Approximate Controllability for Linear Stochastic Differential Equations in Infinite Dimensions

TL;DR

This paper studies the approximate controllability of linear stochastic systems in infinite-dimensional Hilbert spaces, establishing duality with observability and providing necessary conditions via the Hautus test.

Contribution

It extends finite-dimensional controllability results to infinite dimensions and introduces a dual backward stochastic differential equation with unbounded operators.

Findings

Existence and uniqueness of solutions to the dual backward stochastic equation.

Duality between approximate controllability and observability.

Necessary condition for controllability via the generalized Hautus test.

Abstract

The objective of the paper is to investigate the approximate controllability property of a linear stochastic control system with values in a separable real Hilbert space. In a first step we prove the existence and uniqueness for the solution of the dual linear backward stochastic differential equation. This equation has the particularity that in addition to an unbounded operator acting on the Y-component of the solution there is still another one acting on the Z-component. With the help of this dual equation we then deduce the duality between approximate controllability and observability. Finally, under the assumption that the unbounded operator acting on the state process of the forward equation is an infinitesimal generator of an exponentially stable semigroup, we show that the generalized Hautus test provides a necessary condition for the approximate controllability. The paper generalizes former results by Buckdahn, Quincampoix and Tessitore (2006) and Goreac (2007) from the finite dimensional to the infinite dimensional case.