DEGENERATE PARABOLIC OPERATORS OF KOLMOGOROV TYPE WITH A GEOMETRIC CONTROL CONDITION

作者：Beauchard Karine^*; Helffer Bernard; Henry Raphael; Robbiano Luc

来源：ESAIM: Control, Optimisation and Calculus of Variations , 2015, 21(2): 487-512.

DOI：10.1051/cocv/2014035

• Null controllability
• degenerate parabolic equation
• hypoelliptic operator
• geometric control condition

We consider Kolmogorov-type equations on a rectangle domain (x, v) is an element of Omega = T x (-1, 1), that combine diffusion in variable v and transport in variable x at speed v(gamma), gamma is an element of N*, with Dirichlet boundary conditions in v. We study the null controllability of this equation with a distributed control as source term, localized on a subset omega of Omega. When the control acts on a horizontal strip omega = T x (a, b) with 0 < a < b < 1, then the system is null controllable in any time T > 0 when gamma = 1, and only in large time T > T-min > 0 when gamma = 2 (see [K. Beauchard, Math. Control Signals Syst. 26 (2014) 145-176]). In this article, we prove that, when gamma > 3, the system is not null controllable (whatever T is) in this configuration. This is due to the diffusion weakening produced by the first order term. When the control acts on a vertical strip omega = omega(1)x(-1, 1) with (omega) over bar (1) subset of T, we investigate the null controllability on a toy model, where (partial derivative(x), x is an element of T) is replaced by (i(-Delta)(1/2), x is an element of Omega(1)), and Omega(1) is an open subset of R-N. As the original system, this toy model satisfies the controllability properties listed above. We prove that, for gamma = 1, 2 and for appropriate domains (Omega(1), omega(1)), then null controllability does not hold (whatever T > 0 is), when the control acts on a vertical strip omega = omega(1)x(-1, 1) with (omega) over bar (1) subset of Omega(1). Thus, a geometric control condition is required for the null controllability of this toy model. This indicates that a geometric control condition may be necessary for the original model too.

• 出版日期2015-6