Exact controllability for distributed systems. Some trends and some problems. (English) Zbl 0735.93006

Lions, J. L.

Applied and industrial mathematics, Proc. Symp., Venice/Italy 1989, Math. Appl., D. Reidel Publ. Co. 56, 59-84 (1991).

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[For the entire collection see Zbl 0707.00016.]
Motivated by questions in solid and fluid mechanics, the author extends his earlier work on controllability questions of distributed systems. After giving a self-contained review of the Hilbert Uniqueness Method [compare SIAM Rev. 30, No. 1, 1-68 (1988; Zbl 0644.49028)]he applies the method in several classes of examples. Considered are examples of control problems governed by the wave operator \({\partial^ 2 \over \partial t^ 2}-\Delta\) or the diffusion operator \({\partial \over \partial t}-\Delta\). The present methods for those problems are constructive and it is shown how those methods lead in a natural way to numerical algorithms.
In the last section several open problems are indicated which are related to the study of exact controllability for the Navier-Stokes equations.

Reviewer: J.Rosenthal (Notre Dame)

MSC:

93B05	Controllability
35A35	Theoretical approximation in context of PDEs
93C20	Control/observation systems governed by partial differential equations
35B37	PDE in connection with control problems (MSC2000)

Keywords:

Hilbert uniqueness method; numerical algorithms; Navier-Stokes equations

Citations:

Zbl 0707.00016; Zbl 0644.49028