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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.

MSC:
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
Software:
BNDSCO
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References:
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