Controllability of an elliptic equation and its finite difference approximation by the shape of the domain
The authors consider the controllability problem (rarely investigated in the literature) for an elliptic PDE with respect to the shape of the domain where the equation holds. The authors prove the approximate controllability for the linearized system and give a detailed discussion for the controllability of the discrete problem. They point out main difficulties that e.g. do not allow to say anything about convergence of shapes when the discretization step tends to zero. An interesting introduction to problems of controllability for PDE's can be found in \textit{J. L. Lions}'s books [Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués. Tome 1: Contrôlabilité exacte (1988; Zbl 0653.93002); Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués. Tome 2: Perturbations (1988; Zbl 0653.93003)] and in the papers by \textit{E. Zuazua} [``Some problems and results on the controllability of partial differential equations'', Prog. Math. 169, 276--311 (1998; Zbl 1126.35305); Discrete Contin. Dyn. Syst. 8, No. 2, 469--513 (2002; Zbl 1005.35019)].
scientific article; zbMATH DE number 1967716