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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.

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 |