TY - JOUR
T1 - New regularity results and finite element error estimates for a class of parabolic optimal control problems with pointwise state constraints
AU - Christof, Constantin
AU - Vexler, Boris
N1 - Publisher Copyright:
© EDP Sciences, SMAI 2021.
PY - 2021
Y1 - 2021
N2 - We study first-order necessary optimality conditions and finite element error estimates for a class of distributed parabolic optimal control problems with pointwise state constraints. It is demonstrated that, if the bound in the state constraint and the differential operator in the governing PDE fulfil a certain compatibility assumption, then locally optimal controls satisfy a stationarity system that allows to significantly improve known regularity results for adjoint states and Lagrange multipliers in the parabolic setting. In contrast to classical approaches to first-order necessary optimality conditions for state-constrained problems, the main arguments of our analysis require neither a Slater point, nor uniform control constraints, nor differentiability of the objective function, nor a restriction of the spatial dimension. As an application of the established improved regularity properties, we derive new finite element error estimates for the dG(0) - cG(1)-discretization of a purely state-constrained linear-quadratic optimal control problem governed by the heat equation. The paper concludes with numerical experiments that confirm our theoretical findings.
AB - We study first-order necessary optimality conditions and finite element error estimates for a class of distributed parabolic optimal control problems with pointwise state constraints. It is demonstrated that, if the bound in the state constraint and the differential operator in the governing PDE fulfil a certain compatibility assumption, then locally optimal controls satisfy a stationarity system that allows to significantly improve known regularity results for adjoint states and Lagrange multipliers in the parabolic setting. In contrast to classical approaches to first-order necessary optimality conditions for state-constrained problems, the main arguments of our analysis require neither a Slater point, nor uniform control constraints, nor differentiability of the objective function, nor a restriction of the spatial dimension. As an application of the established improved regularity properties, we derive new finite element error estimates for the dG(0) - cG(1)-discretization of a purely state-constrained linear-quadratic optimal control problem governed by the heat equation. The paper concludes with numerical experiments that confirm our theoretical findings.
KW - A priori error estimate
KW - Finite element method
KW - First-order necessary optimality condition
KW - Optimal control
KW - Parabolic partial differential equation
KW - Regularity result
KW - State constraints
UR - http://www.scopus.com/inward/record.url?scp=85099783296&partnerID=8YFLogxK
U2 - 10.1051/cocv/2020059
DO - 10.1051/cocv/2020059
M3 - Article
AN - SCOPUS:85099783296
SN - 1292-8119
VL - 27
JO - ESAIM - Control, Optimisation and Calculus of Variations
JF - ESAIM - Control, Optimisation and Calculus of Variations
M1 - 4
ER -