Time-optimal control of a two-peakon collision

TL;DR

This paper investigates the problem of controlling a two-peakon collision in minimal time, using Hamilton-Jacobi-Bellman equations and dynamic programming, and proposes a method extendable to broader control problems.

Contribution

It introduces a novel approach to solve time-optimal control problems for Hamiltonian systems, including a special maximum principle, with potential for generalization.

Findings

Established controllability of the two-peakon collision problem

Derived the time-optimal control strategy using HJB and dynamic programming

Proposed a new method extendable to other Hamiltonian control systems

Abstract

We study a time-optimal control problem of a two-peakon collision. First, we state the controllability. Next, we find the time-optimal strategy. This is done via the HamiltonJacobi-Bellman equation and the dynamic programming method. We find a tricky way to solve a corresponding Hamilton-Jacobi-Bellman equation. Our approach can be extended to more general control problems. Indeed, a special version of a maximum principle for time-optimal controls of particular Hamiltonian systems is proposed.