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On the controllability of parabolic systems with a nonlinear term involving the state and the gradient. (English) Zbl 1038.93041
This paper studies the controllability of a quasilinear parabolic equation in a bounded domain of $$\mathbb{R}^n$$ with Dirichlet boundary conditions. The controls are considered to be supported on a small open subset of the domain or on a small part of the boundary. The null and approximate controllability of the system at any time is proved if the nonlinear term $$f(y,\nabla y)$$ grows slower than $$| y|\log^{3/2}(1+| y|+|\nabla y|)+|\nabla y|\log^{1/2}(1+| y|+|\nabla y|)$$ at infinity. The proofs use global Carleman estimates, regularity results and fixed point theorems.

##### MSC:
 93C20 Control/observation systems governed by partial differential equations 93B05 Controllability 35K55 Nonlinear parabolic equations 35K05 Heat equation
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