On the Stabilization of Infinite Dimensional Semilinear Systems

This chapter considers the question of the output stabilization for a class of infinite dimensional semilinear system evolving on a spatial domain Ω by controls depending on the output operator. First we study the case of bilinear systems, so we give sufficient conditions for exponential, strong and weak stabilization of the output of such systems. Then, we extend the obtained results for bilinear systems to the semilinear ones. Under sufficient conditions, we obtain controls that exponentially, strongly, and weakly stabilize the output of such systems. The method is based essentially on the decay of the energy and the semigroup approach. Illustrations by examples and simulations are also given.


Introduction
We consider the following semilinear system where A : D A ð Þ⊂ H ! H generates a strongly continuous semigroup of contractions S t ð Þ ð Þ t ≥ 0 on a Hilbert space H, endowed with norm and inner product denoted, respectively, by ∥:∥ and :; : h i, v : ð Þ ∈ V ad (the admissible controls set) is a scalar valued control and B is a nonlinear operator from H to H with B 0 ð Þ ¼ 0 so that the origin be an equilibrium state of system (1).The problem of feedback stabilization of distributed system (1) was studied in many works that lead to various results.In [1], it was shown that the control weakly stabilizes system (1) provided that B be a weakly sequentially continuous operator such that, for all ψ ∈ H, we have and if (3) is replaced by the following assumption then control (2) strongly stabilizes system (1) [2].
In [3], the authors show that when the resolvent of A is compact, B self-adjoint and monotone, then strong stabilization of system (1) is proved using bounded controls.Now, let the output state space Y be a Hilbert space with inner product :; : h i Y and the corresponding norm ∥:∥ Y , and let C ∈ L H; Y ð Þbe an output operator.System (1) is augmented with the output The output stabilization means that w t ð Þ !0 as t !þ∞ using suitable controls.In the case when Y ¼ H and C ¼ I, one obtains the classical stabilization of the state.
If Ω be the system evolution domain and ω ⊂ Ω, when C ¼ χ ω , the restriction operator to a subregion ω of Ω, one is concerned with the behaviour of the state only in a subregion of the system evolution domain.This is what we call regional stabilization.
The notion of regional stabilization has been largely developed since its closeness to real applications, and the existence of systems which are not stabilizable on the whole domain but stabilizable on some subregion ω.Moreover, stabilizing a system on a subregion is cheaper than stabilizing it on the whole domain [4][5][6][7][8].In [9], the author establishes weak and strong stabilization of (5) for a class of semilinear systems using controls that do not take into account the output operator.
In this paper, we study the output stabilization of semilinear systems by controls that depend on the output operator.Firstly we consider the case of bilinear systems, then we give sufficient conditions to obtain exponential, strong and weak stabilization of the output.Secondly, we consider the case of semilinear systems, and then under sufficient conditions, we obtain controls that exponentially, strongly, and weakly stabilize the output of such systems.The method is based essentially on the decay of the energy and the semigroup approach.Illustrations by examples and simulations are also given.
This paper is organized as follows: In Section 2, we discuss sufficient conditions to achieve exponential, strong and weak stabilization of the output (5) for bilinear systems.In Section 3, we study the output stabilization for a class of semilinear systems.Section 4 is devoted to simulations.

Stabilization for bilinear systems
In this section, we develop sufficient conditions that allow exponential, strong and weak stabilization of the output of bilinear systems.Consider system (1) with B : H ! H is a bounded linear operator and augmented with the output (5).
Definition 1.1 The output ( 5) is said to be: 1. weakly stabilizable, if there exists a control v : ð Þ ∈ V ad such that for any initial condition z 0 ∈ H, the corresponding solution z t ð Þ of system (1) is global and satisfies 2. strongly stabilizable, if there exists a control v : ð Þ ∈ V ad such that for any initial condition z 0 ∈ H, the corresponding solution z t ð Þ of system ( 1) is global and verifies ∥Cz t ð Þ∥ Y !0, as t !∞, and 3. exponentially stabilizable, if there exists a control v : ð Þ ∈ V ad such that for any initial condition z 0 ∈ H, the corresponding solution z t ð Þ of system (1) is global and there exist α, β > 0 such that Remark 1.It is clear that exponential stability of ( 5) ) strong stability of ( 5) ) weak stability of (5).

Exponential stabilization
The following result provides sufficient conditions for exponential stabilization of the output (5 hold, then there exists ρ > 0 for which the control exponentially stabilizes the output (5).Proof: System (1) has a unique mild solution z t ð Þ [10] defined on a maximal interval 0; t max ½ by the variation of constants formula From hypothesis 1, we deduce Integrating this inequality, we get It follows that For all z 0 ∈ H and t ≥ 0, we have Using hypothesis 2 and (9), we have It follows that from (7) and condition 2 that Integrating (10) over the interval 0; T ½ and replacing z 0 by z t ð Þ and using (6), we deduce that It follows from the inequality (8) that the sequence ∥Cz n ð Þ∥ Y decreases and that for all n ∈ N, we have Using (11), we deduce Then where which gives the exponential stability of the output (5).
Example 1 On Ω ¼0, 1½, we consider the following system where Let ω be a subregion of Ω. System (12) is augmented with the output where , the restriction operator to ω and χ * ω is the adjoint operator of χ ω .Conditions 1 and 3 of Theorem 1.2 hold, indeed: we have and for T ¼ 2, we have We conclude that for all 0 < ρ < e 4 À1 16e 4 , the control exponentially stabilizes the output (13).

