NULL-EXACT CONTROLLABILITY OF A SEMILINEAR CASCADE SYSTEM OF PARABOLIC-HYPERBOLIC EQUATIONS

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.

We address the following question: again completed with initial and boundary conditions for y and q.
• Cascade heat-wave (or Navier-Stokes-Lamé) systems in different domains. For instance, if Ω = G × (0, L) where G ⊂ IR 2 is a bounded regular domain, we may consider the system where ω ⊂ Ω and O ⊂ G.
Our aim is to understand and explain the control mechanisms for (1)- (2). We believe that this will be useful to deal with similar controllability questions for the previous systems.
Observe that, in (3), we are concerned with a null-exact controllability problem. However, the control acts in the equation satisfied by q indirectly through the variable y and, accordingly, the question under consideration is more intricate than in the standard situation of the exact controllability problem of the classical wave equation. In order to deal with the controllability properties of system (1)- (2), an additional assumption must be imposed on ω ∩ O; see (9). In particular, this assumption implies that ω ∩ O = ∅.
It will be convenient to introduce several functions, sets and spaces. Let where ν(x) denotes the unit outwards normal vector to ∂Ω at x, Let δ > 0 be given. We will consider the sets Then, we will consider the following scalar product in H −1 (Ω): Notice that the norm · H −1 induced by (· , ·) H −1 is also the norm associated to · H 1 0 by duality. We will assume that the function f 1 = f 1 (x, t; s, r) is globally Lipschitz in the variable (s, r) and satisfies |f 1 (x, t; s, r)| ≤ C|s| ∀(x, t; s, r) ∈ Q × IR 2 (6) for some C > 0. We will also assume that the function f 2 = f 2 (x, t; r) satisfies and is globally Lipschitz in the variable r: Our main result is the following: Theorem 1. Assume that, for some x 0 ∈ IR N and some δ > 0, there exists a set of the form G δ (x 0 ) satisfying Assume that T > 2R(x 0 ) and f 1 and f 2 are globally Lipschitz-continuous and satisfy (6)- (8). Then, for any y 0 ∈ H −1 (Ω), (q 0 , q 1 ) ∈ H 1 0 (Ω) × L 2 (Ω) and (r 0 , r 1 ) ∈ Remark 1. In order to prove theorem 1, a fixed point argument will be performed. In particular, we will see that the couple (y, q) satisfies y ∈ C 0 ([0, T ]; H −1 (Ω)), and q ∈ C 0 ([0, T ]; H 1 0 (Ω)) ∩ C 1 ([0, T ]; L 2 (Ω)) and solves the system (11)-(12) for some appropriate h ω and a, b ∈ L ∞ (Q) (which depend on y and q). We will see that y is a solution by transposition of (1) (for the definition of solution by transposition, see subsection 2.1) and the equalities in (3) are satisfied in H −1 (Ω), H 1 0 (Ω) and L 2 (Ω), respectively.
Remark 2. It may seem that the regularity of h ω is not satisfactory. However, it is clear that, in order to get the exact controllability in H 1 0 (Ω) × L 2 (Ω) of (2), y1 O must not be better than L 2 (Q) and consequently h ω must not be better than L 2 (0, T ; D(−∆) ). Accordingly, the previous assertion is reasonable.

Remark 3.
In the particular case f 1 ≡ f 2 ≡ 0, the controllability properties of the cascade system (1)-(2) were analyzed in [4]. There, a result very similar to theorem 1 was proved.
Remark 4. It is well known that, under the assumptions G δ (x 0 ) ⊂ ω and T > 2R(x 0 ), the classical wave equation is exactly controllable with L 2 controls supported by ω × [0, T ]. In other words, for any satisfies This is a consequence of an observability estimate that will be recalled below, see [10]. On the other hand, (10) is not exactly controllable in general. The precise necessary and sufficient conditions on ω and T that guarantee exact controllability are given in [2] (more details will be recalled in Section 4). Therefore, the hypotheses on ω, Ω and T in theorem 1 are, at first sight, appropriate.
The proof of theorem 1 is divided in two parts. We will first prove the null-exact controllability of similar cascade linear systems with potentials a, b ∈ L ∞ (Q) and source g ∈ L 2 (Q): in Ω, in Ω.
More precisely, the following result will be established: such that the corresponding solutions (y, q) to (11)-(12) satisfy (3). In (13), whereq is the solution of the uncontrolled system   In a second step, using a fixed point argument we will obtain the desired controllability result for the nonlinear system. Remark 5. The lack of regularity of the control provided by theorem 2 introduces some technical difficulties in our analysis. To be precise, the fixed point argument will be formulated in L 2 (0, T ; H −1 (Ω) × L 2 (Ω)) and consequently, in order to define a set-valued mapping we need to apply a regularization process. The fixed point argument does not lead directly to the solution. To obtain our result, we still have to absorb the non regular part of the limit in the control (see section 3).
The proof of theorem 2 is based on the existence of a positive constant C = C( a ∞ , b ∞ , ω, O, Ω, T ) such that the observability inequality (z, p, p t )(·, 0) 2 holds true for any solution of the adjoint system   in Ω (17) associated to final data z 0 ∈ H 1 0 (Ω) and (p 0 , p 1 ) ∈ L 2 (Ω) × H −1 (Ω). In (15), ρ ω = ρ ω (x) is an appropriate regular approximation of the characteristic function 1 ω . Among other things, we will assume that ρ ω ∈ C 1 (Ω), ρ ω (x) = 1 for all x ∈ ω ⊂ ω and ρ ω (x) = 0 for all x ∈ ω. As we shall see in Section 4, we must use a smooth approximation of the characteristic function of the set ω in order to guarantee the regularity of the control.
The rest of the paper is organized as follows. In Section 2, we will recall some existence and regularity results for the solutions of the wave and the heat equations and then we will prove theorem 2, i.e. the null-exact controllability of the linear system (11)- (12), assuming that the observability inequality (15) holds true. Section 3 is devoted to prove theorem 1. Finally, Section 4 is devoted to prove (15). This relies mainly on an observability estimate for the solutions of (16), i.e. the exact controllability of (10) with controls in L 2 (ω × (0, T )) and a (global) Carleman estimate for the heat equation taken from [6].

