Nonstandard Linear-Quadratic Decision Making

Abstract

This paper is a compendium to a tutorial session in the 2023 IEEE Conference on Decision and Control (CDC) by the same authors listed here and carrying the same title. The session (as well as this paper) addresses variations around the basic linear-quadratic-Gaussian (LQG) control paradigm, identifies the underlying challenges in such extensions, and discusses some of their resolutions. The paper is organized into four main parts, each one corresponding to a presentation by one of the authors, as the section titles indicate. The first part discusses stochastic control and stochastic games under nonstandard (or nonclassical) information structures, which have elements of signaling and incentive designs, including also variations around the celebrated counterexample in stochastic control due to Witsenhausen. The first part also covers information transmission limitations in control, and rational expectations models arising in economics. The second part considers variations to stochastic LQ Nash and Stackelberg differential games, deriving explicit equilibrium policies expressed by Riccati equations. The third part describes approaches to solving nonstandard LQ control problems to characterize their feedback-type optimal controllers under three different settings, namely (i) irregularity, (ii) delay state equations, and (iii) asymmetric information structures. The last part provides open-loop and closed-loop solvability analyses for LQ control problems, establishing relationships between forward-backward stochastic differential equations, the optimality condition, and (differential/algebraic) Riccati equations.