Controllability of 1-d Coupled Degenerate Parabolic Equations

This article is devoted to the study of null controllability properties for two systems of coupled one dimensional degenerate parabolic equations. The first system consists of two forward equations, while the second one consists of one forward equation and one backward equation. Both systems are in cascade, that is, the solution of the first equation acts as a control for the second equation and the control function only acts directly in the first 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 different kind of systems: the first 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 ω ⊂ (0, 1) and O ⊂ (0, 1) and a number α ∈ [0, 2), we consider the system of equations yt − (xyx)x + c(t, x)y = ξ + hIω in Q = (0, T )× (0, 1) , y(t, 1) = 0 t ∈ (0, T ) , y(t, 0) = 0 if 0 ≤ α < 1, t ∈ (0, T ) , (xyx)(t, 0) = 0 if 1 ≤ α < 2, t ∈ (0, T ) , y(0, ·) = y in (0, 1) , (1.1) and ut − (xux)x + d(t, x)u = yIO in Q , u(t, 1) = 0 t ∈ (0, T ) , u(t, 0) = 0 if 0 ≤ α < 1, t ∈ (0, T ) , (xux)(t, 0) = 0 if 1 ≤ α < 2, t ∈ (0, T ) , u(0, ·) = u in (0, 1) , (1.2) 2000 Mathematics Subject Classification. 35K65, 93C20.