Remarks on input-to-state stability of collocated systems with saturated feedback

We investigate input-to-state stability of infnite-dimensional collocated control systems subject to saturated feedback where the unsaturated closed loop system is dissipative and uniformly globally asymptotically stable. We review recent results from the literature and explore limitations thereof.


Introduction
In this note we continue the study of the stability of systems of the form by the nonlinear feedback law u(t) = −σ(y(t) + d(t)). Here X and U are Hilbert spaces, A : D(A) ⊂ X → X is the generator of a strongly continuous contraction semigroup and B is a bounded linear operator from U to X, i.e. B ∈ L(U, X). The function σ : U → U is Lipschitz continuous and maximal monotone with σ(0) = 0. Of particular interest is the case in which σ is even locally linear. In the following we are interested in stability with respect to both the initial value x 0 , that is internal stability, and the disturbance d; external stability. This is combined in the notion of input-to-state stability (ISS), which has recently been studied for infinite-dimensional systems e.g. in [6,8,18,19] and particularly for semilinear systems in [4,5,22], see also [17] for a survey. The effect of feedback laws acting (approximately) linearly only locally is known in the literature as saturation, and first appeared in [25,23] in the context of stabilization of infinite-dimensional linear systems, see also [9]. There, internal stability of the closedloop system was studied using nonlinear semigroup theory, a natural tool to establish existence and uniqueness of solutions for equations of the above type, see also the more recent works [10,14,15]. The simultaneous study of internal stability and the robustness with respect to additive disturbances in the saturation seems to be rather recent. This notion clearly includes uniform global (internal) stability, which by far is not guaranteed for such nonlinear systems. In [21] this was studied for a wave equation and in [13] Korteweg-de Vries type equation was rigorously discussed, building on preliminary works in [11,12], see also [10]. The combination of saturation and ISS was initiated in [15] and, as for internal stability, complemented in [14]. For the rich finite-dimensional theory on ISS for related semilinear systems, we refer e.g. to [4,5] and the references therein. For (infinite-dimensional) nonlinear systems, ISS is typically assessed by Lyapunov functions, see e.g. [3,7,16,19,22]. These are often constructed by energy-based L 2 norms, but also Banach space methods exist [19], which are much easier to handle in the sense of L ∞ -estimates as present in ISS. We will use some of these constructions here.
In this note we investigate the question whether internal stability of the corresponding linear undisturbed system, that is, (Σ SLD ) with σ(u) = u and d ≡ 0, implies inputto-state stability of (Σ SLD ). In doing so we try to shed light on limitations of existing results. Because the linear system has a bounded input operator, the above question is equivalent to asking whether ISS of the linear system yields that (Σ SLD )is ISS, see e.g. [8]. For nonlinear systems, uniform global (internal) stability is only a necessary condition for ISS, which, however, may fail in presence of saturation. The following system can be seen as a show-case example, (1)

