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Remarques sur la controlabilité approchée. (French) Zbl 0752.93037
Control of distributed systems, Span.-Fr. Days, Málaga/Spain 1990, Grupo Anál. Mat. Apl. Univ. Málaga 3, 77-87 (1990).
[For the entire collection see Zbl 0723.00025.]
The paper deals with the approximate controllability of various distributed parameter systems as for instance $$y_ t+(y\nabla)y-\Delta y=-\nabla\pi+uX_ w$$, $$\text{div }y=0$$ on a domain $$\Omega$$ in $$\mathbb{R}^ 3$$, together with some given boundary conditions. In the above equation $$v=(v_ 1,v_ 2,v_ 3)$$ is a three-dimensional control defined on a subdomain $$w\in\Omega$$. By means of two independent approaches it is shown that a linear equation of the above form is approximately controllable, i.e. for any given function $$z$$ in some function space there exists a sequence of solutions of the distributed system which approaches the given $$z$$. Some remarks on the nonlinear case, including the Navier-Stokes equations complete the paper.

##### MSC:
 93C20 Control/observation systems governed by partial differential equations 93B05 Controllability