2.
Preliminaries and the linear case.
2.1. Preliminaries. We begin this Section by recalling some existence and regularity results for wave and heat equations. For more complete treatises, see for instance [1] and [8].
In the sequel, C, C 1 , C 2 ,. . . stand for generic positive constants, depending on Ω, T , ω, O and maybe the coefficients of the considered equations. We will sometimes (but not always) indicate this dependence explicitly.
For any Banach space X considered below, the usual norm in X will be denoted by · X . In the particular cases of L 2 (Ω), H 1 0 (Ω), etc., the corresponding norms and scalar products will be respectively denoted by where C only depends on Ω. We can also solve (18) when the data In the sequel, for any couple of Banach spaces X and Y satisfying X → Y with a continuous embedding, we will use the following notation: It is then well known that, for any in Ω, satisfies w ∈ W (0, T ; H 1 0 (Ω), H −1 (Ω)) and consequently w ∈ C 0 ([0, T ]; L 2 (Ω)) and the estimate Let us assume that k ∈ L 2 (Q) and w 0 ∈ H 1 0 (Ω). Then the solution satisfies Of course, in (21) and (22) the constants C depend on Ω.
In this paper, we will also have to solve systems of the form (11) with h ∈ L 2 (0, T ; D(−∆) ) and y 0 ∈ H −1 (Ω). The appropriate concept is the solution by transposition.
Thus, assume that h is given in L 2 (0, T ; D(−∆) ), a ∈ L ∞ (Q) and y 0 ∈ H −1 (Ω). By definition, the solution by transposition of is the unique function y ∈ L 2 (Q) satisfying Here, for each g ∈ L 2 (Q), we have denoted by ϕ g the solution to the corresponding In (24) and also in the sequel, · , · and · , · stand for the usual duality pairings associated to D(−∆) and D(−∆) and H −1 (Ω) and H 1 0 (Ω), respectively. Notice that the solution of (25) satisfies . Hence (24) makes sense, y is well defined and one has On the other hand, it is clear that y solves the partial differential equation in (23) in the distributional sense, i.e.
Let G δ (x 0 ) and R(x 0 ) be as in the previous Section (see (4) and (5)). Let us introduce two positive parameters κ, κ 1 ∈ (0, δ) with κ < κ 1 and let ω 0 , ω 1 be the following open sets: Finally, let ρ ω be a function satisfying As mentioned above, the proof of theorem 2 relies on an observability inequality for the adjoint of the linear cascade system (11)- (12). This is given in the following result: The proof of this result is given in Section 4.
Indeed, let us assume that the result is true in this case and let (y 0 , q 0 , given. Let us introduce the solutionq of (14) and let us set (q 0 ,q 1 ) = (q,q t )(·, 0). Then, by (13) holds and the solution (ŷ,q) of (11)-(12) associated to g ≡ 0 and initial data Since (y, q) = (ŷ,q +q) solves (11)- (12) for this h ω and satisfies we deduce that theorem 2 also holds for general g and (r 0 , r 1 ). Thus, let us assume that g ≡ 0 and (r 0 , r 1 ) = (0, 0) and let us consider the null controllability problem for   in Ω, where y 0 ∈ H −1 (Ω) and (q 0 , q 1 ) ∈ H 1 0 (Ω) × L 2 (Ω). There are several ways to deduce the null controllability of (31)-(32) from the observability inequality in proposition 1. We will use here a well known argument which relies on the construction of a sequence of minimal norm controls h n that provide states that converge to the desired target (0, 0, 0) as n → +∞.
3. Proof of theorem 1: The fixed point argument. As mentioned above, for the proof of theorem 1 we will use the controllability result in theorem 2 and a fixed point argument. This strategy was introduced in [12] in the framework of the exact controllability of the semilinear wave equation. Since then, it has been used in several different contexts; for instance, see [13], [3] and [6] for results concerning the approximate and null controllability of semilinear wave and heat equations with Dirichlet or Neumann boundary conditions. Let us also mention the paper [9], where the authors analyzed the null controllability of semilinear abstract systems (and in particular semilinear wave equations) using a global inverse function theorem.

