Carleman Estimates, Observability Inequalities and Null Controllability for Interior Degenerate Non Smooth Parabolic Equations

Abstract

IntroductionIn the last recent years an increasing interest has been devoted to degenerate parabolic equations. Indeed, many problems coming from physics (boundary layer models in [13], models of Kolmogorov type in [7], . . . ), biology (Wright-Fisher models in [50] and Fleming-Viot models in [29]), and economics (Black-MertonScholes equations in [23]) are described by degenerate parabolic equations, whose linear prototype iswith the associated desired boundary conditions, whereIn this paper we concentrate on a special topic related to this field of research, i.e. Carleman estimates for the adjoint problem to (1.1). Indeed, they have so many applications that a large number of papers has been devoted to prove some forms of them and possibly some applications. For example, it is well known that they are a fundamental tool to prove observability inequalities, which lead to global null controllability results for (1.1) also in the non degenerate case: for all T > 0 and for all initial data u 0 ∈ L 2 ((0, T ) × (0, 1)) there is a suitable control h ∈ L 2 ((0, T ) × (0, 1)), supported in a subset ω of [0, 1], such that the solution u of (1.1) satisfies u(T, x) = 0 for all x ∈ [0, 1] (see, for instance, The common point of all the previous papers dealing with degenerate equations, is that the function a degenerates at the boundary of the domain. For example, as a, one can consider the double power functionwhere k and α are positive constants. For related systems of degenerate equations we refer to [1], [2] and [14]. However, the papers cited above deal with a function a that degenerates at the boundary of the spatial domain. To our best knowledge, [51] is the first paper treating the existence of a solution for the Cauchy problem associated to a parabolic equation which degenerates in the interior of the spatial domain, while degenerate parabolic problems modelling biological phenomena and related optimal control problems are later studied in [43] and [9]. Recently, in [32] the authors analyze in detail the degenerate operator A in the space L 2 (0, 1), with or without weight,

Null controllability for a degenerate population model in divergence form via Carleman estimates

“…for a strictly positive constant C. Proceeding, for example, as in [16] Thus, for a strictly positive constant C, Adding (4.9), (4.12) and (4.13), the thesis follows.…”

Carleman estimates and null controllability for a degenerate population model

“…Observe that, if k is nondegenerate, the spaces L 2 1 k (0, 1), H 1 1 k (0, 1) and H 2 1 k (0, 1) (or H 2 1 k ,x0 (0, 1)) coincide, respectively, with L 2 (0, 1), H 1 0 (0, 1) and H 2 (0, 1) ∩ H 1 0 (0, 1). Denoting by H 2 1 k (0, 1) the space H 2 1 k (0, 1) or H 2 1 k ,x0 (0, 1), we have, as in [13], [14] or [31], that the operator…”

“…for a strictly positive constant C. Imposing this condition on k x , for k(x) = x α , gives α ≥2. This necessary condition that ensures the well posedness of (1.1) makes it not null controllable (see [31] for the interior degeneracy). For this reason, in this paper as in [13], [14], [26], [27] or [31], we prove null controllability for (1.1) without deducing it by the previous results for the problem in divergence form.…”

“…If the function k in (1.1) degenerates in the interior of (0, 1) and the problem is in divergence form, related results can be founded in [10]. To our best knowledge, as written before, this is the first paper where the problem in nondivergence form is considered allowing the diffusion coefficient to degenerate at the boundary or in the interior of (0, 1) (when y is independent of a we refer, for example, to [31]). We underline that in [10] the authors assume that, if x 0 ∈ (0, 1) is the degenerate point, the function…”

Identification of a diffusion coefficient in strongly degenerate parabolic equations with interior degeneracy

“…If k is a constant or a strictly positive function, null controllability for (1.1) is studied, for example, in [3]. If k degenerates at the boundary or at an interior point of the domain and y is independent of a we refer, for example, to [2], [10], [11] and to [12], [13], [14] if µ is singular at the same point of k. Actually, [1] is the first paper where y depends on t, a and x and the dispersion coefficient k degenerates. In particular, in [1], k degenerates at the boundary of the domain (for example k(x) = x α , being x ∈ (0, 1) and α > 0).…”