Linear-Quadratic Optimal Controls in Finite Horizons

This chapter is devoted to a study of stochastic linear-quadratic optimal control problems in a finite horizon from two points of view: open-loop and closed-loop solvabilities. A simple example shows that these two solvabilities are essentially different. Open-loop solvability is established by studying the solvability of a constrained linear forward-backward stochastic differential equation. Closed-loop solvability is reduced to the existence of a regular solution to the associated differential Riccati equation, which is implied by the uniform convexity of the quadratic cost functional. The relation between open-loop and closed-loop solvabilities, as well as some other aspects, such as conditions ensuring the convexity of the cost functional, finiteness of the problem and construction of minimizing sequences, are also discussed.