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Controllability for blowing up semilinear parabolic equations. (English. Abridged French version) Zbl 0952.93061
The paper studies controllability problems (exact and approximate) of a class of semilinear parabolic systems in a bounded domain $\Omega\subset \bbfR^d$ described by $$y_t- \Delta y+f(y)= v(x,t)1_\omega, \quad\text{in }\Omega\times (0,T),$$ $$y=0,\quad \text{on }\partial\Omega \times (0,T), \qquad y(x,\cdot)= y_0\in L^2(\Omega), \quad\text{in }\Omega.$$ Here, $y(x,t)$ denotes the state; $f:\bbfR\to \bbfR$ locally Lipschitz continuous; $1_\omega$ the characteristic function of the nonempty subset $\omega\subset \Omega$; and $v(x,t)\in L^\infty (\omega \times (0,T))$ the control. Exact controllability of the system is defined as having the following property: Given ${}^\forall y_0$ and ${}^\forall y^* (\cdot,t)$ (corresponding to $y_0^*$ and $v^*(x,t)$), there exists a control $v(x,t)$ such that $y(x,T)= y^* (x,T)$. If $f'(s)$ is of polynomial growth order and $f(s)$ is bounded from above by $|s|\log^{3/2} (1+|s|)$ at infinity, the exact controllability as well as the approximate controllability are guaranteed. Also the existence of $f$ behaving like $|s|\log^p (1+|s|)$ at infinity with $p>2$ such that the system fails to be exactly and approximately controllable is shown.
Reviewer: T.Nambu (Kobe)

93C20Control systems governed by PDE
35K20Second order parabolic equations, initial boundary value problems
35B37PDE in connection with control problems (MSC2000)
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