Optimal control of quasilinear parabolic PDEs with gradient terms and pointwise constraints on the gradient of the state

Bonifacius L, Hoppe F, Meinlschmidt H, Neitzel I (2025)

Publication Language: English

Publication Type: Journal article

Publication year: 2025

Journal

Mathematical Control and Related Fields American Institute of Mathematical Sciences (AIMS)

Book Volume: 15

Pages Range: 1284-1319

Journal Issue: 4

DOI: 10.3934/mcrf.2025037

Abstract

We derived existence results and first order necessary optimality conditions for optimal control problems governed by quasi-linear parabolic partial differential equations (PDEs) with a class of first order nonlinearities that included, for instance, quadratic gradient terms. Pointwise in space and time or averaged in space and pointwise in time constraints on the gradient of the state controlled the growth of the nonlinear terms. We relied on and extended the improved regularity analysis for quasilinear parabolic PDEs on a whole scale of function spaces from [29]. In case of integral in space gradient-constraints, we derived first-order optimality conditions under rather general regularity assumptions for domain, coefficients, and boundary conditions, similar to e.g. [8]. In the case of pointwise in time and space gradient-constraints, we used slightly stronger regularity assumptions leading to a classical smoother W^2,p-setting similar to [11].

Authors with CRIS profile

Hannes Meinlschmidt Juniorprofessur für Applied Analysis

Involved external institutions

Rheinische Friedrich-Wilhelms-Universität Bonn

Germany (DE) Deutsches Zentrum für Luft- und Raumfahrt e.V. (DLR) / German Aerospace Center

Germany (DE)

How to cite

APA:

Bonifacius, L., Hoppe, F., Meinlschmidt, H., & Neitzel, I. (2025). Optimal control of quasilinear parabolic PDEs with gradient terms and pointwise constraints on the gradient of the state. Mathematical Control and Related Fields, 15(4), 1284-1319. https://doi.org/10.3934/mcrf.2025037

MLA:

Bonifacius, Lucas, et al. "Optimal control of quasilinear parabolic PDEs with gradient terms and pointwise constraints on the gradient of the state." Mathematical Control and Related Fields 15.4 (2025): 1284-1319.

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