Observability inequalities for the backward stochastic evolution equations and their applications

TL;DR

This paper establishes observability inequalities for backward stochastic evolution equations using spectral and interpolation inequalities, enabling null controllability with minimal control inputs for various stochastic PDEs.

Contribution

It introduces a novel approach combining spectral, interpolation inequalities, and the telegraph series method to directly derive observability inequalities for backward stochastic evolution equations.

Findings

Null controllability achieved with one control in the drift term

Applicable to stochastic degenerate, fourth order parabolic, and heat equations

Establishes new observability inequalities for backward stochastic PDEs

Abstract

The present article delves into the investigation of observability inequalities pertaining to backward stochastic evolution equations. We employ a combination of spectral inequalities, interpolation inequalities, and the telegraph series method as our primary tools to directly establish observability inequalities. Furthermore, we explore three specific equations as application examples: a stochastic degenerate equation, a stochastic fourth order parabolic equation and a stochastic heat equation. It is noteworthy that these equations can be rendered null controllability with only one control in the drift term to each system.