Galerkin approximations for the optimal control of nonlinear delay differential equations

TL;DR

This paper develops a Galerkin approximation method using Koornwinder polynomials for solving optimal control problems in nonlinear delay differential equations, providing error estimates and demonstrating efficiency near bifurcation points.

Contribution

It introduces a novel Galerkin-Koornwinder approximation scheme with error bounds for optimal control of nonlinear DDEs, applicable to bifurcation analysis and control.

Findings

Low-dimensional controls effectively reduce oscillation amplitude.

Controls from PMP and HJB agree well with full solutions.

Reduced HJB provides a good approximation of the full value function.

Abstract

Optimal control problems of nonlinear delay differential equations (DDEs) are considered for which we propose a general Galerkin approximation scheme built from Koornwinder polynomials. Error estimates for the resulting Galerkin-Koornwinder approximations to the optimal control and the value function, are derived for a broad class of cost functionals and nonlinear DDEs. The approach is illustrated on a delayed logistic equation set not far away from its Hopf bifurcation point in the parameter space. In this case, we show that low-dimensional controls for a standard quadratic cost functional can be efficiently computed from Galerkin-Koornwinder approximations to reduce at a nearly optimal cost the oscillation amplitude displayed by the DDE's solution. Optimal controls computed from the Pontryagin's maximum principle (PMP) and the Hamilton-Jacobi-Bellman equation (HJB) associated with the corresponding ODE systems, are shown to provide numerical solutions in good agreement. It is finally argued that the value function computed from the corresponding reduced HJB equation provides a good approximation of that obtained from the full HJB equation.