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Persistent regional null controllability for a class of degenerate parabolic equations. (English) Zbl 1063.35092

It is considered the parabolic problem \[ u_t - (a(x)u_x)_x + c(x)u = f(t,x)\chi_{(\alpha,\beta)}(x),\quad (t,x)\in (0,T)\times(0,1) \]
\[ \lim_{x\rightarrow 0}a(x)u_x(t,x) = 0,\quad t\in (0,T) \]
\[ u(t,1) = 0,\;t\in (0,T);\quad u(0,x) = u_0(x),\;x\in(0,1) \] with \(u_0\in L^2(0,1)\), \(f\in L^2((0,T)\times(0,1))\). Because of the second condition the system is considered degenerate from the point of view of the null controllability. Regional and persistent regional null controllability are defined and corresponding criteria (controllability inequalities) proved. Further, the case of the unbounded domain \((0,\infty)\) is considered for the heat equation and controllability obtained for both regular and degenerate case (the Crocco equation).

MSC:

35K65 Degenerate parabolic equations
93B05 Controllability
93C20 Control/observation systems governed by partial differential equations
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