Optimal management and spatial patterns in a distributed shallow lake model.

*(English)*Zbl 1354.49003Summary: 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.

##### MSC:

49J20 | Existence theories for optimal control problems involving partial differential equations |

49N90 | Applications of optimal control and differential games |

35B32 | Bifurcations in context of PDEs |

91B76 | Environmental economics (natural resource models, harvesting, pollution, etc.) |

##### Keywords:

optimal control; Pontryagin’s maximum principle; bioeconomics; canonical steady states; connecting orbits
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\textit{D. Grass} and \textit{H. Uecker}, Electron. J. Differ. Equ. 2017, Paper No. 01, 21 p. (2017; Zbl 1354.49003)

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