Constructive proof of the exact controllability for semi-linear wave equations
aa r X i v : . [ m a t h . A P ] J a n Constructive proof of the exact controllability for semi-linearwave equations
J´erˆome Lemoine ∗ Arnaud M¨unch † January 19, 2021
Abstract
The exact distributed controllability of the semilinear wave equation ∂ tt y − ∆ y + g ( y ) = f ω posed over multi-dimensional and bounded domains, assuming that g ∈ C ( R ) satisfies the growthcondition lim sup r →∞ g ( r ) / ( | r | ln / | r | ) = 0 has been obtained by Fu, Yong and Zhang in 2007. Theproof based on a non constructive Leray-Schauder fixed point theorem makes use of precise estimatesof the observability constant for a linearized wave equation. Assuming that g ′ does not grow fasterthan β ln / | r | at infinity for β > g ′ is uniformly H¨older continuous on R with exponent s ∈ (0 , s after a finite number ofiterations. AMS Classifications:
Keywords:
Semilinear wave equation, exact controllability, least-squares approach.
Let Ω be a bounded domain of R d , d ∈ { , } with C , boundary and ω ⊂⊂ Ω be a non empty openset. Let
T > Q T := Ω × (0 , T ), q T := ω × (0 , T ) and Σ T := ∂ Ω × (0 , T ). We consider thesemilinear wave equation ∂ tt y − ∆ y + g ( y ) = f ω , in Q T ,y = 0 , on Σ T , ( y ( · , , y t ( · , u , u ) , in Ω , (1)where ( u , u ) ∈ V := H (Ω) × L (Ω) is the initial state of y and f ∈ L ( q T ) is a control function. Here andthroughout the paper, g : R → R is a function of class C such that | g ( r ) | ≤ C (1 + | r | ) ln(2 + | r | ) for every r ∈ R and some C >
0. Then, (1) has a unique global weak solution in C ([0 , T ]; H (Ω)) ∩ C ([0 , T ]; L (Ω))(see [3],[2]).The exact controllability for (1) in time T is formulated as follows: for any ( u , u ) , ( z , z ) ∈ V , finda control function f ∈ L ( q T ) such that the weak solution of (1) satisfies ( y ( · , T ) , ∂ t y ( · , T )) = ( z , z ).Assuming a growth condition on the nonlinearity g at infinity, this problem has been solved in [11]. Theorem 1. [11] For any x ∈ R d \ Ω , let Γ = { x ∈ ∂ Ω , ( x − x ) · ν ( x ) > } and, for any ǫ > , O ǫ (Γ ) = { y ∈ R d | | y − x | ≤ ǫ for x ∈ Γ } . Assume ( H ) T > x ∈ Ω | x − x | and ω ⊆ O ǫ (Γ ) ∩ Ω for some ǫ > . ∗ Laboratoire de Math´ematiques Blaise Pascal, Universit´e Clermont Auvergne, UMR CNRS 6620, Campus universitairedes C´ezeaux, 3, place Vasarely, 63178, Aubi`ere, France. E-mail: [email protected]. † Laboratoire de Math´ematiques Blaise Pascal, Universit´e Clermont Auvergne, UMR CNRS 6620, Campus universitairedes C´ezeaux, 3, place Vasarely, 63178, Aubi`ere, France. E-mail: [email protected]. f g satisfies ( H ) lim sup | r |→∞ | g ( r ) || r | ln / | r | = 0 then (1) is exactly controllable in time T . This result improves [19] where a stronger condition of the support ω is made, namely that ω is aneighborhood of ∂ Ω and that
T > diam(Ω \ ω ). In Theorem 1, Γ is the usual star-shaped part of thewhole boundary of Ω introduced in [20].A special case of Theorem 1 is when g is globally Lipschitz continuous, which gives the main result of[28], later generalized to an abstract setting in [14] using a global version of Inverse Function Theoremand improved in [26] for control domains ω satisfying the classical multiplier method of Lions [20].Theorem 1 extends to the multi-dimensional case the result of [29] devoted to the one dimensionalcase under the condition lim sup | r |→∞ | g ( r ) || r | ln | r | = 0, relaxed later on in [2], following [9], and in [21]. Theexact controllability for subcritical nonlinearities is obtained in [7] assuming the sign condition rg ( r ) ≥ r ∈ R . This latter assumption has been weakened in [12] to an asymptotic sign condition leadingto a semi-global controllability result in the sense that the final data ( z , z ) is prescribed in a precisesubset of V . In this respect, we also mention in the one dimensional case [6] where a positive boundarycontrollability