7.1 Introduction

In Chapter 6, the state and co-state canonical matrix was used to obtain the optimal functions x* and λ*. The optimal control u* was then derived in terms of λ*. In this chapter, for the linear quadratic (LQ) optimal control problem, the co-state function λ is considered to be a linear feedback function of x, leading to the matrix Riccati equation and, subsequently, to the optimal pair x* and u*. In Section 7.3, the Pontryagin principle for finding the optimal control for systems with constraints on the control function is discussed. Applications are provided in Section 7.4.

7.2 Feedback Control

Consider the following LQ problem: Find the optimal control u∗ and the corresponding optimal state x∗ for the linear time-varying system