On the linear quadratic problem for systems with time reversed Markov jump parameters and the duality with filtering of Markov jump linear systems

TL;DR

This paper investigates linear quadratic control problems for systems with parameters driven by a Markov chain in reverse time, establishing duality with filtering of Markov jump linear systems and providing stability and optimal control formulas.

Contribution

It extends classical control-filtering duality to systems with time-reversed Markov jump parameters, offering new theoretical insights and formulas.

Findings

Recursive second moment matrix characterization

Spectral radius test for mean square stability

Formulas for optimal control in reversed Markov systems

Abstract

We study a class of systems whose parameters are driven by a Markov chain in reverse time. A recursive characterization for the second moment matrix, a spectral radius test for mean square stability and the formulas for optimal control are given. Our results are determining for the question: is it possible to extend the classical duality between filtering and control of linear systems (whose matrices are transposed in the dual problem) by simply adding the jump variable of a Markov jump linear system. The answer is positive provided the jump process is reversed in time.