ISS for saturated systems
ii) σ is Lipschitz continuous, i.e. there exists a k > 0 such that If additionally a Banach space S is continuously, densely embedded in U with we call σ a saturation function. Here U ⊂ S ′ in the sense of Gelfand triples.
Example 1. Let sat R be the function from (1). It is easy to see that the function is an admissible feedback function. Moreover, for S = L ∞ (0, 1) we have As Property (v) from Definition 1 follows similarly, sat is a saturation function.
Let σ be an admissible feedback function. In the rest of the paper we will be interested in the following two types of systems: The unsaturated system, and the disturbed saturated system By Lumer-Phillips theorem, A generates a strongly continuous semigroup of contractions ( T (t)) t≥0 as −BB * ∈ L(X) is dissipative. Clearly, (Σ L ) is a special case of (Σ SLD ) with d = 0, as σ(u) = u is an admissible feedback function.
is called a mild solution of (Σ SLD ). If x is differentiable almost everywhere such that x ′ ∈ L 1 (0, ∞; X), x(0) = x 0 and (Σ SLD ) holds for almost every t ≥ 0, we say that x is a strong solution.
By our assumptions, (Σ SLD ) has a unique mild solution for any x 0 ∈ X and u ∈ L ∞ (0, ∞; U ), [2,Prop. 4.3.3]. In order to introduce the external stability notions, the following well-known comparison functions are needed, is called semi-globally exponentially stable if for d = 0 and any r > 0 there exist two µ(r) > 0 and K(r) > 0 such that for any mild solution x with initial value is called locally input-to-state stable (LISS) if there exist r > 0, β ∈ KL and ρ ∈ K ∞ such that for every mild solution x with initial value satisfying x 0 X ≤ r and all t ≥ 0, we have If (2) holds for (Σ SLD ) with d ≡ 0 and r = ∞, the system is called uniformly globally asymptotically stable (UGAS).
Compared to the other notions, semi-global exponential stability seems to be less common in the literature, but appeared already in the context of saturated systems in [14]. Note that for the linear System (Σ L ) UGAS is equivalent to the existence of constants M, ω > 0 such that T (t) X ≤ M e −ωt for all t ≥ 0. Clearly, if (Σ SLD ) is UGAS, then it is globally asymptotically stable. Moreover, (Σ L ) is UGAS if and only if it is semi-globally exponentially stable, and semi-global exponential stability implies global asymptotical stability.
Next we investigate the question whether (semi)-global exponential stability or UGAS of system (Σ L ) implies (semi)-global exponential stability or UGAS of System (Σ SLD ).
Under the assumption that σ is an admissible feedback function with the additional properties that for all u ∈ U , ℜ u, σ(u) = 0 implies u = 0 and D(A) equipped with the norm · D(A) = · X + A · X is a Banach space compactly embedded in X, in [10,Theorem 2] it is shown that global asymptotic stability of (Σ L ) implies global asymptotic stability of (Σ SLD ). Note, that by [ Here we are interested in results for general admissible feedback functions and saturation functions. The following result was proved in [14] and [15]. ii) If S = U and then (Σ SLD ) is semi-globally exponentially stable.
Thus System (4) is neither semi-globally exponentially stable nor UGAS.
The following theorem shows that UGAS of System (Σ L ) together with Condition (3) is not sufficient to guarantee UGAS of System (Σ SLD ). We note, that System (Σ SLD ) of Theorem 1 equals (Σ sat ). Further, in [15, Theorem 1] it has been wrongly stated that the saturated system is UGAS.
Proof. It is easy to see that system (Σ L ) is UGAS. System (Σ SLD ) is given by (Σ sat ) in the introduction. Condition (3) is fulfilled, because H 1 (0, 1) is continuously embedded in L ∞ (0, 1). Hence, (Σ SLD ) is semi-globally exponentially stable by Proposition 1. Note that A generates the periodic shift semigroup on L 2 (0, 1). By extending the initial function f periodically to R + , the unique mild solution y ∈ C([0, ∞); L 2 (0, 1)) of (Σ sat ) is given by where x is defined in (5). By the particular form of (5), this implies that holds for all t ≥ 0. We can therefore choose the same sequence (f n ) n ∈ L 2 (0, 1) with f n L 2 (0,1) = 1 as in the proof of Proposition 2 in order to conclude This shows that System (Σ sat ) is not UGAS and thus not ISS.
An important tool for the verification of ISS of System (Σ SLD ) are ISS Lyapunov functions.
If r = ∞, then V is called an ISS Lyapunov function. For System (Σ SLD ) with d ≡ 0 we will call an ISS Lyapunov function a UGAS Lyapunov function.
In [16,Theorem 4] it is shown that system (Σ SLD ) with an admissible feedback function is LISS if and only if there exists an LISS Lyapunov function for (Σ SLD ) which is Lipschitz continuous. Moreover, system (Σ SLD ) with an admissible feedback function is ISS if and only if there exists an ISS Lyapunov function for (Σ SLD ) which is locally Lipschitz continuous [19,Theorem 5].
In this paper, we are mainly interested in the contruction of ISS Lyapunov functions. In the setting of Theorem 1 the operator A generates a semigroup which is not exponentially stable. The following result shows that this is not accidental if the saturated system is not UGAS.
Proposition 3. Suppose that there exists α > 0 such that T (t) ≤ e −αt for all t > 0 and let σ be an admissible feedback function. Then the function is an UGAS Lyapunov function for (Σ SLD ) and thus System (Σ SLD ) is UGAS.
Proof. Assume first that x 0 ∈ D(A). Then by [20, Theorem 1.6, p. 189] there exists a strong solution x for (Σ SLD ). We obtain for a.e. t ≥ 0 by using Definition 1.iii). For x 0 ∈ X there exists a sequence (x 0,n ) in D(A) with x 0,n → x 0 . Denoting by x the mild solution of (Σ SLD ) with initial value x 0 and by x n the strong solution of (Σ SLD ) with initial value x 0,n we have Application of Gronwall's Lemma yields for n → ∞ uniformly on bounded intervalls. We can therefore conclude for n → ∞. UGAS of (Σ SLD ) now follows from [19,Theorem 5] with d ≡ 0.
Note that the assumption on the semigroup made in Theorem 3 is strictly stronger than the condition that (T (t)) t≥0 is an exponentially stable contraction semigroup as can be seen e.g. for a nilpotent shift-semigroup on X = L 2 (0, 1). However, note that by the Lumer-Phillips theorem the following assertions are equivalent for a semigroup (T (t)) t≥0 generated by A.
Next we study the question whether UGAS of (Σ L ), implies that System (Σ SLD ) has an ISS Lyapunov function. In [19] the following ISS Lyapunov function was shown to be an ISS Lyapunov function for System (Σ L ). By adapting the proof we obtain the following.
Proof. We can rewrite (Σ SLD ) in the form Hence, the mild solution satisfies Denoting the integral by I h , we have, using that σ is admissible, .
for every ε > 0 due to the continuity of the mild solution.
Example 2. Let X = U = R and A = 0. Then for every B ∈ R\{0} the operator A − BB * = −B 2 generates the semigroup (e −B 2 t ) t≥0 . By choosing σ = 0 and x 0 = 0, the constant function x(t) = x 0 solves System (Σ SLD ). In this case ω = B 2 = M 2 B 2 (k+1) holds, as k = 0 and M = 1, and the system is not ISS. Thus, the lower bound for ω required in Theorem 2 is optimal.
Locally linear admissible feedback functions yield LISS Lyapunov functions. for every ε > 0. The Lipschitz continuity of V can be shown by using the same argumentation as in the proof of Theorem 2. Application of [16,Theorem 4] yields local input-to-state stability of (Σ SLD ).
Note that property iii) of Definition 1 has not been used in the proofs of Theorems 2 and 3.

Conclusion
A general assumption of this article is the contractivity of the underlying C 0 -semigroup generated by A. It is an open question whether the results of the paper also hold in the case of bounded semigroups. The following example shows that for general strongly continuous semigroup it may happen that the nonlinear system (Σ SLD ) is not uniformly globally asymptotically stable, but the underlying linear system (Σ L ) is UGAS.