3.1.
The case in which f 1 and f 2 are C 1 . Let us introduce the functions g i with Under the assumptions imposed in theorem 1 on the functions f 1 and f 2 , one has and the functions G 1 = g 1 (x, t; v ε , ξ) and G 2 = g 2 (x, t; ξ). Observe that G 1 and G 2 belong to L ∞ (Q).
Recall that ω satisfies (9) for some δ > 0. Let us choose δ 1 and δ 2 such that 0 < δ 1 < δ 2 < δ and let us set ω i = G δ i (x 0 ) for i = 1 and i = 2. In view of theorem 2, there exist controls h ε for some C only depending on L 1 , L 2 , ω, O, Ω and T and the associated solutions in Ω, satisfy (3). In (43), we have denoted by (q 0 ,q 1 ) the couple (q,q t )(·, T ), whereq is the solution of (14) with b(x, t) ≡ g 2 (x, t; ξ) and g(x, t) ≡ f 2 (x, t; 0). Consequently, we also have We will denote by U ε (v, ξ) the set of these controls. Let us introduce the set-valued mapping Λ ε : We have the following result: Proposition 3. Under the assumptions of theorem 1, there exists a compact set K ⊂ Z such that, for every (v, ξ) ∈ Z, one has Λ ε (v, ξ) ⊂ K. Furthermore, for every (v, ξ), Λ ε (v, ξ) is a non-empty convex compact subset of Z and the mapping Λ ε is upper hemicontinuous, that is to say, for each linear continuous form µ ∈ Z , the real-valued function is upper semicontinuous.
Proof: Observe that, for each (v, ξ) ∈ Z, the solution to (44)-(45) associated to a control h ∈ U ε (v, ξ) is such that , where C only depends on f 1 , f 2 , ω, O, Ω and T and W is the space ). Since W → Z with a compact embedding, there exists a compact set K such that On the other hand, from theorem 2 and the properties satisfied by ω 1 and T , we know that Λ ε (v, ξ) is non-empty. Since U ε (v, ξ) is convex, the fact that the system (44)-(45) is linear implies that Λ ε (v, ξ) is also a convex set.
Since Λ ε (v, ξ) ⊂ K for some compact set K of Z, in order to prove that Λ ε (v, ξ) is compact, we only need to check that it is closed.
Finally, let us prove that Λ ε is upper hemicontinuous. We have to check that the set is closed for every α ∈ IR and every µ ∈ Z .
As a consequence of proposition 3, Kakutani's theorem can be applied for every ε > 0 and there exists a fixed point (y ε , q ε ) of the mapping Λ ε . If we denote by y ε ε the solution to the linear problem (42) with v = y ε , then (y ε , q ε ) verifies in Ω. Observe that, for a positive constant C independent of ε and which only depends on f 1 , f 2 , ω, O, Ω and T , one has as ε → 0.
For every ε > 0, let us put in Ω.
Then we have the following: • Y is a fixed function in L 2 (Q).
• On the other hand, the unique reason for the lack of regularity of w ε is the lack of regularity of h ε ω 1 . For every p ∈ [1, ∞), let us introduce the spaces and the associated norms Since the support of h ε ω 1 is contained in ω 1 × [0, T ], as a consequence of the regularizing effect of the heat equation and the choice we have made of ω 1 and ω 2 , we have w ε ∈ X 2 and w ε X 2 ≤ C ( y 0 H −1 , (q 0 , q 1 ) H 1 0 ×L 2 , (r 0 , r 1 ) H 1 0 ×L 2 ) for some C > 0 independent of ε (see for instance [7]).
For every n we can argue as in the previous subsection and find a control h n ∈ L 2 (0, T ; D(−∆) ) with Supp h n ⊂ ω 2 × [0, T ] such that the system    y t,n − ∆y n + f 1,n (x, t; y n , q n ) = h n in Q, y n = 0 on Σ, y n (x, 0) = y 0 (x) in Ω,    q tt,n − ∆q n + f 2,n (x, t; q n ) = y n 1 O in Q, q n = 0 on Σ, q n (x, 0) = q 0 (x), q t,n (x, 0) = q 1 (x) in Ω.
This ends the proof of theorem 1.
This result is proved in [6] (see Ch. I, lemma 1.2 for the proof in a more general context; see also the Appendix of [5] for a simplified proof). In fact, a similar inequality holds for any T > 0 (with other appropriate ζ and C 1 ) if G κ (x 0 ) is replaced in (48) by an arbitrary nonempty open set D ⊂ Ω. Furthermore, as noticed in [5], the way the function ζ and the constant C 1 depend on c ∞ can be found explicitly.
We will also need an observability inequality for the wave equation (here, the quantity R(x 0 ) is as in Section 1):