Linear Degenerate Parabolic Equations in Bounded Domains: Controllability and Observability

Abstract

In this paper we study controllability properties of linear degenerate parabolic equations. Due to degeneracy, classical null controllability results do not hold in general. Thus we investigate results of 'regional null controllability', showing that we can drive the solution to rest at time T on a subset of the space domain, contained in the set where the equation is nondegenerate.keywords: linear degenerate equations, regional null controllability, persistent regional null controllability.

Carleman estimates for parabolic equations with interior degeneracy and Neumann boundary conditions

“…11) for some positive constant C. To complete the proof it is sufficient to prove a similar inequality on the interval [0,λ 1 ]. To this aim, we follow a reflection procedure as before considering the problem (4.13).…”

Controllability results for a class of one-dimensional degenerate parabolic problems in nondivergence form

“…Then, the null controllability of (1.2) on the full interval [0, 1] is derived by standard arguments. Several results have also been obtained for a semilinear version of (1.2), see, for example, [1], [4] or [5].…”

Approximate controllability of a class of semilinear systems with boundary degeneracy

“…Particularly, it has been shown that the approximate controllability is a consequence of the null controllability for the control systems governed by nondegenerate linear parabolic equations [11,12]. However, the study on the controllability of degenerate parabolic equations just began several years ago and very few results have been known [1,[3][4][5][6][7][8]14,17,[20][21][22]24]. Among these, some authors have investigated the null controllability of one-dimensional linear and semilinear equations with boundary degeneracy.…”

“…As we know, the well-posed problems for parabolic equations with boundary degeneracy are different from the common ones [23,26]. In [1,[3][4][5][6][7][8]22], the degeneracy of Eq. (1.1) is divided into weak one and strong one according to the value of α, and different boundary conditions are proposed for the two cases.…”

“…(1.7) since there is a restriction a ∈ W 2,∞ (Q T ) in the classical theory [23]. Different from [1,[3][4][5][6][7][8]22], in the present paper we do not prescribe the Neumann boundary condition for Eq. (1.7) on the boundary where the equation is strongly degenerate, but describe this boundary condition via restricting the solution space just as done in [24].…”