Manuscript submitted to Website: http://AIMsciences.org AIMS’ Journals Volume X, Number 0X, XX 200X pp. X–XX NULL-EXACT CONTROLLABILITY OF A SEMILINEAR CASCADE SYSTEM OF PARABOLIC-HYPERBOLIC EQUATIONS Enrique Fern´ andez-Cara, Manuel Gonz´ alez-Burgos Dpto. E.D.A.N., Universidad de Sevilla Aptdo. 1106, 41080 Sevilla, Spain Luz de Teresa Instituto de Matem´aticas, UNAM Circuito Exterior, C.U. 04510 D.F., M´ exico (Communicated by Martino Bardi) Abstract. This paper is concerned with the null-exact controllability of a cascade system formed by a semilinear heat and a semilinear wave equation in a cylinder Ω×(0,T ). More precisely, we intend to drive the solution of the heat equation (resp. the wave equation) exactly to zero (resp. exactly to a prescribed but arbitrary final state). The control acts only on the heat equation and is supported by a set of the form ω × (0,T ), where ω ⊂ Ω. In the wave equation, the restriction of the solution to the heat equation to another set O× (0,T ) appears. The nonlinear terms are assumed to be globally Lipschitz-continuous. In the main result in this paper, we show that, under appropriate assumptions on T , ω and O, the equations are simultaneously controllable. 1. Introduction. The main result. Let Ω ⊂ IR N be a bounded domain of class C 2 (N ≥ 1). Let ω and O be two nonempty open subsets of Ω. Let T> 0 and set Q =Ω × (0,T ) and Σ = ∂ Ω × (0,T ). We will consider the following cascade system:    y t − Δy + f 1 (x, t; y,q)= h ω in Q, y =0 on Σ, y(x, 0) = y 0 (x) in Ω, (1)    q tt − Δq + f 2 (x, t; q)= y 1 O in Q, q =0 on Σ, q(x, 0) = q 0 (x), q t (x, 0) = q 1 (x) in Ω. (2) Here, y 0 and (q 0 ,q 1 ) are given, h ω is a control with support in ω × [0,T ], 1 O is the characteristic function of the set O and f 1 and f 2 are appropriate Carath´ eodory functions (measurable in (x, t) and continuous in the other variables). We address the following question: 2000 Mathematics Subject Classification. 35M20, 93B05, 93B07. Key words and phrases. Semilinear systems, parabolic-hyperbolic equations, controllability, observability inequalities. Supported by grant BFM2003-06446 of the D.G.E.S. (Spain) and by project IN102799 of D.G.A.P.A. (Mexico). 1