Regional controllability of semilinear degenerate parabolic equations in bounded domains

Abstract

In this paper we study controllability properties of semilinear 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.

Null controllability of degenerate parabolic equations of Grushin and Kolmogorov type

“…Several extensions of the above results are available in one space dimension, see [1,37] for equations in divergence form, [11,10] for nondivergence form operators, and [9,25] for cascade systems. Fewer results are available for multidimensional problems, mainly in the case of two dimensional parabolic operators which simply degenerate in the normal direction to the boundary of the space domain, see [15].…”

Linear Degenerate Parabolic Equations in Bounded Domains: Controllability and Observability

“…Using the fact that o is nondegenerate on (a, 1) and a classical result known for linear nondegenerate parabolic equations in bounded domains (see for example [10,8]), we will now give a direct proof of regional null controllability for the linear degenerate problem (9). This result was proved in [6] and in [4] in the case 6 = 0 and b j^ 0, respectively, and a G C^ [0,1]. In particular, in [6], the proof was based on suitable regional observability inequalities which constituted the major technical part of the paper.…”

“…In [6] and in [4], problem (1) is considered, under different assumptions on a, in the special case & = 0 and 5^0, c{t, x)u = f{t, x, u), respectively. In both cases the following notion of regional null controllability has been developed.…”

“…Such an inequality is obtained by an appropriate use of cut-off functions and Carleman estimates (see, e.g., [1], [9], or [12]) for nondegenerate parabolic operators. In [4] and in the present paper the main feature of our approach is that we use a new method of proof. Indeed, instead of deducing null controllability from observability, we derive the result directly, using cut-off functions and the fact that equation of (1) is null controllable when x varies in any subinterval / CC (0,1], where a in nondegenerate.…”

“…More precisely, if u(T) = 0 in (0,1) and if we stop controlling the system at time T, then for all t > T, u{t) = 0 in (0,1). On the contrary, regional null controllability is a weaker property: due to the uncontrolled part on (0, a + S), (3) is no more preserved with time if we stop controlling at time T. Thus, it is important to improve the previous result, as shown in [6] or [4], proving that the solution can be forced to vanish identically on {a + 5,1) during a given time interval (T, T'), i.e. that the solution is persistent regional null controllable.…”