## 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

### Keywords:

distributed controllability; wave equation

Zbl 0728.00013