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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 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|. 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 3/2p2. Results of the same kind are proved in the case of approximate controllability.

MSC:
93C20Control systems governed by PDE
93B05Controllability
93C10Nonlinear control systems
35B37PDE in connection with control problems (MSC2000)