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Exponential Stability of Linear Systems Under a Class of Desch

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Exponential Stability of Linear Systems Under a Class of Desch Exponential stability of linear systems under a class of Desch-Schappacher perturbations S. El Alaoui and M. Ouzahra M2PA Laboratory, University of Sidi Mohamed Ben Abdellah P.O. Box 5206, Bensouda, Fès, Morocco [email protected] Abstract ( ) In this paper we investigate the uniform exponential stability of the system dx t = Ax(t) dt − ρBx(t), where ρ is a real parameter, the system’s operator A is the infinitesimal generator of a linear C0 semigroup of contractions (S(t))t 0 on a Hilbert space X, and B is a Desch-Schappacher operator. − ≥ We provide sufficient conditions for exponential stability of the above system. Our stability result is then applied to show the uniform exponential stabilization of bilinear partial differential equations. Keywords: Exponential stabilization, Desch-Schappacher operator, linear system, bilinear con- trol, unbounded control operator I. Introduction In this paper we consider the following abstract system: x˙(t) = Ax(t) ρBx(t), t > 0, x(0) = x , (1) − 0 where x(.) is the state of the system and takes values in a Hilbert state space X with inner product .,. and corresponding norm . , the unbounded operator (A, D(A)) generates a h iX k kX C0 semigroup (S(t))t 0 on X. Here, B is an unbounded linear operator of X in the sense that − ≥ it is bounded from X to some extrapolating space of X. In several practical situations the mod- B arXiv:2107.12195v1 [math.OC] 26 Jul 2021 eling gives rise to systems of the form (1), where the perturbation operator is of type Desch- Schappacher. This fact appears for instance in the case of bilinear control systems when the control action is applied through the boundary or a point for systems governed by partial differ- ential equations and in many situation of internal control (see [4, 7, 8] and the references therein). In general, due to the unbounded aspect of the operator B, the solution of (1) does not exist with values in X. Thus, to confront this difficulty the concept of admissibility is needed, which requires the introduction of interpolating and extrapolating spaces of the state space X. Our aim in this paper is to investigate the uniform exponential stability of the system (1). This consists on looking for a set of parameters ρ for which there exists a global X valued mild − solution x(t) of (1) and is such that x(t) Ke σt x , t 0 for some constants K, σ > 0. k k ≤ − k 0k ∀ ≥ As an application, one can consider the stabilization of bilinear systems by means of switching controllers, which leads to a closed-loop system like (1). This problem has been considered in [11] for a bounded operator B. The case of a Miyadera-Voigt type operator has been investigated in 1 [15]. Moreover, in [2] the case of 1 admissibility in Banach space has been considered. However, − this admissibility condition excludes the case of Hilbert state space where several applications are also available. In this paper, we will rather use the p admissibility property with p 1. Then − ≥ we introduce new sufficient conditions for uniform exponential stability of system (1), which are easily checkable. In the sequel, we proceed as follows: The main results of this paper are contained in Section 2. In Section 3, applications to feedback stabilization of bilinear heat and transport equations are considered. II. Exponential stability In this section, we provide our two main stabilization results. We start by introducing the nec- essary tools regarding the notion of admissibility in connection with the generation results and then provide some a priori estimations of the solution. i. Preliminary As pointed out in the introduction, the unbounded aspect of the operator B do not guarantee the existence of an X valued solution x(t) of (1). However, one may extend the system at hand in a − larger (extrapolating) space X 1 of the state space X in which the existence of