Stabilization of control systems associated with a strongly continuous group

Sorbonne Université, Universitsé Paris Cité, CNRS, INRIA, Laboratoire Jacques-Louis Lions, LJLL, 75005 Paris, France

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Received: 20 June 2024
Accepted: 14 December 2024

Abstract

This paper is devoted to the stabilization of a linear control system y' = Ay + Bu and its suitable non-linear variants where (A, 𝒟(A)) is an infinitesimal generator of a strongly continuous group in a Hilbert space ℍ, and B defined in a Hilbert space U is an admissible control operator with respect to the semigroup generated by A. Let λ ϵ ℝ and assume that, for some positive symmetric, invertible Q = Q(λ) ϵ ℒ(ℍ), for some non-negative, symmetric R = R(λ) ϵ ℒ(ℍ), and for some non-negative, symmetric W = W (λ) ∈ ℒ(U), it holds

AQ + QA^* − BW B^* + QRQ + 2λQ= 0.

We then present a new approach to study the stabilization of such a system and its suitable nonlinear variants. Both the stabilization using dynamic feedback controls and the stabilization using static feedback controls in a weak sense are investigated. To our knowledge, the nonlinear case is out of reach previously when B is unbounded for both types of stabilization.