U)\mathrm {d}s \right) \end{aligned}$$(5.1)for all \(x_0 \in D(A)\), \(u \in {\mathcal {U}}\), and \(t \ge 0\).
In particular, if there exists \(\alpha >0\) such that for all \(r > 0\), \(\kappa (r,t) = O(t^{1/\alpha })\) as \(t \rightarrow \infty \), then \(\varSigma (A,F)\) is called polynomially integral inputtostate stable (polynomially iISS) with parameter \(\alpha >0\).
Note that the integral \(\int ^t_0 \mu (\Vert u(s)\Vert _U)\mathrm {d}s\) in the righthand side of the inequality (5.1) may be infinite. In that case, the inequality (5.1) trivially holds.
For every generator A of a semiuniformly stable semigroup and every bounded input operator B, the linear system \(\varSigma _{\mathrm {lin}}(A,B)\) is semiuniform iISS. Moreover, if the linear system \(\varSigma _{\mathrm {lin}}(A,B)\) is semiuniform iISS, then \(\varSigma _{\mathrm {lin}}(A,B)\) is strong iISS in the sense of Definition 4 in [23]. This can be seen by using the equality \(\phi (t,x_0,u) = \phi (t,x_0,0) +\phi (t,0,u)\) as in the case of semiuniform ISS discussed in Theorem 3.11.
The aim of this section is to give a sufficient condition for bilinear systems satisfying Assumption 3.2 to be polynomially iISS for \({\mathcal {U}} = L^{\infty }({\mathbb {R}}_+,U)\). We prove that if the nonlinear operator additionally satisfies a certain smoothness assumption, then the bilinear system is polynomially iISS. To this end, we use a nonLyapunov method devised in Theorem 4.2 of [20] for uniform iISS.
Theorem 5.3
Let A be the generator of a polynomially stable semigroup \((T(t))_{t\ge 0}\) with parameter \(\alpha >0\) on a Banach space X. Suppose that the nonlinear operator F satisfies Assumption 3.2 for another Banach space U and that there exist \(K>0\) and \(\chi \in {\mathcal {K}}\) such that for all \(\xi \in X\), \(v \in U\), and \(t \ge 0\),
Then, the bilinear system \(\varSigma (A,F)\) is polynomially iISS with parameter \(\alpha \) for \({\mathcal {U}} = L^{\infty }({\mathbb {R}}_+,U)\).
Proof
Since \((T(t))_{t\ge 0}\) is polynomially stable with parameter \(\alpha >0\), there exists \(M \ge 1\) such that
By Gronwall’s inequality (see Appendix A of [24] for a simple proof), we have that for all \(x_0 \in X\), \(u \in L^{\infty }({\mathbb {R}}_+,U)\), and \(t \ge 0\),
as long as x is a mild solution of \(\varSigma (A,F)\) on [0, t]. Hence, \(\varSigma (A,F)\) is forward complete by the remark following Assumption 3.2. Moreover, \(F(\xi ,0) = 0\) for all \(\xi \in X\) under Assumption 3.2. Therefore, if \(u(t) \equiv 0\), then the mild solution x of \(\varSigma (A,F)\) satisfies
for all \(x_0 \in X\) and \(t \ge 0\), which implies that \(\varSigma (A,F)\) is uniformly globally stable at zero.
Take \(x_0 \in D(A)\) and \(u \in L^{\infty }({\mathbb {R}}_+,U)\). The mild solution x of \(\varSigma (A,F)\) satisfies
for all \(t \ge 0\). Define \(z(t) := (t+1)^{1/\alpha } \Vert x(t)\Vert \) for \(t \ge 0\). Then,
for all \(t \ge 0\). Since
Gronwall’s inequality implies that for all \(t \ge 0\),
which is equivalent to
Using the inequality
we obtain
for all \(t \ge 0\). Since
it follows that
for all \(t \ge 0\). The inverse function of \(q(r) := \ln (1+r)\), \(r \ge 0\), is given by \(q^{1}(r) = e^{r}1\). Using the inequality
twice, we obtain
for all \(t \ge 0\). Thus, the bilinear system \(\varSigma (A,F)\) is polynomially iISS with parameter \(\alpha \), where \(\kappa \in \mathcal {KL}\), \(\theta \in {\mathcal {K}}_{\infty }\), and \(\mu \in {\mathcal {K}}\) are given by
for the estimate (5.1). \(\square \)
Conclusion
We have introduced the notion of semiuniform ISS and have established its characterization based on attractivity properties. We have given sufficient conditions for linear systems to be polynomially ISS. In the sufficient conditions, the range of the input operator is restricted, depending on the polynomial decay rate of the product of the \(C_0\)semigroup and the resolvent of its generator. We have also shown that a class of bilinear systems are polynomially iISS if the nonlinear operator satisfies a smoothness assumption like the range condition of input operators for polynomial ISS of linear systems. Important directions for future research are to explore the relation between semiuniform ISS and semiuniform iISS and to construct Lyapunov functions for polynomial ISS and polynomial iISS.
