Finite Element Methods for Elliptic Distributed Optimal Control Problems with Pointwise State Constraints (Survey)

Brenner, Susanne C.

doi:10.1007/978-3-030-42687-3_1

Cited by 4 publications

(3 citation statements)

References 63 publications

(85 reference statements)

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“…Note that, in this case, the obtained system of equations involves the bi-Laplacian so that some care has to be taken regarding the choice of the used finite element spaces. Compare, e.g., with [12] in this context where a similar technique is applied to state-constrained problems. We do not discuss this alternative discretization here to avoid overloading this paper.…”

Section: Numerical Experimentsmentioning

confidence: 99%

Multiobjective optimal control of a non-smooth semilinear elliptic partial differential equation

Christof

Müller

2021

ESAIM: COCV

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This paper is concerned with the derivation and analysis of first-order necessary optimality conditions for a class of multiobjective optimal control problems governed by an elliptic non-smooth semilinear partial differential equation. Using an adjoint calculus for the inverse of the non-linear and non-differentiable directional derivative of the solution map of the considered PDE, we extend the concept of strong stationarity to the multiobjective setting and demonstrate that the properties of weak and proper Pareto stationarity can also be characterized by suitable multiplier systems that involve both primal and dual quantities. The established optimality conditions imply in particular that Pareto stationary points possess additional regularity properties and that mollification approaches are - in a certain sense - exact for the studied problem class. We further show that the obtained results are closely related to rather peculiar hidden regularization effects that only reveal themselves when the control is eliminated and the problem is reduced to the state. This observation is also new for the case of a single objective function. The paper concludes with numerical experiments that illustrate that the derived optimality systems are amenable to numerical solution procedures.

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Section: Numerical Experimentsmentioning

confidence: 99%

Multiobjective optimal control of a non-smooth semilinear elliptic partial differential equation

Christof

Müller

2021

ESAIM: COCV

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“…Fast solvers for (1.1)-(1.3) are also studied in [15,10]. We refer to [9] for a more detailed survey about finite element methods for (1.1)- (1.3). Overall, as pointed out in [9], optimal control problems with pointwise state constraints can be analyzed using known finite element methods for fourth order boundary value problems.…”

Section: Introductionmentioning

confidence: 99%

“…We refer to [9] for a more detailed survey about finite element methods for (1.1)- (1.3). Overall, as pointed out in [9], optimal control problems with pointwise state constraints can be analyzed using known finite element methods for fourth order boundary value problems. This crucial observation opens doors to many possible numerical methods for optimal control problems with pointwise state constraints.…”

Section: Introductionmentioning

confidence: 99%

Discontinuous Galerkin Methods for an Elliptic Optimal Control Problem with a General State Equation and Pointwise State Constraints

Liu¹,

Tan²,

Zhang³

2023

Preprint

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We investigate discontinuous Galerkin methods for an elliptic optimal control problem with a general state equation and pointwise state constraints on general polygonal domains. We show that discontinuous Galerkin methods for general second-order elliptic boundary value problems can be used to solve the elliptic optimal control problems with pointwise state constraints. We establish concrete error estimates and numerical experiments are shown to support the theoretical results.

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Discontinuous Galerkin methods for an elliptic optimal control problem with a general state equation and pointwise state constraints

Liu,

Tan,

Zhang

2024

Journal of Computational and Applied Mathematics

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Finite Element Methods for Elliptic Distributed Optimal Control Problems with Pointwise State Constraints (Survey)

Abstract: Finite element methods for a model elliptic distributed optimal control problem with pointwise state constraints are considered from the perspective of fourth order boundary value problems.

Cited by 4 publications

References 63 publications

Multiobjective optimal control of a non-smooth semilinear elliptic partial differential equation

Multiobjective optimal control of a non-smooth semilinear elliptic partial differential equation

Discontinuous Galerkin Methods for an Elliptic Optimal Control Problem with a General State Equation and Pointwise State Constraints

Discontinuous Galerkin methods for an elliptic optimal control problem with a general state equation and pointwise state constraints

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