Constrained asymptotic null-controllability for semi-linear infinite dimensional systems
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Constrained asymptotic null-controllability for semi-linear infinite dimensional systems Lahoucine Boujallal1 · Khalid Kassara1 Received: 21 April 2020 / Revised: 8 October 2020 / Accepted: 12 October 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract This paper investigates asymptotic controllability of systems governed by semi-linear partial differential equations, under mixed input-state constraints. A unified approach based on viability theory and set-valued analysis, is provided, requiring that the linear part of the system generates a strongly continuous compact semigroup. In the case of convex constraints, it is shown that Michael selection theorem can be used to get existence of the needed feedback control laws. Examples are numerically treated in order to illustrate the results established. Keywords Semi-linear systems · Null-controllability · Lyapunov functions · Mixed constraints · Viability theory · Contingent cone Mathematics Subject Classification 93B03 · 93C10 · 93C20 · 35K58
1 Introduction and statement of the problem Controllability is one of the old and fundamental concepts in control theory. Its study, for infinite dimensional systems, has been in focus because of its outstanding importance not solely of theoretical interest, but also for numerous applications, such as spaceflight, aviation, communication, robot, bioengineering; cf. [1–4]. Out of their omnipresence in real word applications, constraints receive much attention, notably in association with optimal control problems. These may concern state and/or control, as in [5–8] and [9,10], and may even be mixed, ie. involving both state and control as in [11], which deals with an optimal control problem for age-structured systems. Saturating control case also have been treated in a series of studies, among them one can cite [12–14]. The present paper continues the research started in [15]. Its objective is to extend the unified set-valued approach, developed there, to the case of systems described by semi-linear
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Khalid Kassara [email protected] Lahoucine Boujallal [email protected]
1
Department of Mathematics and Computer Sciences, Faculty of Sciences Ain Chock, University Hassan II of Casablanca, Casablanca, Morocco
partial differential equations. More precisely, one is concerned with the question of asymptotic null-controllability of the following constrained semi-linear control system, z˙ + Az = f (z, u), (z, u) ∈ K, t ≥ 0,
(1)
where −A stands for an unbounded densely defined linear operator, which generates a C0 semigroup (S(t))t≥0 on a real Hilbert space Z . According to the semi-linear infinite dimensional setting, mapping f is assumed to be a nonlinear operator which maps Z 0 × R p into Z , with Z 0 , a possibly dense subspace of Z . Constraints subset K stands for a subset of Z × R p , satisfying, K ⊂ dom(A) ∩ Z 0 × R p . If no ambiguity holds, inner products on Hilbert spaces are denoted ·, · and corresponding norms, | · |, while balls centred at the origin with radius ρ are denoted by Bρ
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