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A time-domain decomposition iterative method for the solution of distributed linear quadratic optimal control problems. (English) Zbl 1075.65091
The subject of this paper is the numerical solution of large-scale linear quadratic optimal control problems governed by parabolic partial differential equations. A time-domain decomposition is used and the problem is reformulated as a discrete-time optimal control (DTOC) one using a multiple shooting approach which is matrix free. It is based on the observation that the optimality conditions for the DTOC problem lead to a block tridiagonal linear system. Moreover the diagonal blocks are invertible and are related to the original linear quadratic optimal control problem restricted to smaller time-subintervals. A block Gauss-Seidel method is therefore used which is preconditioned by a Krylov-subspace method. Moreover some instantaneous control techniques can be interpreted as the application of one step of the preconditioned Gauss-Seidel method. Numerical experiments are presented such as Neumann control for the 1D heat equation and Dirichlet control for the 2D heat equation.

65K10 Numerical optimization and variational techniques
49M27 Decomposition methods
49J20 Existence theories for optimal control problems involving partial differential equations
49N10 Linear-quadratic optimal control problems
65F10 Iterative numerical methods for linear systems
65F35 Numerical computation of matrix norms, conditioning, scaling
Full Text: DOI
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