Generalized differential Riccati equation and indefinite stochastic LQ control with cross term

Abstract

A stochastic indefinite linear-quadratic (LQ) optimal control problem with cross term in a finite time horizon is studied. A new generalized differential Riccati equation (GDRE) is introduced. It is shown that the solvability of the GDRE is equivalent to the solvability of the indefinite stochastic LQ problem. Furthermore, all of the optimal controls including feedback and open-loop can be identified via the solution to the GDRE. An example is presented to illustrate the theory obtained.