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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