Strong stabilization
The following result will be used to prove strong stabilization of the output (5).
allows the estimate Proof: From hypothesis 1 of Theorem 1.3, we have 1 2 In order to make the energy nonincreasing, we consider the control so that the resulting closed-loop system is where Since f is locally Lipschitz, then system (16) has a unique mild solution z t ð Þ [10] defined on a maximal interval 0; t max ½ by Because of the contractions of the semigroup, we have Integrating this inequality, we get From hypothesis 1 of Theorem 1.3, we have We deduce Using (17) and Schwartz inequality, we get Since B is bounded and C continuous, we have where K is a positive constant.Replacing z 0 by z t ð Þ in (20) and (21), we get Integrating this relation over 0; T ½ and using Cauchy-Schwartz, we deduce Nonlinear Systems -Theoretical Aspects and Recent Applications which achieves the proof.
The following result gives sufficient conditions for strong stabilization of the output (5) From ( 15) and ( 22), we have where β ¼ Taking s k ¼ ∥Cz kT ð Þ∥ 2 Y , the inequality (24) can be written as Since s kþ1 ≤ s k , we obtain we deduce Finally the inequality s k ≤ x k ð Þ, together with the fact that ∥Cz t ð Þ∥ Y decreases, we deduce the estimate (23).
Example 2 Let us consider a system defined on Ω ¼0, 1½ by where The operator A generates a semigroup of contractions on strongly stabilizes the output (26) with decay estimate , as t !þ∞:

Weak stabilization
The following result provides sufficient conditions for weak stabilization of the output (5).
Theorem 1.5 Let A generate a semigroup S t ð Þ ð Þ t ≥ 0 of contractions on H and B is a compact operator.If the conditions: 14) weakly stabilizes the output (5).
Proof: Let us consider the nonlinear semigroup Γ t ð Þz 0 ≔ z t ð Þ and let t n ð Þ be a sequence of real numbers such that t n !þ∞ as n !þ∞.
From (18), Γ t n ð Þz 0 is bounded in H, then there exists a subsequence Since B is compact and C continuous, we have For all n ≥ , we set Hence, by the dominated convergence Theorem, we have We conclude that Using condition 3 of Theorem 1.5, we deduce that On the other hand, it is clear that (27) holds for each subsequence Example 3 Consider a system defined in Ω ¼0, þ ∞½, and described by where We have so, the assumption 1 of Theorem 1.5 holds.The operator B is compact and verifies Then, the control

Stabilization for semilinear systems
In this section, we give sufficient conditions for exponential, strong and weak stabilization of the output (5).Consider the semilinear system (1) augmented with the output (5).

Exponential stabilization
In this section, we develop sufficient conditions for exponential stabilization of the output (5).
The following result concerns the exponential stabilization of (5).Theorem 1.6 Let A generate a semigroup S t ð Þ ð Þ t ≥ 0 of contractions on H and B be locally Lipschitz.If the conditions: hold, then the control exponentially stabilizes the output (5).
Integrating this inequality, and using hypothesis 2 of Theorem 1.6, it follows that For all z 0 ∈ H and t ≥ 0, we have Since B is locally Lipschitz, there exists a constant positive L that depends on ∥z 0 ∥ such that where α is a positive constant.Using (33), we deduce While from the variation of constants formula and using Schwartz's inequality, we obtain Integrating (34) over the interval 0; T ½ and taking into account (35) and (36), we get Now, let us consider the nonlinear semigroup U t ð Þz 0 ≔ z t ð Þ [1].Replacing z 0 by U t ð Þz 0 in (37), and using the superposition properties of the semigroup Thus, by using (31) and (37), it follows that where is a non-negative constant depending on ∥z 0 ∥ and T. From hypothesis 1 of Theorem 1.6, we have Using (38), (39) and the fact that By recurrence, we show that ∥CU nT the integer part of t T , we deduce that > 0, which achieves the proof.

Strong stabilization
The following result provides sufficient conditions for strong stabilization of the output (5).
Theorem 1.7 Let A generate a semigroup S t ð Þ ð Þ t ≥ 0 of contractions on H and B be locally Lipschitz.If the conditions: strongly stabilizes the output (5).Proof: From hypothesis 1 of Theorem 1.7, we obtain Integrating this inequality, gives 2 Thus From the variation of constants formula and using Schwartz's inequality, we deduce Integrating (34) over the interval 0; T ½ and taking into account (44), we obtain Replacing z 0 by z t ð Þ and using the superposition property of the solution, we get From ( 40) and (48), we deduce that where ϱ ¼ 1 γ θ: Integrating the above inequality gives We obtain Let us introduce the sequence r n ¼ ∥CU nT ð Þz 0 ∥ 2 Y , ∀n ≥ 0: Using (50), we deduce that Since the sequence r n ð Þ decreases, we get and also We deduce that hold, then the control weakly stabilizes the output (5).
Proof: Let us consider ψ ∈ Y and t n ð Þ≥ 0 be a sequence of real numbers such that t n !þ∞, as n !þ∞.
Example 4 Let us consider the system defined in Ω ¼0, þ ∞½ by where weakly stabilizes the output (54).
For ω ¼0, 2½, we have Figure 1 shows that the output (54) is weakly stabilized on ω with error equals 6:8 Â 10 À4 and the evolution of control is given by Figure 2.
For ω ¼0, 3½, we have Figure 3 shows that the output (54) is weakly stabilized on ω with error equals 9:88 Â 10 À4 and the evolution of control is given by Figure 4.

Conclusions
In this work, we discuss the question of output stabilization for a class of semilinear systems.Under sufficient conditions, we obtain controls depending on the output operator that strongly and weakly stabilizes the output of such systems.This work gives an opening to others questions; this is the case of output stabilization for hyperbolic semilinear systems.This will be the purpose of a future research paper.

Remark 2 .
It is clear that the control (55) stabilizes the state on ω, but do not take into account the residual part Ω ω.