Applied Mathematics E-Notes, 10(2010), 36-39 c  ISSN 1607-2510 Available free at mirror sites of http://www.math.nthu.edu.tw/∼amen/ On The Controllability Under Constraints On The Control For Hyperbolic Equations ∗ Belhassen Dehman † , Abdennebi Omrane ‡ Received 19 March 2009 Abstract Null-controllability for the wave equation is studied in the context of dis- tributed linear constraints on the control. 1 Introduction Let Ω be a bounded open subset of R d , d ∈ N * with boundary Γ of class C 2 . For T> 0, we set Q = (0,T ) × Ω, Σ = (0,T ) × Γ, and we consider the wave system       a0 q = g in Q, q = 0 on Σ,  q(T ), ∂q ∂t (T )  = (q 0 ,q 1 ) in Ω, (1) where  a0 = ∂ 2 ∂t 2 − d  j =1 ∂ 2 ∂x 2 j + a 0 I, (2) is the d’Alembertian with potential a 0 . Here a 0 lies in L ∞ (Q) and is real valued. It is well known that given g ∈ L 1 ([0,T ]; L 2 (Ω)) and (q 0 ,q 1 ) ∈ H 1 0 (Ω) × L 2 (Ω), the problem (1) admits a unique solution q in the space C([0,T ]; H 1 0 (Ω)) ∩C 1 ([0,T ]; L 2 (Ω)). Here, we state the problem of exact controllability for solutions of system (1). Let ω be an open subset of Ω; denote by Q ω = (0,T ) × ω the interior cylinder and χ ω its characteristic function. Given (q 0 ,q 1 ) ∈ H 1 0 (Ω) × L 2 (Ω), the goal is to find a source v in L 2 (Q ω ) such that the unique solution q of (1) with g = χ ω v satisfies q(0) = 0 and ∂q ∂t (0) = 0 in Ω. (3) * Mathematics Subject Classifications: 35K05, 37N30, 49J20, 65P99, 80M20, 93B05. † Universit´ e de Tunis El Manar, Facult´ e des Sciences de Tunis, D´ epartement de Math´ ematiques, Campus Universitaire, 1060 Tunis, email : belhassen.dehman@fst.rnu.tn ‡ Universit´ e des Antilles et de la Guyane, UFR Sciences Exactes et Naturelles, D´ epartement de Math´ ematiques et Informatique, Campus de Fouillole, 97159 Pointe ` a Pitre, Guadeloupe (FWI), email : aomrane@univ-ag.fr 36