Doubova, A.; Fernández-Cara, E.; González-Burgos, M.; Zuazua, E. On the controllability of parabolic systems with a nonlinear term involving the state and the gradient. (English) Zbl 1038.93041 SIAM J. Control Optimization 41, No. 3, 798-819 (2002). 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. Reviewer: Sebastian Aniţa (Iaşi) Cited in 93 Documents MSC: 93C20 Control/observation systems governed by partial differential equations 93B05 Controllability 35K55 Nonlinear parabolic equations 35K05 Heat equation Keywords:distributed control; controllability; nonlinear parabolic equations; nonlinear gradient terms; Dirichlet boundary conditions; Carleman estimates; fixed point theorems PDFBibTeX XMLCite \textit{A. Doubova} et al., SIAM J. Control Optim. 41, No. 3, 798--819 (2002; Zbl 1038.93041) Full Text: DOI