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Zbl 1036.49033
Lagnese, J. E.; Leugering, G.
Domain decomposition in optimal control problems for distributed parameter systems. (English)
[J] Jahresber. Dtsch. Math.-Ver. 104, No. 3, 91-124 (2002). ISSN 0012-0456

Summary: In this survey article we consider methods of domain decomposition for static and more importantly for dynamic partial differential equations on complicated domains, like graph-type or heterogenous domains, which are subjected to controls. In this context one is either interested in solving the corresponding optimality conditions, or in establishing gradient or even higher-order informations. Therefore, one is led to coupled `direct' and `adjoint' equations. Instead of just using domain decomposition methods as a convenient computational tool for solving the system equations, we consider mainly methods that can be applied to the corresponding optimality systems. Moreover, we emphazise that methods from optimal control theory of partial differential equations are important in developing domain decomposition methods by the way of `virtual controls'. We also point out that decomposition schemes with optimized transmission properties are very useful in the context of real-time large-scale optimal control problems. For that reason, we believe that the material presented here is of general interest, and, therefore, we kept the presentation in a general context.

MSC 2000:: *49M27 Decomposition methods
49J20 Optimal control problems with PDE (existence)
49-02 Research monographs (calculus of variations)
49M20 Methods of relaxation type
35L20 Second order hyperbolic equations, boundary value problems
58J45 Hyperbolic equations
65M55 Multigrid methods; domain decomposition (IVP of PDE)
65N55 Multigrid methods; domain decomposition (BVP of PDE)

Keywords: distributed parameter systems; domain decomposition; dynamic partial differential equations; optimality conditions; real-time large-scale optimal control

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