Deterministic control of stochastic reaction-diffusion equations

Wilhelm Stannat; Lukas Wessels

doi:10.3934/eect.2020087

Article Contents

2021, Volume 10, Issue 4: 701-722. Doi: 10.3934/eect.2020087

This issue Previous Article Solvability in abstract evolution equations with countable time delays in Banach spaces: Global Lipschitz perturbation Next Article Lifespan of solutions to a parabolic type Kirchhoff equation with time-dependent nonlinearity

Deterministic control of stochastic reaction-diffusion equations

Wilhelm Stannat and
Lukas Wessels^,

Technische Universität Berlin, Institut für Mathematik, Straße des 17. Juni 136, 10623 Berlin, Germany

^*Corresponding author
^*Corresponding author

Received: January 31, 2020

Revised: May 31, 2020

Early access: August 2020

Published: December 2021

The authors gratefully acknowledge financial support from the Deutsche Forschungsgemeinschaft (DFG) through the Collaborative Research Center 910 "Control of self-organizing nonlinear systems: Theoretical methods and concepts of application" (project A10)

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Abstract

We consider the control of semilinear stochastic partial differential equations (SPDEs) via deterministic controls. In the case of multiplicative noise, existence of optimal controls and necessary conditions for optimality are derived. In the case of additive noise, we obtain a representation for the gradient of the cost functional via adjoint calculus. The restriction to deterministic controls and additive noise avoids the necessity of introducing a backward SPDE. Based on this novel representation, we present a probabilistic nonlinear conjugate gradient descent method to approximate the optimal control, and apply our results to the stochastic Schlögl model. We also present some analysis in the case where the optimal control for the stochastic system differs from the optimal control for the deterministic system.

Keywords:

Stochastic reaction-diffusion equations,
variational approach,
optimal control,
stochastic Schlögl model,
stochastic Nagumo equation,
nonlinear conjugate gradient descent.

Mathematics Subject Classification: Primary: 93E20, 49K45; Secondary: 49M05, 60H15, 35K57.

Citation:

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Figure 1. Solution without Control in the Stochastic Case

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Figure 2. Solution with Optimal Control, $ \sigma = 0.5 $

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Figure 3. Solution without Control, $ \sigma = 1 $

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Figure 4. Optimal Control

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Figure 5. Potential

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Figure 6. Solution without Control, σ = 1

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Figure 7. Optimal Control, σ = 0.5

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Figure 8. Solution with Optimal Control, σ = 0.5

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Figure 9. Optimal Control, σ = 1

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Figure 10. Solution with Optimal Control, σ = 1

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