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Controllability for blowing up semilinear parabolic equations. (English) Zbl 0952.93061

The paper studies controllability problems (exact and approximate) of a class of semilinear parabolic systems in a bounded domain Ω d described by

y t -Δy+f(y)=v(x,t)1 ω ,inΩ×(0,T),
y=0,onΩ×(0,T),y(x,·)=y 0 L 2 (Ω),inΩ·

Here, y(x,t) denotes the state; f: locally Lipschitz continuous; 1 ω the characteristic function of the nonempty subset ωΩ; and v(x,t)L (ω×(0,T)) the control. Exact controllability of the system is defined as having the following property: Given y 0 and y * (·,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)
MSC:
93C20Control systems governed by PDE
35K20Second order parabolic equations, initial boundary value problems
93B05Controllability
35B37PDE in connection with control problems (MSC2000)