The pde2path add-on toolbox p2pOC for solving infinite time-horizon spatially distributed optimal control problems - Quickstart Guide -

TL;DR

p2pOC is a Matlab toolbox extension for numerically solving infinite horizon optimal control problems in PDE systems, utilizing canonical system analysis, bifurcation detection, and boundary value problem solving techniques.

Contribution

It introduces a novel approach combining stationary solution branches and canonical paths to solve high-dimensional infinite horizon optimal control problems.

Findings

Demonstrated bifurcations of patterned canonical steady states.

Applied method to shallow lake and grazing system models.

Showed effectiveness in computing optimal control paths.

Abstract

p2pOC is an add-on toolbox to the Matlab package pde2path. It is aimed at the numerical solution of optimal control (OC) problems with an infinite time horizon for parabolic systems of PDE over 1D or 2D spatial domains. The basic idea is to treat the OC problem via the associated canonical system in two steps. First we use pde2path to find branches of stationary solutions of the canonical system, also called canonical steady states (CSS). In a second step we use the results and the spatial discretization of the first step to calculate the objective values of time-dependent canonical paths ending at a CSS with the so called saddle point property. This is a (typically very high dimensional) boundary value problem (BVP) in time, which we solve by combining a modification of the BVP solver TOM with a continuation algorithm in the initial states. We explain the design and usage of the package via two example problems, namely the optimal management of a distributed shallow lake model, and of a semi-arid grazing system. Both show interesting bifurcations of so called patterned CSS, and in particular the latter also a variety of patterned optimal steady states. The package (library and demos) can be downloaded at www.staff.uni-oldenburg.de/hannes.uecker/pde2path.