Exact Controllability of Linear Stochastic Differential Equations and Related Problems

TL;DR

This paper introduces the concept of $L^p$-exact controllability for linear stochastic differential equations, establishing conditions and equivalences with observability and optimal control problems.

Contribution

It defines $L^p$-exact controllability for stochastic equations and proves its equivalence to observability and control optimization problems.

Findings

Established sufficient conditions for $L^p$-exact controllability.

Proved equivalence between controllability, observability, and optimal control.

Provided theoretical framework linking controllability and control problems.

Abstract

A notion of $L^p$-exact controllability is introduced for linear controlled (forward) stochastic differential equations, for which several sufficient conditions are established. Further, it is proved that the $L^p$-exact controllability, the validity of an observability inequality for the adjoint equation, the solvability of an optimization problem, and the solvability of an $L^p$-type norm optimal control problem are all equivalent.