CONTROLLABILITY OF 1-D COUPLED DEGENERATE PARABOLIC EQUATIONS

By Piermarco Cannarsa, Luz de Teresa. 2009

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This paper is devoted to the study of null controllability proper- ties for two systems of coupled one dimensional degenerate parabolic equations. The rst system consists of two forward equations, while the second one con- sists of one forward equation and one backward equation. Both systems are in cascade, that is, the solution of the rst equation acts as a control for the sec- ond equation and the control function only acts directly in the rst equation. We prove positive null controllability results when the control and coupling sets have nonempty intersection and 0 does not belong to the coupling set. 1. Statement of the problem In this paper we are concerned with the controllability properties of systems of coupled degenerate parabolic equations. We are going to consider two dierent kind of systems: the rst one consists of two forward equations and the second one, consists of one forward equation and one backward equation. More precisely, given two non empty open sets ! (0; 1) andO (0; 1) and a number 2 (0; 2), we consider the system of equations yt (x yx)x +c(t;x)y = +hI! in Q = (0;T ) (0; 1); y(t; 1) = 0 t2 (0;T ); y(t; 0) = 0 if 0 < 1; t2 (0;T ); (x yx)(t; 0) = 0 if 1

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Citations in the corpus

2d Grushin-type equations: Minimal time and null controllable data (2015, background): “In particular, null controllability still holds for weak degeneracies and fails for too strong ones, see [12, 13, 11, 1, 29, 10, 9, 8, 20, 34].”
Carleman estimates and observability inequalities for parabolic equations with interior degeneracy (2012, background): “ry of the domain. For example, as a, one can consider the double power function a(x) = xk(1 −x)α, x∈ [0,1], where kand αare positive constants. For related systems of degenerate equations we refer to [9] and [1]. Moreover, in [3] the authors study the null controllability of the parabolic equation a”
DEGENERATE PARABOLIC OPERATORS OF KOLMOGOROV TYPE WITH A GEOMETRIC CONTROL CONDITION (2015, background): “Several extensions of the above results are available in one space dimension, see [2, 49] for equations in divergence form, [20, 19] for operators in nondivergence form, and [18, 35] for cascade systems.”
Null controllability of Grushin-type operators in dimension two (2011, background): “Several extensions of the above results are available in one space dimension; see [1, 37] for equations in divergence form, [10, 9] for nondivergence form operators, and [8, 24] for cascade systems.”