## Finite dimensional null controllability for the semilinear heat equation.(English)Zbl 0872.93014

Summary: We study a finite-dimensional version of the null controllability problem for semilinear heat equations in bounded domains $$\Omega$$ of $$\mathbb{R}^n$$ with Dirichlet boundary conditions. The control acts on any open and non-empty subset of $$\Omega$$. The question under consideration is the following: given an initial state, a control time $$t=T$$ and a finite-dimensional subspace $$E$$ of $$L^2(\Omega)$$, is there a control such that the orthogonal projection over $$E$$ of the solution at time $$t=T$$ vanishes? Under rather natural growth conditions on the nonlinearity we show that this can be done provided the initial data is sufficiently small. The method of proof combines the implicit function theorem with a constructive method to solve this finite controllability problem in the linear case. We then consider nonlinearities with the “good sign”. Using the decay properties of solutions and the fact that the problem is solvable for small data, we show that the problem is solvable for large data too. When analyzing the linear heat equation we will prove that with one sole control one can obtain simultaneously approximate controllability and exact reachability of a finite number of constraints. The same result holds when the nonlinearity is globally Lipschitz.

### MSC:

 93B05 Controllability 93C10 Nonlinear systems in control theory 35K05 Heat equation 93C20 Control/observation systems governed by partial differential equations
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### References:

 [1] Brezis, H., Analyse fonctionnelle, () · Zbl 0511.46001 [2] Cazenave, T.; Haraux, A., Introduction aux problèmes d’évolution semi-linéaires, () · Zbl 0786.35070 [3] Fabre, C.; Puel, J.P.; Zuazua, E., Approximate controllability of the semilinear heat equation, (), 31-61 · Zbl 0818.93032 [4] Fabre, C.; Puel, J.P.; Zuazua, E., Contrôlabilité approchée de l’équation de la chaleur linéaire avec des contrôles de norme L∞ minimale, C. R. acad. sci. Paris, 316, 679-684, (1993) · Zbl 0799.35094 [5] Fursikov, A.; Imanuvilov, O.Yu., Controllability of evolution equations, () · Zbl 0862.49004 [6] Henry, J., Étude de la contrôlabilité de certaines équations paraboliques, () [7] Lebeau, G.; Robbiano, L., Contrôle exact de l’équation de la chaleur, Comm. PDE, 20, 335-356, (1995) · Zbl 0819.35071 [8] Lin Guo, Y.-J.; Littman, W., Null boundary controllability for semilinear heat equations, Appl. math. optim., 32, 281-316, (1995) · Zbl 0835.35075 [9] Lions, J.-L., Contrôlabilité exacte, stabilisation et perturbations de systèmes distribués, () · Zbl 0653.93002 [10] Lions, J.L.; Malgrange, B., Sur l’unicité rétrograde dans LES problèmes mixtes paraboliques, Math. scand., 8, 277-286, (1960) · Zbl 0126.12202 [11] Mizohata, S., Unicité du prolongement des solutions pour quelques opérateurs différentiels paraboliques, Mem. coll. sci. univ. Kyoto, 31, 3, 219-239, (1958), Ser. A · Zbl 0087.09303 [12] Rubio, J.E., The global control of nonlinear diffusion equations, SIAM J. cont. optim., 33, 1, 308-322, (1995) · Zbl 0820.93014 [13] Saut, J.C.; Scheurer, B., Unique continuation for some evolution equations, J. diff. equations, 66, 1, (1987) · Zbl 0631.35044 [14] Zuazua, E., Exact boundary controllability for the semilinear wave equation, (), 357-391 [15] Zuazua, E., Exact controllability for the semilinear wave equation, J. math. pures et appl., 69, 1, 33-55, (1990)
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