References
Angeli D, Sontag ED (1999) Forward completeness, unboundedness observability, and their Lyapunov characterizations. Syst. Control Lett. 38:209–217
Arendt W, Batty CJK, Hieber M, Neubrander F (2001) Vectorvalued Laplace transforms and Cauchy problems. Birkhäuser, Basel
Bátkai A, Engel KJ, Prüss J, Schnaubelt R (2006) Polynomial stability of operator semigroups. Math Nachr 279:1425–1440
Batty CJK, Duyckaerts T (2008) Nonuniform stability for bounded semigroups on Banach spaces. J Evol Equ 8:765–780
Borichev A, Tomilov Y (2010) Optimal polynomial decay of functions and operator semigroups. Math Ann 347:455–478
Cazenave T, Haraux A (1998) An introduction to semilinear evolution equations. Oxford Univ. Press, New York
Chill R, Paunonen L, Seifert D, Stahn R, Tomilov Y (2019) Nonuniform stability of damped contraction semigroups. Anal PDE (to appear). https://arxiv.org/pdf/1911.04804.pdf
Chill R, Seifert D, Tomilov Y (2020) Semiuniform stability of operator semigroups and energy decay of damped waves. Philos Trans Roy Soc A 378:20190614
Curtain RF, Zwart HJ (2020) An introduction to infinitedimensional systems: a state space approach. Springer, New York
Dashkovskiy S, Mironchenko A (2013) Inputtostate stability of infinitedimensional control systems. Math Control Signals Syst 25:1–35
Engel KJ, Nagel R (2000) Oneparameter semigroups for linear evolution equations. Springer, New York
Haase M (2006) The functional calculus for sectorial operators. Birkhäuser, Basel
Hosfeld R, Jacob B, Schwenninger F (2022) Integral inputtostate stability of unbounded bilinear control systems. Math Control Signals Syst. https://doi.org/10.1007/s00498021003089
Jacob B, Mironchenko A, Partington JR, Wirth F (2020) Noncoercive Lyapunov functions for inputtostate stability of infinitedimensional systems. SIAM J Control Optim 58:2952–2978
Jacob B, Nabiullin R, Partington JR, Schwenninger FL (2018) Infinitedimensional inputtostate stability and Orlicz spaces. SIAM J Control Optim 56:868–889
Jacob B, Partington JR, Pott S, Rydhe E, Schwenninger FL (2021) Laplace–Carleson embeddings and infinitynorm admissibility. https://arxiv.org/pdf/2109.11465.pdf
Jayawardhana B, Logemann H, Ryan EP (2008) Infinitedimensional feedback systems: the circle criterion and inputtostate stability. Commun Inf Syst 8:413–444
Karafyllis I, Krstic M (2016) ISS with respect to boundary disturbances for 1D parabolic PDEs. IEEE Trans Automat Control 61:3712–3724
Liu Z, Rao B (2005) Characterization of polynomial decay rate for the solution of linear evolution equation. Angew Math Phys 56:630–644
Mironchenko A, Ito H (2016) Characterizations of integral inputtostate stability for bilinear systems in infinite dimensions. Math Control Relat Fields 6:447–466
Mironchenko A, Prieur C (2020) Inputtostate stability of infinitedimensional systems: recent results and open questions. SIAM Rev 62:529–614
Mironchenko A, Wirth F (2018) Characterizations of inputtostate stability for infinitedimensional systems. IEEE Trans Automat Control 63:1692–1707
Nabiullin R, Schwenninger FL (2018) Strong inputtostate stability for infinitedimensional linear systems. Math Control Signals Syst 30:4
Pata V (2011) Uniform estimates of Gronwall type. J Math Anal Appl 373:264–270
Paunonen L (2011) Perturbation of strongly and polynomially stable Rieszspectral operators. Syst Control Lett 60:234–248
Paunonen L (2012) Robustness of strongly and polynomially stable semigroups. J Funct Anal 263:2555–2583
Paunonen L (2013) Robustness of polynomial stability with respect to unbounded perturbations. Syst Control Lett 62:331–337
Paunonen L (2014) Polynomial stability of semigroups generated by operator matrices. J Evol Equ 14:885–911
Rozendaal J, Seifert D, Stahn R (2019) Optimal rates of decay for operator semigroups on Hilbert spaces. Adv Math 346:359–388
Schmid J (2019) Weak inputtostate stability: characterizations and counterexamples. Math Control Signals Syst 31:433–454
Sontag ED (1989) Smooth stabilization implies coprime factorization. IEEE Trans Automat Control 34:435–443
Sontag ED (1998) Comments on integral variants of ISS. Syst Control Lett 34:93–100
Sontag ED, Wang Y (1996) New characterizations of inputtostate stability. IEEE Trans Automat Control 41:1283–1294
Su P, Tucsnak M, Weiss G (2020) Stabilizability properties of a linearized water waves system. Syst Control Lett 138:104672
Tucsnak M, Weiss G (2009) Observation and control of operator semigroups. Birkhäuser, Basel
Weiss G (1989) Admissibility of unbounded control operators. SIAM J Control Optim 27:527–545
Xu CZ, Sallet G (1996) On spectrum and Riesz basis assignment of infinitedimensional linear systems by bounded linear feedbacks. SIAM J Control Optim 34:521–541
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The author would like to thank the editor and anonymous reviewers for their careful reading of the manuscript and many insightful comments, which, in particular, make the argument in Example 4.4 simpler.
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Wakaiki, M. Semiuniform inputtostate stability of infinitedimensional systems. Math. Control Signals Syst. 34, 789–817 (2022). https://doi.org/10.1007/s00498022003261
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DOI: https://doi.org/10.1007/s00498022003261
Keywords
 Infinitedimensional systems
 Inputtostate stability
 Polynomial stability
 \(C_0\)semigroups
Mathematics Subject Classification
 47D06
 47N70
 93C25
 93D09