Virtual and effective control for distributed systems and decomposition of everything

Abstract

For the numerical approximation of the solution of boundary value problems (BVP), decomposition techniques are very important, in particular in view of parallel computations. The same is true, in principle, for optimal control of distributed systems, i.e., systems governed (modelled) by partial differential equations (PDE). Very many techniques have been studied for the approximation of BVP, such as DDM (domain decomposition method), decomposition of operators (splitting up, for instance). In contrast, not so many techniques of decomposition have been used in control problems for distributed systems, as pointed out in the contributions of Benamou [1], Benamou and Després [2], and Lagnese and Leugering [11]. However, it has been observed by Pironneau and Lions [24], [26] that by using so-calledvirtual controls, systematic DDM can be obtained, and that problems of optimal control and analysis of BVP can be considered in the same framework. We then deal with virtual control problems for BVP, virtual and effective control problems for the control of PDE (cf. Pironneau and Lions [25]). Using the idea of virtual control in other guises, Glowinski, Lions and Pironneau [9] have shown how to obtain new decomposition methods for the energy spaces (cf. Section 3), and Pironneau and Lions [27] have shown how to obtain systematically operator decomposition in BVP. In the present paper, we show (without assuming prior knowledge) how to apply the virtual control ideas in several different guises to the “decomposition of everything” for PDE of evolution and for their control. In this way, one can decompose the geometrical domain, the energy space and the operator. This is briefly presented in Sections 2, 3 and 4.

We show in Section 5 how one can simultaneously apply two of the decomposition techniques and also indicate briefly how virtual control ideas can be used in case of bilinear control.

The content of this paper is presented here for the first time. It is part of a systematic program which is in progress, developed with several colleagues. I wish to thank particularly F. Hecht, R. Glowinski, J. Periaux, O. Pironneau, H. Q. Chen and T. W. Pan.

Of course we do not claim by any means that the methods based on “virtual control” are “better” than the many decomposition techniques already available (no attempt has been made to compile a Bibliography on these topics). Numerical works in progress show that the methods are “not bad”, but no serious benchmarking has been made yet. Possible interest lies in the fact thatone technique, with some variants, seems to lead to all possible Decompositions of Everything.