Characterization by observability inequalities of controllability and stabilization properties

Abstract

Given a linear control system in a Hilbert space with a bounded control operator, we establish a characterization of exponential stabilizability in terms of an observability inequality. Such dual characterizations are well known for exact (null) controllability. Our approach exploits classical Fenchel duality arguments and, in turn, leads to characterizations in terms of observability inequalities of approximately null controllability and of α-null controllability. We comment on the relationships between those various concepts, at the light of the observability inequalities that characterize them.

Stabilization and approximate null-controllability for a large class of diffusive equations from thick control supports

“…It is well-known, see e.g. [6] (Theorem 2.43, page 56) or [21] (Remark 16), that the approximate null-controllability (without uniform cost) of the system (E F ) is equivalent to a unique continuation property of the adjoint system. More precisely, the adjoint system…”

“…Theorem 3.1 (Theorem 1 in [21]). The following assertions are equivalent: (i) The evolution system (E F ) is exponentially stabilizable from ω.…”

“…The study of the (rapid) stabilization and the (approximate) null-controllability of evolution equations of the form (E F ) has been much addressed recently [2,3,9,12,16,17,19,21]. The Schrödinger counterparts of these equations and the same equations posed on bounded domains have also been studied, respectively in [18] and [15,23].…”

“…The study of the (rapid) stabilization of the control system (E F ), as for it, has been addressed very recently in the works [9,17,21]. It has been proven in [9] (Theorem 1.1) that for all s > 0, the fractional heat equation (E t 2s ) is exponentially stabilizable from the support ω if and only if ω is thick.…”

Feedback law to stabilize linear infinite-dimensional systems

“…Our feedback law is based on the integration of a mutated Gramian operator-valued function. In the structure of the aforementioned mutated Gramian operator, we utilize the weak observability inequality in [21,14] and borrow some idea used to construct generalized Gramian operators in [11,23,24]. Unlike most related works where the exact controllability is required, we only assume the above-mentioned weak observability inequality which is equivalent to the stabilizability of the system.…”