Domain decomposition in optimal control problems for distributed parameter systems.

*(English)*Zbl 1036.49033Summary: 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 heterogeneous 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:

49M27 | Decomposition methods |

49J20 | Existence theories for optimal control problems involving partial differential equations |

49-02 | Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control |

49M20 | Numerical methods of relaxation type |

35L20 | Initial-boundary value problems for second-order hyperbolic equations |

58J45 | Hyperbolic equations on manifolds |

65M55 | Multigrid methods; domain decomposition for initial value and initial-boundary value problems involving PDEs |

65N55 | Multigrid methods; domain decomposition for boundary value problems involving PDEs |