Linear quadratic optimal control for a class of continuous-time nonhomogeneous Markovian jump linear systems in infinite time horizon

Abstract

In this paper, the infinite time horizon linear quadratic optimal control problem is investigated for the continuous-time Itô stochastic Markovian jump linear systems (MJLSs) with time-varying transition rates. It is assumed that the time-varying transition rates of the MJLSs possess piecewise homogeneous time-varying property, which implies that the transition rates are time-varying in the whole time domain but they are time-invariant in some small time intervals. The variations of the transition rates in these small time intervals are considered to be in two cases: arbitrary variation and stochastic variation. With this, the considered system becomes a piecewise homogeneous Itô stochastic MJLS. The main contribution of this paper is that two linear quadratic optimal controllers in infinite time horizon are proposed for the above modeled continuous-time piecewise homogeneous Itô stochastic MJLS in the sense of arbitrary variation and stochastic variation, respectively. Moreover, the sufficient and necessary conditions for the existence of the designed controllers are established based on the existence of the unique positive definite solution of two coupled algebraic Riccati matrix equations. Finally, a simulation example is provided to illustrate the effectiveness of the proposed linear quadratic optimal controllers.