New blow-up rates for fast controls of Schrodinger and heat equations

[摘要] We consider the null-controllability problem for the Schrodinger and heat equations with boundary control. We concentrate on short-time, or fast, controls. We improve recent estimates (see [L. Miller, Geometric bounds on the growth rate of null-control]ability cost for the heat equation in small time, J. Differential Equations 204 (2004) 202-226; L. Miller, How violent are fast controls for Schrodinger and plate vibrations?, Arch. Ration. Mech. Anal. 172 (2004) 429-456; L. Miller, Controllability cost of conservative systems: Resolvent condition and transmutation, J. Funct. Anal. 218 (2005) 425-444; L. Miller, The control transmutation method and the cost of fast controls, SIAM J. Control Optim. 45 (2006) 762-772]) on the norm of the operator associating to any initial state the minimal norm control driving the system to zero. Our main results concern the Schrodinger and heat equations in one space dimension. They yield new estimates concerning window problems for series of exponentials as described in [T.I. Seidman, The coefficient map for certain exponential sums, Nederl. Akad. Wetensch. Indag. Math. 48 (1986) 463-478] and in [T.I. Seidman, S.A. Avdonin, S.A. Ivanov, The window problem for series of complex exponentials, J. Fourier Anal. Appl. 6 (2000) 233-254]. These results are used, following [L. Miller, The control transmutation method and the cost of fast controls, SIAM J. Control Optim. 45 (2006) 762-772], to deal with the case of several space dimensions. (c) 2007 Elsevier Inc. All rights reserved.

[发布日期] 2007-12-01

[关键词] null-controllability;Schrodinger equation;heat equation;series of exponentials