Note on the cost of the approximate controllability for the heat equation with potential

Niveau: Supérieur, Licence, Bac+2
J. Math. Anal. Appl. 295 (2004) 527–538 Note on the cost of the approximate controllability for the heat equation with potential K.-D. Phung Received 2 September 2003 Available online 28 May 2004 Submitted by R. Triggiani Abstract We prove the approximate controllability for the heat equation with potential with a cost of order ec/? when the target is in H 10 (?) with a precision in L 2(?) norm. Also a quantification estimate of the unique continuation for initial data in L2(?) of the heat equation with potential is established. ? 2004 Elsevier Inc. All rights reserved. 1. Introduction and main results Throughout this paper, ? is a bounded domain in Rn, n 1, with a boundary ∂? of class C2, ? is a non-empty open subset of ? and T > 0 is a real number. Further, we denote ? · ? ∞ the usual norm in L∞(? ? (0, T )) and we consider a = a(x, t) a function in L∞(? ? (0, T )). In this paper we study the following heat equation with a potential a in L∞(? ?(0, T )): ? ? ? ∂tu ? ∆u + au = f · 1|? in ? ? (0, T ), u = 0 on ∂? ? (0, T

• ∂tu ?


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• ud ?


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• heat equation


• quantitative estimate


• approximate controllability