Linear Quadratic Optimal Control Problems

We have studied general nonlinear optimal control problems for both the deterministic and stochastic cases in previous chapters. In this chapter we are going to investigate a special case of optimal control problems where the state equations are linear in both the state and control with nonhomogeneous terms, and the cost functionals are quadratic. Such a control problem is called a linear quadratic optimal control problem (LQ problem, for short). The LQ problems constitute an extremely important class of optimal control problems, since they can model many problems in applications, and more importantly, many nonlinear control problems can be reasonably approximated by the LQ problems. On the other hand, solutions of LQ problems exhibit elegant properties due to their simple and nice structures.