Exponential stabilization of distributed semilinear systems by optimal control

Abstract

This paper studies exponential stabilization of distributed semilinear systems. The paper (i) gives a constrained feedback control that ensure the exponential stabilizability and (ii) shows that this control is the unique solution of an appropriate minimization problem. Examples of hyperbolic partial equations are provided.

Stabilisation and polynomial decay estimate for distributed semilinear systems

“…k yðtÞk M ffi ffi t p , M 4 0, for t large enough), has been obtained (Berrahmoune 1999;Ouzahra 2008). The case r ¼ 2 has been considered bilinear/semilinear systems in Ouzahra (2010) and Ouzahra (2011), where exponential stabilisation results have been established using p 2 (t) provided that the observation assumption (3) holds.…”

Necessary and sufficient conditions for exponential stabilization of systems affine in controls

“…where n : H → H is a (possibly) nonlinear operator. Let us also consider the perturbed system on A and B : Then, 1) the control (17) uniformly exponentially stabilizes the system (35) for any perturbation n ∈ P N such that L n < σ M , where M, σ are given by ( 18), 2) the control (17) uniformly exponentially stabilizes (36) under any perturbation (a, b) ∈ P 2 N such that :…”

“…However, in this way the convergence of the resulting closed loop state is not better than z(t) = O( 1 √ t ). The problem of exponential stabilization of the system (1) has been considered in [11,34,36] with the following bounded feedback v(t) = −ρ z(t), Bz(t) z(t) 2 1 {t≥0; z(t) =0} , (5) where ρ > 0 is the gain control. Moreover, under the assumption (4), the exponential stabilization of (1) has been studied in [37] using the switching control w(t) = −ρ sign( z(t), Bz(t) ), ρ > 0.…”

Feedback stabilization for a bilinear control system under weak observability inequalities

“…In addition, others polynomial estimates was provided in [27] using the control p r (t) with r < 2. The case r = 2 has been considered in [26], where exponential stabilization results have been established using p 2 (t) under the observation assumption (3). Note that the inequality (3) is necessary for uniform stabilization of conservative systems, so we can not expect such a degree of stability under a weaker observability assumption.…”