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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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