Second order optimality conditions for optimal control of quasilinear parabolic equations

Lucas Bonifacius; Ira Neitzel

doi:10.3934/mcrf.2018001

Article Contents

2018, Volume 8, Issue 1: 1-34. Doi: 10.3934/mcrf.2018001

This issue Previous Article Preface: A tribute to professor Eduardo Casas on his 60th birthday Next Article Optimal voltage control of non-stationary eddy current problems

Second order optimality conditions for optimal control of quasilinear parabolic equations

Lucas Bonifacius^1, , and
Ira Neitzel^2,

1.
Technische Universität München, Fakultät für Mathematik, Boltzmannstr. 3, 85748 Garching, Germany
2.
Rheinische Friedrich-Wilhelms-Universität Bonn, Institut für Numerische Simulation, Wegelerstr. 6, 53115 Bonn, Germany

* Corresponding author: Ira Neitzel
* Corresponding author: Ira Neitzel

Received: March 2017

Revised: September 2017

Published: January 2018

The first author is supported by the International Research Training Group IGDK, funded by the German Science Foundation (DFG) and the Austrian Science Fund (FWF).

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Abstract

We discuss an optimal control problem governed by a quasilinear parabolic PDE including mixed boundary conditions and Neumann boundary control, as well as distributed control. Second order necessary and sufficient optimality conditions are derived. The latter leads to a quadratic growth condition without two-norm discrepancy. Furthermore, maximal parabolic regularity of the state equation in Bessel-potential spaces $H_D^{-\zeta,p}$ with uniform bound on the norm of the solution operator is proved and used to derive stability results with respect to perturbations of the nonlinear differential operator.

Keywords:

Optimal control,
second order optimality conditions,
quasilinear parabolic partial differential equation,
nonautonomous equation,
maximal parabolic regularity.

Mathematics Subject Classification: 35K59, 49K20, 90C48.

Citation:

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Table 1. Summary of differentiability and integrability exponents

Variable	Description
$p$	Given by isomorphism $-\nabla\cdot\mu\nabla + 1 \colon W_D^{1,p}\rightarrow W_D^{-1,p}$ and close to the spatial dimension $d$ ; see Assumption 3.
$\zeta$	Differentiability exponent close to one defining $H_D^{-\zeta,p}$ .
$s$	Integrability exponent for the controls determined by $p$ , $\zeta$ , and $d$ , possibly large; see Assumption 4.
$r$ , $r'$	Integrability exponents for linearized and adjoint state equation introduced in Section 4.

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