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HUM and the wave equation with variable coefficients. (English) Zbl 0848.93031
The following boundary value problem is considered $\begin{cases} Lw= w''- \sum^n_{j,i= 1} {\partial\over \partial x_i} \Biggl(a_{ij}(x, t) {\partial w\over \partial x_j}\Biggr)+\\ + \sum^n_{i= 1} b_i(x, t) {\partial w'\over \partial x_i}+ \sum^n_{i= 1} d_i(x, t) {\partial w\over \partial x_i}= 0\text{ in }Q,\\ w= v\quad \text{on} \quad \Sigma,\\w(0)= w^0,\;w'(0)= w^1,\end{cases}\tag{$$*$$}$ where $$Q$$ is the finite cylinder $$\Omega\times ]0, T[$$ with lateral boundary $$\Sigma= \Gamma\times ]0, T[$$ and $$\Omega$$ is a bounded domain of $$\mathbb{R}^n$$. For this problem the solution by transposition is defined and used.
The purpose is to establish, with some appropriate hypotheses, the following result concerning the boundary exact controllability: if $$T> T_0$$, where $$T_0$$ is a special time, then for every initial data $$\{w^0, w^1\}$$ belonging to $$L^2(\Omega)\times H^{- 1}(\Omega)$$, there exists a control $$v\in L^2(\Sigma(x^0))$$, where $$\Sigma(x^0)$$ is a special subset of $$\Sigma$$, such that the solution by transposition $$w$$ of problem $$(*)$$ satisfies: $$w(T)= 0$$, $$w'(T)= 0$$.
This is the main result of the paper, for which the author uses the Hilbert uniqueness method, which was introduced by J. L. Lions. At first, he establishes an inverse inequality, a direct inequality and proves the regularity of the solutions by transposition.
Finally, we note that in the particular case $$a_{ij}= \delta_{ij} a(t)$$, $$b_i= d_i= 0$$, one obtains the problem studied by J. L. Lions and J. Muñez Rivera, and in the case $$a_{ij}= \delta_{ij} a(x)$$, $$b_i= d_i= 0$$, the problem studied by E. Zuazua.

##### MSC:
 93C20 Control/observation systems governed by partial differential equations 93C99 Model systems in control theory 93B05 Controllability