## Sur le contrôle ponctuel de systèmes hyperboliques ou du type Petrowski.(French)Zbl 0549.49018

Sémin. Goulaouic-Schwartz 1979-1980, Équat. dériv. part., Exposé No. 20, 18 p. (1984).
The author considers the problem of minimizing $$\int (y(x,T)-z_ a(x))^ 2 dx+N\int v^ 2(t) dt$$ subject to $$v\in U\subset L^ 2$$ and $$y''-\Delta y=v(t)\delta (x-b)$$ in $$\Omega\times (\cdot,T)$$, $$\Omega\subset {\mathbb{R}}^ n$$, $$b\in\Omega$$, $$\delta$$ Dirac distribution, as well as some variants of this problem. For the case above he proves necessary optimality conditions in the form of an adjoint equation and a variational inequality. A major technical problem is the regularity of $$y(\cdot,T)$$. In an addendum it is shown that $$y(\cdot,T)\in L^ 2$$ if $$v\in L^ 2$$. To prove the necessary conditions, the first integral in the cost functional is regularized in order to obtain a regular adjoint, and a passage to the limit is carried out.
Reviewer: M.Brokate

### MSC:

 49K20 Optimality conditions for problems involving partial differential equations 35L05 Wave equation 93C20 Control/observation systems governed by partial differential equations 35B37 PDE in connection with control problems (MSC2000)
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