Numerical Study of Polynomial Feedback Laws for a Bilinear Control Problem

TL;DR

This paper develops a numerical approach to derive polynomial feedback laws for infinite-dimensional bilinear control problems, using generalized Lyapunov equations and tensor calculus, and demonstrates their effectiveness on a Fokker-Planck control problem.

Contribution

It introduces a novel numerical method combining generalized Lyapunov equations and tensor calculus to compute polynomial feedback laws for bilinear control systems.

Findings

Polynomial feedback laws are effective near equilibrium.

The method reduces the curse of dimensionality.

Numerical experiments confirm the approach's efficiency.

Abstract

An infinite-dimensional bilinear optimal control problem with infinite-time horizon is considered. The associated value function can be expanded in a Taylor series around the equilibrium, the Taylor series involving multilinear forms which are uniquely characterized by generalized Lyapunov equations. A numerical method for solving these equations is proposed. It is based on a generalization of the balanced truncation model reduction method and some techniques of tensor calculus, in order to attenuate the curse of dimensionality. Polynomial feedback laws are derived from the Taylor expansion and are numerically investigated for a control problem of the Fokker-Planck equation. Their efficiency is demonstrated for initial values which are sufficiently close to the equilibrium.