Finite Codimensional Controllability, and Optimal Control Problems with Endpoint State Constraints

TL;DR

This paper introduces the concept of finite codimensional controllability for infinite-dimensional systems, providing criteria and links to classical control conditions, with applications to wave equations with endpoint constraints.

Contribution

It defines finite codimensional controllability, establishes criteria, and connects it to classical geometric control conditions for wave equations, advancing control theory for infinite-dimensional systems.

Findings

Finite codimensional controllability criteria are established.

A G{ a}rding type inequality is derived for the adjoint system.

Finite codimensional exact controllability is linked to geometric control conditions.

Abstract

In this paper, motivated by the study of optimal control problems for infinite dimensional systems with endpoint state constraints, we introduce the notion of finite codimensional (exact/approximate) controllability. Some equivalent criteria on the finite codimensional controllability are presented. In particular, the finite codimensional exact controllability is reduced to deriving a G{\aa}rding type inequality for the adjoint system, which is new for many evolution equations. This inequality can be verified for some concrete problems (and hence applied to the corresponding optimal control problems), say the wave equations with both time and space dependent potentials. Moreover, under some mild assumptions, we show that the finite codimensional exact controllability of this sort of wave equations is equivalent to the classical geometric control condition.