Journal ArticleDOI
Using piecewise linear functions in the numerical approximation of semilinear elliptic control problems
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It is proved that the L2-error estimates are of order o(h), which is optimal according with the C0,1 -regularity of the optimal control.Abstract:
We study the numerical approximation of distributed optimal control problems governed by semilinear elliptic partial differential equations with pointwise constraints on the control. Piecewise linear finite elements are used to approximate the control as well as the state. We prove that the L
2-error estimates are of order o(h), which is optimal according with the $C^{0,1}(\overline{\Omega})$
-regularity of the optimal control.read more
Citations
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Journal ArticleDOI
Error Estimates for the Numerical Approximation of Boundary Semilinear Elliptic Control Problems
TL;DR: The uniform convergence of discretized controls to optimal controls is proven under natural assumptions by taking piecewise constant controls.
Journal ArticleDOI
Second Order Optimality Conditions and Their Role in PDE Control
Eduardo Casas,Fredi Tröltzsch +1 more
TL;DR: In this paper, a survey of the second order sufficient optimality condition for nonlinear partial differential equations in infinite dimensions is presented. But the authors focus on the case of linear normed spaces.
Journal ArticleDOI
Second Order Analysis for Optimal Control Problems: Improving Results Expected From Abstract Theory
Eduardo Casas,Fredi Tröltzsch +1 more
TL;DR: It is proved that, in contrast to a widespread common belief, the standard second-order conditions formulated for these control problems imply strict local optimality of the controls not only in the sense of L^\infty, but also of $L^2$.
Book ChapterDOI
Optimal Control of Partial Differential Equations
Eduardo Casas,Mariano Mateos +1 more
TL;DR: In this paper, an optimal control problem governed by a semilinear elliptic equation, the control being subject to bound constraints, is considered and the methods to prove the existence of a solution, derive the first and second order optimality conditions, and approximate the control problem by discrete problems are presented.
Journal ArticleDOI
Error estimates of fully discrete mixed finite element methods for semilinear quadratic parabolic optimal control problem
Yanping Chen,Zuliang Lu +1 more
TL;DR: By applying some error estimates techniques of mixed finite element methods, this paper derives a priori error estimates both for the coupled state and the control approximation and presents a numerical example which confirms the theoretical results.
References
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Book
The Finite Element Method for Elliptic Problems
Philippe G. Ciarlet,J. T. Oden +1 more
TL;DR: The finite element method has been applied to a variety of nonlinear problems, e.g., Elliptic boundary value problems as discussed by the authors, plate problems, and second-order problems.
Book
Finite Element Method for Elliptic Problems
TL;DR: In this article, Ciarlet presents a self-contained book on finite element methods for analysis and functional analysis, particularly Hilbert spaces, Sobolev spaces, and differential calculus in normed vector spaces.
Book
Elliptic Problems in Nonsmooth Domains
TL;DR: Second-order boundary value problems in polygons have been studied in this article for convex domains, where the second order boundary value problem can be solved in the Sobolev spaces of Holder functions.
Book
Introduction à l'analyse numérique des équations aux dérivées partielles
TL;DR: Theoretique and approximation numerique des problemes aux limites lineaires for les equations aux derivees partielles are discussed in this article, where the authors also propose an approximation variationnelle for problemes paraboliques.
Book
Optimal Control of Partial Differential Equations
TL;DR: This chapter introduces the basic concepts of optimal control for linear elliptic partial differential equations and shows two different numerical approaches for control problems, based on the Galerkin finite element method.