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Null and approximate controllability for weakly blowing up semilinear heat equations. (English) Zbl 0970.93023
Summary: We consider the semilinear heat equation in a bounded domain of \(\mathbb{R}^d\), with control on a subdomain and homogeneous Dirichlet boundary conditions. We prove that the system is null-controllable at any time provided a globally defined and bounded trajectory exists and the nonlinear term \(f(y)\) is such that \(|f(s)|\) grows slower than \(|s|\log^{3/2}(1+|s|)\) as \(|s|\to\infty\). For instance, this condition is fulfilled by any function \(f\) growing at infinity like \(|s|\log^p(1+|s|)\) with \(1< p<3/2\) (in this case, in the absence of control, blow-up occurs). We also prove that, for some functions \(f\) that behave at infinity like \(|s|\log^p(1+|s|)\) with \(p> 2\), null controllability does not hold. The problem remains open when \(f\) behaves at infinity like \(|s|\log^p(1+|s|)\), with \(3/2\leq p\leq 2\). Results of the same kind are proved in the case of approximate controllability.

MSC:
93C20 Control/observation systems governed by partial differential equations
93B05 Controllability
93C10 Nonlinear systems in control theory
35B37 PDE in connection with control problems (MSC2000)
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