Optimal control of discrete-time bilinear systems with applications to switched linear stochastic systems

This paper aims at characterizing the most destabilizing switching law based on the second-order moment of the state to analyze uniform mean-square stability of discrete-time switched linear stochastic systems under arbitrary switching. To approach this purpose, a Mayer-type optimal control problem is investigated for a generalized discrete-time bilinear system governed by a set of linear positive operators, and a second-order necessary optimality condition is derived by using variational approach. To associate optimal control laws with switching laws, optimal equivalence between the bilinear system and a switched system is then analyzed. We surprisingly find that any optimal control law can be equivalently expressed as a switching law of the switched system. Intuitively, this switching law is most unstable not only for the switched system, but also for the corresponding switched linear stochastic system in the meaning of the second-order moment norm. An algorithm of checking the uniform mean-square stability is provided, and numerical simulations are presented to verify the usefulness of the theoretic results.