# American Institute of Mathematical Sciences

September  2018, 8(3&4): 607-623. doi: 10.3934/mcrf.2018025

## Analysis and optimal control of some quasilinear parabolic equations

 1 Departamento de Matemática Aplicada y Ciencias de la Computación, E.T.S.I. Industriales y de Telecomunicación, Universidad de Cantabria, 39005 Santander, Spain 2 Department of Mathematics, School of Applied Mathematics and Physical Sciences, National Technical University of Athens, Zografou Campus, 15780 Athens, Greece

Corresponding author: Eduardo Casas

Dedicated to Prof. Jiongmin Yong on the occasion of his 60th birthday

Received  May 2017 Revised  January 2018 Published  September 2018

Fund Project: The first author was partially supported by the Spanish Ministerio de Economía, Industria y Competitividad under projects MTM2014-57531-P and MTM2017-83185-P

In this paper, we consider optimal control problems associated with a class of quasilinear parabolic equations, where the coefficients of the elliptic part of the operator depend on the state function. We prove existence, uniqueness and regularity for the solution of the state equation. Then, we analyze the control problem. The goal is to get first and second order optimality conditions. To this aim we prove the necessary differentiability properties of the relation control-to-state and of the cost functional.

Citation: Eduardo Casas, Konstantinos Chrysafinos. Analysis and optimal control of some quasilinear parabolic equations. Mathematical Control & Related Fields, 2018, 8 (3&4) : 607-623. doi: 10.3934/mcrf.2018025
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