7.3 Riccati equations and inequalities in robust control

This section builds on Chapter 6, with the goal of demonstrating that the tools and results of that chapter--in particular, optimality conditions expressed in terms of algebraic Riccati equations--continue to play a central role in the context of more general control problems. The first problem studied here is a special case of the infinite-horizon LQR problem except that the matrix $Q$ in the cost functional is no longer positive semidefinite, which leads to its interpretation as the problem of characterizing the disturbance-to-output $\mathcal L_2$ gain. Afterwards, we take a brief look at the $\mathcal H_\infty$ control problem, which has been an important benchmark problem in robust control theory.