result is proved for a steady-state initial and final data specific class of initial and finaldata and for T large enough by a quasi-static deformation approach.The proof given in [19, 11] is based on a fixed point argument introduced in [27, 29] that reducesthe exact controllability problem to the obtention of suitable a priori estimates for the linearized waveequation with a potential (see Proposition 6 in appendix A). More precisely, it is shown that the operator K : L ∞ (0 , T ; L d (Ω)) → L ∞ (0 , T ; L d (Ω)) where y ξ := K ( ξ ) is a controlled solution through the controlfunction f ξ of the linear boundary value problem ∂ tt y ξ − ∆ y ξ + y ξ b g ( ξ ) = − g (0) + f ξ ω , in Q T ,y ξ = 0 , on Σ T , ( y ξ ( · , , ∂ t y ξ ( · , u , u ) , in Ω , b g ( r ) := g ( r ) − g (0) r r = 0 ,g ′ (0) r = 0 , (2)satisfying ( y ξ ( · , T ) , y ξ,t ( · , T )) = ( z , z ) has a fixed point. The control f ξ is chosen in [19] as the one ofminimal L ( q T )-norm. The existence of a fixed point for the compact operator K is obtained by usingthe Leray-Schauder’s degree theorem. Precisely, it is shown under the growth assumption ( H ) thatthere exists a constant M = M ( k u , u k V , k z , z k V ) such that K maps the ball B L ∞ (0 ,T ; L d (Ω)) (0 , M )into itself.The main goal of this article is to design an algorithm providing an explicit sequence ( f k ) k ∈ N thatconverges strongly to an exact control for (1). A first idea that comes to mind is to consider the Picarditerations ( y k ) k ∈ N associated with the operator K defined by y k +1 = K ( y k ), k ≥ y ∈ L ∞ (0 , T ; L d (Ω)). The resulting sequence of controls ( f k ) k ∈ N is then so that f k +1 ∈ L ( q T )is the control of minimal L ( q T ) norm for y k +1 solution of ∂ tt y k +1 − ∆ y k +1 + y k +1 b g ( y k ) = − g (0) + f k +1 ω , in Q T ,y k +1 = 0 , on Σ T , ( y k +1 ( · , , ∂ t y k +1 ( · , y , y ) , in Ω . (3)Such a strategy usually fails since the operator K is in general not contracting, even if g is globallyLipschitz. We refer to [10] providing numerical evidence of the lack of convergence in parabolic cases (seealso Remark 5 in appendix A). A second idea is to use a Newton type method in order to find a zero ofthe C mapping e F : Y W defined by e F ( y, f ) := (cid:18) ∂ tt y − ∆ y + g ( y ) − f ω , y ( · , − u , ∂ t y ( · , − u , y ( · , T ) − z , ∂ t y ( · , T ) − z (cid:19) (4)2or some appropriates Hilbert spaces Y and W (see further): given ( y , f ) in Y , the sequence ( y k , f k ) k ∈ N is defined iteratively by ( y k +1 , f k +1 ) = ( y k , f k ) − ( Y k , F k ) where F k is a control for Y k solution of ∂ tt Y k − ∆ Y k + g ′ ( y k ) Y k = F k ω + ∂ tt y k − ∆ y k + g ( y k ) − f k ω , in Q T ,Y k = 0 , on Σ T ,Y k ( · ,
0) = u − y k ( · , , ∂ t Y k ( · ,
0) = u − ∂ t y k ( · ,
0) in Ω (5)such that Y k ( · , T ) = − y k ( · , T ) and ∂ t Y k ( · , T ) = − ∂ t y k ( · , T ) in Ω. This linearization makes appear anoperator K N , so that y k +1 = K N ( y k ) involving the first derivative of g . However, as it is well known,such a sequence may fail to converge if the initial guess ( y , f ) is not close enough to a zero of F (see[10] where divergence is observed numerically for large data).The controllability of nonlinear partial differential equations has attracted a large number of worksin the last decades (see the monography [5] and references therein). However, as far as we know, feware concerned with the approximation of exact controls for nonlinear partial differential equations, andthe construction of convergent control approximations for controllable nonlinear equations remains achallenge.In this article, given any initial data ( u , u ) ∈ V , we design an algorithm providing a sequence( y k , f k ) k ∈ N converging to a controlled pair for (1), under assumptions on g that are slightly strongerthan the one done in Theorem 1. Moreover, after a finite number of iterations, the convergence is super-linear. This