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On a completion problem in the theory of distribute control of wave equations. (English) Zbl 0731.93056

Nonlinear partial differential equations and their applications, Lect. Coll. de France Semin., Paris/Fr. 1987-88, Vol. X, Pitman Res. Notes Math. Ser. 220, 241-271 (1991).
Summary: [For the entire collection see Zbl 0728.00013.]
In the theory of distributed controllability for the wave equation or systems of such equations in a bounded domain \(\Omega\) of \({\mathbb{R}}^ N\), a basic role is played by the completion F of the usual energy space relative to the \(L^ 2\) norm of the trace of solutions on (0,T)\(\times \Omega\), where \(T>0\) is sufficiently large and \(\omega\) is an open subdomain. It is established, for a single equation with Dirichlet homogeneous boundary conditions that in some important cases, \(F=L^ 2(\Omega)\times H^{-1}(\Omega),\) a result which implies the exact controllability of all initial states in \(H^ 1_ 0(\Omega)\times L^ 2(\Omega)\) by means of an \(L^ 2\) control supported by (0,T)\(\times \omega\). On the other hand counterexamples are given to establish that \(L^ 2(\Omega)\times H^{-1}(\Omega)\) is generally a strict subspace of F. Similar results are derived for some systems, and the corresponding controllability results are explained.

MSC:

93C20 Control/observation systems governed by partial differential equations

Citations:

Zbl 0728.00013