Constrained null controllability for distributed systems and applications to hyperbolic-like equations

Berrahmoune, Larbi

ESAIM: Control, Optimisation and Calculus of Variations, Tome 25 (2019), article no. 32.

Suite au passage du modèle économique de la revue en S20, le texte intégral des articles des années 2019 et 2020 est accessible uniquement sur le site de l'éditeur et est réservé aux abonnés.

We consider linear control systems of the form y′(t) = Ay(t) + Bu(t) on a Hilbert space Y . We suppose that the control operator B is bounded from the control space U to a larger extrapolation space containing Y . The aim is to study the null controllability in the case where the control u is constrained to lie in a bounded subset Γ ⊂ U. We obtain local constrained controllability properties. When ($$)$$ is a group of isometries, we establish necessary conditions and sufficient ones for global constrained controllability. Moreover, when the constraint set Γ contains the origin in its interior, the local constrained property turns out to be equivalent to a dual observability inequality of L¹ type with respect to the time variable. In this setting, the study is focused on hyperbolic-like systems which can be reduced to a second order evolution equation. Furthermore, we treat the problem of determining a steering control for general constraint set Γ in nonsmooth convex analysis context. In the case where Γ contains the origin in its interior, a steering control can be obtained by minimizing a convenient smooth convex functional. Applications to the wave equation and Euler-Bernoulli beams are presented.

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