# Crank-Nicolson Schemes for Optimal Control Problems with Evolution Equations

@article{Apel2012CrankNicolsonSF, title={Crank-Nicolson Schemes for Optimal Control Problems with Evolution Equations}, author={Thomas Apel and Thomas G. Flaig}, journal={SIAM J. Numer. Anal.}, year={2012}, volume={50}, pages={1484-1512} }

Crank--Nicolson methods are often used for the simulation of initial boundary value problems for parabolic partial differential equations. In this paper we present a family of discretizations for parabolic optimal control problems based on Crank--Nicolson schemes with different time discretizations for state $y$ and adjoint state $p$ so that discretization and optimization commute. One of these methods can also be explained as a Stormer--Verlet scheme in the context of geometric numerical…

## 31 Citations

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A Priori Error Analysis of the Petrov-Galerkin Crank-Nicolson Scheme for Parabolic Optimal Control Problems

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A finite element discretization of an optimal control problem governed by the heat equation is considered and a discrete solution is obtained for which error estimates of optimal order are proven.

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A mixed finite element discretization of an optimal control problem for the wave equation with homogeneous Dirichlet boundary condition is considered. For the temporal discretization, a…

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A second-order leapfrog finite difference scheme in time is proposed to solve the first-order necessary optimality systems arising from parabolic optimal control problems. Different from classical…

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## References

SHOWING 1-10 OF 36 REFERENCES

A Priori Error Analysis of the Petrov-Galerkin Crank-Nicolson Scheme for Parabolic Optimal Control Problems

- Mathematics, Computer ScienceSIAM J. Control. Optim.
- 2011

A finite element discretization of an optimal control problem governed by the heat equation is considered and a discrete solution is obtained for which error estimates of optimal order are proven.

Time Discretization of Parabolic Problems by the HP-Version of the Discontinuous Galerkin Finite Element Method

- Computer Science, MathematicsSIAM J. Numer. Anal.
- 2000

The discontinuous Galerkin finite element method (DGFEM) for the time discretization of parabolic problems is analyzed in the context of the hp-version of theGalerkin method and it is shown that the hp's spectral convergence gives spectral convergence in problems with smooth time dependence.

Multigrid methods for parabolic distributed optimal control problems

- Mathematics
- 2003

Multigrid schemes that solve parabolic distributed optimality systems discretized by finite differences are investigated. Accuracy properties of finite difference approximation are discussed and…

A Priori Error Estimates for Space-Time Finite Element Discretization of Parabolic Optimal Control Problems Part II: Problems with Control Constraints

- Computer ScienceSIAM J. Control. Optim.
- 2008

A priori error estimates are derived for space-time finite element discretizations of parabolic optimal control problems with pointwise inequality constraints on the control variable.

A Priori Error Estimates for Space-Time Finite Element Discretization of Parabolic Optimal Control Problems Part I: Problems Without Control Constraints

- Mathematics, Computer ScienceSIAM J. Control. Optim.
- 2008

A priori error analysis for Galerkin finite element discretizations of optimal control problems governed by linear parabolic equations and error estimates of optimal order with respect to both space and time discretization parameters are developed.

A space-time multigrid solver for distributed control of the time-dependent Navier-Stokes system

- 2008

We present a space-time hierarchical solution concept for optimization problems governed by the time-dependent Navier–Stokes system. Discretisation is carried out with finite elements in space and a…

Geometric Numerical Integration: Structure Preserving Algorithms for Ordinary Differential Equations

- Mathematics
- 2004

Numerical methods that preserve properties of Hamiltonian systems, reversible systems, differential equations on manifolds and problems with highly oscillatory solutions are the subject of this book.…

Adaptivity with Dynamic Meshes for Space-Time Finite Element Discretizations of Parabolic Equations

- Mathematics, Computer ScienceSIAM J. Sci. Comput.
- 2008

An error estimator and an adaptive algorithm for efficient solution of parabolic partial differential equations are developed and the space and time discretization errors are equilibrated, leading to an efficient method.

Strategies for time-dependent PDE control using an integrated modeling and simulation environment. Part one: problems without inequality constraints

- Computer Science
- 2007

This work presents the treatment of the coupled optimality system in the space-time cylinder, and the iterative approach by sequentially solving state and adjoint system and updating the controls, taking advantage of built-in discretization, solver and post-processing technologies.

The role of symplectic integrators in optimal control

- Mathematics
- 2009

SUMMARY For general optimal control problems, Pontryagin’s maximum principle gives necessary optimality conditions, which are in the form of a Hamiltonian differential equation. For its numerical…