Linear quadratic optimization problems

Abstract

In this chapter several problems of the optimization of a quadratic cost functional along the trajectories of a discrete-time linear stochastic system affected by jumping Markov perturbations are independent random perturbations are investigated. In Section 6.2 we deal with the classical problem of the linear quadratic optimal regulator which means the minimization of a quadratic cost functional with definite sign along the trajectories of a controlled linear system. Also in Section 6.3 the general case of a linear quadratic optimization problem with a cost functional without sign is treated. It is shown that in the case of a linear quadratic optimal regulator, the optimal control is constructed via the minimal solution of a system of discrete-time Riccati-type equations, whereas in the general case of the linear quadratic optimization problem without sign, the optimal control, if it exists, is constructed based on the stabilizing solution of a system of discrete-time Riccati-type equations. In Section 6.4 we deal with the problem of the optimization of a quadratic cost functional of a discrete-time affine stochastic system affected by jumping Markov perturbations and independent random perturbations. Both the case of finite time horizon as well as the infinite time horizon are considered. Optimal control is constructed using the stabilizing solution for a system of discrete-time Riccati-type equations. A set of necessary and sufficient conditions ensuring the existence of the desired solutions of the discrete-time Riccati equations involved in this chapter were given in Chapter 5. A tracking problem is also solved.