Hamilton-Jacobi-Bellman Equation for Control Systems with Friction

TL;DR

This paper develops a novel framework for control systems with friction, modeling them as controlled differential inclusions with discontinuities, and derives the associated Hamilton-Jacobi-Bellman equation to characterize optimal control solutions.

Contribution

It introduces a new approach to model friction in control systems using differential inclusions and establishes the HJB equation for the resulting optimal control problem.

Findings

Existence and uniqueness of solutions for the proposed model

Derivation of the Hamilton-Jacobi-Bellman equation for systems with friction

Characterization of the value function as a viscosity solution

Abstract

This paper proposes a new framework to model control systems in which a dynamic friction occurs. The model consists in a controlled differential inclusion with a discontinuous right hand side, which still preserves existence and uniqueness of the solution for each given input function $u(t)$. Under general hypotheses, we are able to derive the Hamilton-Jacobi-Bellman equation for the related free time optimal control problem and to characterise the value function as the unique, locally Lipschitz continuous viscosity solution.