Background results for robust minmax control of linear dynamical systems

TL;DR

This paper summarizes key arguments and results for robust minmax control of linear systems with quadratic costs, including corrections, extensions, and generalizations of previous propositions and formulas.

Contribution

It provides a revised, extended set of results and formulas crucial for robust minmax control of linear dynamical systems, including new propositions and corrected optimal control formulas.

Findings

Corrected optimal control formulas in key propositions

Combined related propositions for clarity

Extended results to more general cases

Abstract

The purpose of this note is to summarize the arguments required to derive the results appearing in robust minmax control of linear dynamical systems using a quadratic stage cost. The main result required in robust minmax control is Proposition 20.a. Moreover, the solution to the trust-region problem given in Proposition 15 and Lemma 16 may be of more general interest. This revised (second) version provides the following corrections and extensions of the previous (first) version. 1. The optimal u and w formulas in the original Corollary 13, Proposition 14, Corollary 19, and Proposition 20 have been corrected in this revision. 2. Corollary 13 and Proposition 14 are combined in the revised Proposition 14.a. 3. Corollary 19 and Proposition 20 are combined in the revised Proposition 20.a. 4. The revised Proposition 12.a is a generalization of the previous Proposition 12. 5. Propositions 5.a, 5.b, and 5.c are new in this revision. 6. Figure 1 has been revised to illustrate the revised Proposition 14.a.