Systems of matrix rational differential equations arising in connection with linear stochastic systems with Markovian jumping

In this paper, a class of systems of matrix nonlinear differential equations containing as particular cases the systems of coupled Riccati differential equations arising in connection with control of some linear stochastic systems is considered. The system of differential equations considered in this paper are converted in a suitable nonlinear differential equation on a finite-dimensional Hilbert space adequately choosen. This allows us to use the positivity properties of the linear evolution operator defined by the linear differential equations of Lyapunov type. Our aim is to investigate properties of stabilizing and bounded solutions of the considered differential equations and to obtain some conditions ensuring the existence of such solutions. Conditions providing the existence of a maximal solution (minimal solution respectively) with respect to some classes of global solutions are presented. It is shown that if the coefficients of the equations are periodic functions all these special solutions (stabilizing, maximal, minimal) are periodic functions, too. Whenever possible the probabilistic arguments were avoided and so the results proved in the paper appear as results in the field of differential equations with interest in themselves.