INDEFINITE STOCHASTIC LQ CONTROLS WITH MARKOVIAN JUMPS IN A FINITE TIME HORIZON

This paper is concerned with a stochastic linear-quadratic (LQ) control problem over a finite time horizon with Markovian jumps in the problem parameters. The problem is indefinite in that the cost weighting matrices for the state and control are allowed to be indefinite. A system of coupled generalized (dierential) Riccati equations (CGREs) is introduced to cope with the indefi- niteness of the problem. Specifically, it is proved that the solvability of the CGREs is sucient for the well-posedness of the stochastic LQ problem. Moreover, it is shown that the solvability of the CGREs is necessary for the well-posedness of the stochastic LQ problem and the existence of optimal (feedback/open-loop) controls via the dynamic programming approach. An example is presented to illustrate the results established.