Controllability of the heat equation with a control acting on a measurable set

The paper deals with the controllability of a heat equation. It is well-known that the heat equation yt − Δy = uχE in (0, T)×Ω with homogeneous Dirichlet boundary conditions is null controllable for any T > 0 and any open nonempty subset E of Ω. In this note, the author studies the case that E is an arbitrary measurable set with positive measure.

[1]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .

[2]  D. Russell Controllability and Stabilizability Theory for Linear Partial Differential Equations: Recent Progress and Open Questions , 1978 .

[3]  G. Lebeau,et al.  Contróle Exact De Léquation De La Chaleur , 1995 .

[4]  Gengsheng Wang,et al.  Linfinity-Null Controllability for the Heat Equation and Its Consequences for the Time Optimal Control Problem , 2008, SIAM J. Control. Optim..

[5]  Luc Miller,et al.  Geometric bounds on the growth rate of null-controllability cost for the heat equation in small time , 2003, math/0307158.

[6]  Xu Zhang,et al.  A Remark on Null Exact Controllability of the Heat Equation , 2001, SIAM J. Control. Optim..

[7]  Chi-Cheung Poon Qnique Continuation for , 1996 .

[8]  J. Lions Exact controllability, stabilization and perturbations for distributed systems , 1988 .

[9]  Enrique Zuazua,et al.  Null controllability of the heat equation as singular limit of the exact controllability of dissipative wave equations , 2000 .

[10]  Carlos Castro,et al.  Unique Continuation and Control for the Heat Equation from an Oscillating Lower Dimensional Manifold , 2004, SIAM J. Control. Optim..

[11]  Oleg Yu. Imanuvilov,et al.  Controllability of Evolution equations , 1996 .

[12]  Enrique Zuazua,et al.  Null‐Controllability of a System of Linear Thermoelasticity , 1998 .

[13]  Chi Y Lo Linear Partial Differential Equations , 2000 .

[14]  C. Bardos,et al.  Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary , 1992 .

[15]  O Yu Emanuilov,et al.  Controllability of parabolic equations , 1995 .