Tail-Constraining Stochastic Linear-Quadratic Control: Large Deviation and Statistical Physics Approach

TL;DR

This paper introduces new methods for risk-sensitive stochastic LQ control that focus on tail behavior of the cost distribution, providing static optimization solutions and analyzing large deviation asymptotics.

Contribution

It proposes two alternative approaches to classical RS-LQ control that eliminate the need for an exogenous risk parameter and reduce the problem to static optimization.

Findings

Solutions for scalar and 1D chain systems are illustrated.

Large deviation asymptotics of the cost functional PDF are analyzed.

The methods provide stable control strategies based on tail distribution constraints.

Abstract

Standard definition of the stochastic Risk-Sensitive Linear-Quadratic (RS-LQ) control depends on the risk parameter, which is normally left to be set exogenously. We reconsider the classical approach and suggest two alternatives resolving the spurious freedom naturally. One approach consists in seeking for the minimum of the tail of the Probability Distribution Function (PDF) of the cost functional at some large fixed value. Another option suggests to minimize the expectation value of the cost functional under constraint on the value of the PDF tail. Under assumption of the resulting control stability, both problems are reduced to static optimizations over stationary control matrix. The solutions are illustrated on the examples of scalar and 1d chain (string) systems. Large Deviation self-similar asymptotic of the cost functional PDF is analyzed.