Electron. J. Differential Equations, Vol. 2017 (2017), No. 01, pp. 1-21.

Optimal management and spatial patterns in a distributed shallow lake model

Dieter Grass, Hannes Uecker

We present a numerical framework to treat infinite time horizon spatially distributed optimal control problems via the associated canonical system derived by Pontryagin's maximum principle. The basic idea is to consider the canonical system in two steps. First we perform a bifurcation analysis of canonical steady states using the continuation and bifurcation package pde2path, yielding a number of so called flat and patterned canonical steady states. In a second step we link pde2path to the two point boundary value problem solver TOM to study time dependent canonical paths to steady states having the so called saddle point property. As an example we consider a shallow lake model with diffusion.

Submitted June 10, 2015. Published January 4, 2017.
Math Subject Classifications: 49J20, 49N90, 35B32.
Key Words: Optimal control; Pontryagin's maximum principle; bioeconomics; canonical steady states; connecting orbits.

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Dieter Grass
ORCOS, Institute of Mathematical Methods in Economics
Vienna University of Technology
A-1040 Vienna, Austria
email: dieter.grass@tuwien.ac.at
Hannes Uecker
Institut für Mathematik
Universität Oldenburg
D26111 Oldenburg, Germany

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