A unified controllability/observability theory for some stochastic and deterministic partial differential equations

TL;DR

This paper introduces a universal framework for analyzing controllability and observability in both stochastic and deterministic PDEs, utilizing weighted identities and Carleman estimates to unify various control-related problems.

Contribution

It develops a unified approach based on weighted identities to study controllability, observability, stabilization, and inverse problems for stochastic and deterministic PDEs.

Findings

Established fundamental weighted identities for PDE operators.

Derived global Carleman estimates applicable to multiple PDE types.

Unified treatment of control, stabilization, and inverse problems.

Abstract

The purpose of this paper is to present a universal approach to the study of controllability/observability problems for infinite dimensional systems governed by some stochastic/deterministic partial differential equations. The crucial analytic tool is a class of fundamental weighted identities for stochastic/deterministic partial differential operators, via which one can derive the desired global Carleman estimates. This method can also give a unified treatment of the stabilization, global unique continuation, and inverse problems for some stochastic/deterministic partial differential equations.