2d Grushin-type equations: Minimal time and null controllable data

Abstract

We study internal null controllability for degenerate parabolic equations of Grushin-type G γ = ∂ 2 xx + |x| 2γ ∂ 2 yy (γ > 0), in the rectangle (x, y) ∈ = (−1, 1) × (0, 1). Previous works proved that null controllability holds for weak degeneracies (γ small), and fails for strong degeneracies (γ large). Moreover, in the transition regime and with strip shaped control domains, a positive minimal time is required.In this paper, we work with controls acting on two strips, symmetric with respect to the degeneracy. We give the explicit value of the minimal time and we characterize some initial data that can be steered to zero in time T (when the system is not null controllable): their regularity depends on the control domain and the time T .We also prove that, with a control that acts on one strip, touching the degeneracy line {x = 0}, then Grushin-type equations are null controllable in any time T > 0 and for any degeneracy γ > 0.Our approach is based on a precise study of the observability property for the one-dimensional heat equations satisfied by the Fourier coefficients in variable y. This precise study is done, through a transmutation process, on the resulting one-dimensional wave equations, by lateral propagation of energy method.

Non-null-controllability of the Grushin operator in 2D

“…We proved the non-null controllability of the Grushin equation on some special control domain, and if we combine our result with the previous ones [5,7], all of the following situations can happen depending on the control domain ω:…”

Precise estimates for biorthogonal families under asymptotic gap conditions

“…is certainly to be taken into account. • In 2D problems such as the Grushin equation (see [2,3]), where the solution is decomposed into Fourier modes, one has to give uniform bounds for a certain sequence of elliptic problems, the eigenvalues of which satisfy (1. 1) and (1.…”

Control of the Grushin equation: non-rectangular control region and minimal time

“…The null-controllability in the critical time T = a 2 /2 in Corollary 3.5 also remains an open problem. Since for ω := [(−b, −a) ∪ (a, b)] × (0, 1) with 0 < a < b ≤ 1 the minimal time is equal to a 2 /2 and the Grushin equation is not null-controllable in this time [9], we can conjecture that it is also the case in Corollary 3.5. Finally, we have a positive result for the generalized Grushin equation…”