7 An optimal control problem for equations with p-structure and its finite element discretization
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Adrian Hirn
Abstract
We analyze a finite element approximation of an optimal control problem that involves an elliptic equation with p-structure (e. g., the p-Laplace) as a constraint. As the nonlinear operator related to the p-Laplace equation mapping the space W01,p (Ω) to its dual (W01,p (Ω))* is not Gâteaux differentiable, first-order optimality conditions cannot be formulated in a standard way. Without using adjoint information, we derive novel a priori error estimates for the convergence of the cost functional for both variational discretization and piecewise constant controls.
Abstract
We analyze a finite element approximation of an optimal control problem that involves an elliptic equation with p-structure (e. g., the p-Laplace) as a constraint. As the nonlinear operator related to the p-Laplace equation mapping the space W01,p (Ω) to its dual (W01,p (Ω))* is not Gâteaux differentiable, first-order optimality conditions cannot be formulated in a standard way. Without using adjoint information, we derive novel a priori error estimates for the convergence of the cost functional for both variational discretization and piecewise constant controls.
Kapitel in diesem Buch
- Frontmatter I
- Preface V
- Contents VII
- 1 Reduced basis model order reduction in optimal control of a nonsmooth semilinear elliptic PDE 1
- 2 Pointwise moving control for the 1-D wave equation 33
- 3 Limits of stabilizability for a semilinear model for gas pipeline flow 59
- 4 Minimal cost-time strategies for mosquito population replacement 73
- 5 The sterile insect technique used as a barrier control against reinfestation 91
- 6 Variational discretization approach applied to an optimal control problem with bounded measure controls 113
- 7 An optimal control problem for equations with p-structure and its finite element discretization 137
- 8 Unstructured space-time finite element methods for optimal sparse control of parabolic equations 167
- 9 An adaptive finite element approach for lifted branched transport problems 189
- 10 High-order homogenization of the Poisson equation in a perforated periodic domain 237
- 11 Least-squares approaches for the 2D Navier–Stokes system 285
- 12 Numerical issues and turnpike phenomenon in optimal shape design 343
- 13 Feedback stabilization of Cahn–Hilliard phase-field systems 367
- 14 Ensemble Kalman filter for neural network-based one-shot inversion 393
- 15 Deep learning in high dimension: ReLU neural network expression for Bayesian PDE inversion 419
- Index 463
Kapitel in diesem Buch
- Frontmatter I
- Preface V
- Contents VII
- 1 Reduced basis model order reduction in optimal control of a nonsmooth semilinear elliptic PDE 1
- 2 Pointwise moving control for the 1-D wave equation 33
- 3 Limits of stabilizability for a semilinear model for gas pipeline flow 59
- 4 Minimal cost-time strategies for mosquito population replacement 73
- 5 The sterile insect technique used as a barrier control against reinfestation 91
- 6 Variational discretization approach applied to an optimal control problem with bounded measure controls 113
- 7 An optimal control problem for equations with p-structure and its finite element discretization 137
- 8 Unstructured space-time finite element methods for optimal sparse control of parabolic equations 167
- 9 An adaptive finite element approach for lifted branched transport problems 189
- 10 High-order homogenization of the Poisson equation in a perforated periodic domain 237
- 11 Least-squares approaches for the 2D Navier–Stokes system 285
- 12 Numerical issues and turnpike phenomenon in optimal shape design 343
- 13 Feedback stabilization of Cahn–Hilliard phase-field systems 367
- 14 Ensemble Kalman filter for neural network-based one-shot inversion 393
- 15 Deep learning in high dimension: ReLU neural network expression for Bayesian PDE inversion 419
- Index 463
