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

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.