Infinite time horizon spatially distributed optimal control problems with pde2path algorithms and tutorial examples

TL;DR

This paper presents a numerical continuation approach using pde2path to analyze infinite time horizon optimal control problems for PDE systems, including algorithms, implementation, and examples like lake management and vegetation control.

Contribution

It introduces a two-step continuation method for solving high-dimensional boundary value problems in optimal control of PDEs, with practical algorithms and example applications.

Findings

Identification of bifurcations in patterned steady states

Demonstration of the method on lake and vegetation models

Validation of the approach with a toy ODE model

Abstract

We use the continuation and bifurcation package pde2path to numerically analyze infinite time horizon optimal control problems for parabolic systems of PDEs. The basic idea is a two step approach to the canonical systems, derived from Pontryagin's maximum principle. First we find branches of steady or time-periodic states of the canonical systems, i.e., canonical steady states (CSS) respectively canonical periodic states (CPS), and then use these results to compute time-dependent canonical paths connecting to a CSS or a CPS with the so called saddle point property. This is a (high dimensional) boundary value problem in time, which we solve by a continuation algorithm in the initial states. We first explain the algorithms and then the implementation via some example problems and associated pde2path demo directories. The first two examples deal with the optimal management of a distributed shallow lake, and of a vegetation system, both with (spatially, and temporally) distributed controls. These examples show interesting bifurcations of so called patterned CSS, including patterned optimal steady states. As a third example we discuss optimal boundary control of a fishing problem with boundary catch. For the case of CPS-targets we first focus on an ODE toy model to explain and validate the method, and then discuss an optimal pollution mitigation PDE model.