the solution x(t) is − ensured and then give the required admissibility conditions of B so that the solution x(t) lies in X. Classically, the spaces X1 and X 1 are defined as follows: X1 :=(D(A), 1), where x 1 := − k · k k k (λI A)x X, x D(A) for some λ in the resolvent set ρ(A) of A, and X 1 is the completion of X k − k ∈ 1 − with respect to the norm x 1 := (λI A)− x X, x X. These spaces are independent of the k k− k − k ∈ .choice of λ and are related by the following continuous and dense embedding: X1 ֒ X ֒ X 1 → → − That way the unbounded operator B becomes bounded from X to the extrapolating space X 1, i.e, − B (X, X 1). Thus, in order to give a meaning to solutions of (1), we have to use the fact that ∈ L − the semigroup (S(t))t 0 can be extended to a C0 semigroup (S 1(t))t 0 on X 1 whose generator ≥ − − ≥ − A 1 has D(A 1) = X as domain and is such that A 1x = Ax, for any x D(A). Moreover, if the − − − ∈ semigroup S(t) is a contraction, then so is S 1(t). Indeed, having in mind that for all λ ρ(A 1), 1 − 1 1 ∈ − we have (λI A 1)− x X, x X 1 and (λI A 1)− x = (λI A)− x, x X, we can see − − ∈ ∀ ∈ − − − − ∀ ∈ that all for x X we have ∈ 1 1 S 1(t)x 1 = (λI A)− S(t)x X (λI A)− x X = x 1, k − k− k − k ≤ k − k k k− thus, by density of X in X 1, we conclude that the semigroup S 1(t) is a contraction on X 1. Recall that for any given initial− state x X, a mild solution of− (1) is an X valued continuous− 0 ∈ − function x on [0, T] satisfying the following variation of parameters formula: t x(t) = S(t)x0 ρ S 1(t s)Bx(s)ds, t 0, − Z0 − − ∀ ≥ which always makes sense in X 1. The system (1) can be rewritten in the large space X 1 in the following abstract form: − − x˙(t) = A 1x(t) ρBx(t), x(0) = x0. (2) − − which is well-posed in X whenever A ρB is a generator of a C semigroup on X (cf. [6], − 0− Section II.6). The well-posedeness of systems like (1) has been studied in many works using different approaches (see e.g. [1, 3, 5, 9, 13]). The next result provides sufficient conditions on a Desch-Schappacher perturbation B to guar- antee the existence and uniqueness of the mild solution of (1) (see [1] & ([6], p. 183)). 2 Theorem II.1 Let A be the generator of a C0 semigroup S(t) onX andlet B (X, X 1) be p admissible − ∈L − − for some 1 p < ∞, i.e., there is a T > 0 such that ≤ T p S 1(T t)Bu(s)ds X, u L (0, T; X). (3) Z0 − − ∈ ∀ ∈ Then for any ρ, the operator (A 1 ρB) X defined on the domain D((A 1 ρB) X) := x X : (A 1 ρB)x X − − | − − | { ∈ − − ∈ } by (A 1 ρB) Xx := A 1x ρBx, x D((A 1 ρB) X) (4) − − | − − ∀ ∈ − − | is the generator of a C0 semigroup (T(t))t 0 on X, which verifies the variation of parameters formula − ≥ t T(t)x = S(t)x ρ S 1(t s)BT(s)xds, t 0, x D((A 1 ρB) X). − Z0 − − ∀ ≥ ∀ ∈ − − | An operator B (X, X 1) satisfying the condition (3) is called a Desch-Schappacher opera- ∈ L − tor or perturbation. Moreover, the operator defined by (4) is the part (A 1 ρB) X of (A 1 ρB) − − | − − on X see ([16], p. 39) and ([6], p. 147) . Remark 1 Notice that since W1,p(0, T; X) is dense in Lp(0, T; X), the rang condition (3) is equivalent to the existence of some M > 0 such that T 1,p S 1(T s)Bu(s)ds M u Lp(0,T;U), u W (0, T; X), (5) − − ≤ k k ∀ ∈ Z0 X 1 T p p with u Lp(0,T;X) = u(t) Xdt . k k 0 k k Z Remark 2 Note that if the operator B (X, X 1) is p admissible in [0, T], then it is so in [0, t] for ∈ L − − any t [0, T]. In other words, if (5) holds then for all t [0, T] we have the following inequality ∈ ∈ t S 1(t r)Bu(r)dr M u Lp(0,t;X) (6) 0 − − ≤ k k · Z X Indeed, for all t [0, T] and u Lp(0, T; X) we define u Lp(0, T; X) by ∈ ∈ t ∈ 0, for r [0, T t] ∈ − u (r) := t u(r + t T), for r (T t, T] − ∈ − t T Then, observing that 0 S 1(t r)Bu(r)dr = 0 S 1(T r)But(r)dr X, it comes from the p admissibility t − − − − ∈ t − of B that S 1(t r)Bu(r)dr M ut Lp(0,T;X) Thus for all t [0, T], we have S 1(t r)Bu(r)dr 0 − −R X ≤ k kR · ∈ 0 − − X ≤ M u p L ( 0,Rt;X) R k k · Let us now show the two following lemmas that will be needed in the sequel. < < 1 Lemma II.2 Let assumptions of Theorem II.1 hold.
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