is done by introducing a quadratic functional measuring how much a pair ( y, f ) ∈ Y is closeto a controlled solution for (1) and then by determining a particular minimizing sequence enjoying theannounced property. A natural example of an error (or least-squares) functional is given by e E ( y, f ) := k e F ( y, f ) k W to be minimized over Y . Exact controllability for (1) is reflected by the fact that the globalminimum of the nonnegative functional e E is zero, over all pairs ( y, f ) ∈ Y solutions of (1). In the lineof recent works on the Navier-Stokes system (see [16, 17]), we determine, using an appropriate descentdirection, a minimizing sequence ( y k , f k ) k ∈ N converging to a zero of the quadratic functional.The paper is organized as follows. In Section 2, we define the (nonconvex) least-squares functional E and the corresponding (nonconvex) optimization problem (6). We show that E is Gateaux-differentiableand that any critical point ( y, f ) for E such that g ′ ( y ) ∈ L ∞ (0 , T ; L d (Ω)) is also a zero of E . This is doneby introducing an adequate descent direction ( Y , F ) for E at any ( y, f ) for which E ′ ( y, f ) · ( Y , F )is proportional to p E ( y, f ). This instrumental fact compensates the failure of convexity of E and is atthe base of the global convergence properties of the least-squares algorithm. The design of this algorithmis done by determining a minimizing sequence based on ( Y , F ), which is proved to converge to acontrolled pair for the semilinear wave equation (1), in our main result (Theorem 2), under appropriateassumptions on g . Moreover, we prove that, after a finite number of iterations, the convergence is super-linear. Theorem 2 is proved in Section 3. We show in Section 4 that our least-squares approach coincideswith the classical damped Newton method applied to a mapping similar to e F , and we give a number ofother comments. In Appendix A, we state some a priori estimates for the linearized wave equation withpotential in L ∞ (0 , T ; L d (Ω)) and source term in L ( Q T ) and we show that the operator K is contractingif k ˆ g ′ k L ∞ ( R ) is small enough.As far as we know, the method introduced and analyzed in this work is the first one providing anexplicit, algorithmic construction of exact controls for semilinear wave equations with non Lipschitznonlinearity and defined over multi-dimensional bounded domains. It extends the one-dimensional studyaddressed in [24]. For parabolic equations with Lipschitz nonlinearity, we mention [15]. These worksdevoted to controllability problems takes their roots in earlier works, namely [16, 17], concerned with theapproximation of solution of Navier-Stokes type problem, through least-square methods: they refine theanalysis performed in [18, 22] inspired from the seminal contribution [1]. Notations.
Throughout, we denote by k · k ∞ the usual norm in L ∞ ( R ), by ( · , · ) X the scalar product of X (if X is a Hilbert space) and by h· , ·i X,Y the duality product between X and Y . The notation k · k ,q T k · k L ( q T ) and k · k p for k · k L p ( Q T ) , p ∈ N ⋆ .Given any s ∈ [0 , g ∈ C ( R ) the following hypothesis :( H s ) [ g ′ ] s := sup a,b ∈ R a = b | g ′ ( a ) − g ′ ( b ) || a − b | s < + ∞ meaning that g ′ is uniformly H¨older continuous with exponent s . For s = 0, by extension, we set[ g ′ ] := 2 k g ′ k ∞ . In particular, g satisfies ( H ) if and only if g ∈ C ( R ) and g ′ ∈ L ∞ ( R ), and g satisfies( H ) if and only if g ′ is Lipschitz continuous (in this case, g ′ is almost everywhere differentiable and g ′′ ∈ L ∞ ( R ), and we have [ g ′ ] s ≤ k g ′′ k ∞ ).We also denote by C a positive constant depending only on Ω and T that may vary from lines to lines.In the rest of the paper, we assume that the open set ω and the time T satisfy ( H ). We define the Hilbert space HH = (cid:26) ( y, f ) ∈ L ( Q T ) × L ( q T ) | ∂ tt y − ∆ y ∈ L ( Q T ) , ( y ( · , , ∂ t y ( · , ∈ V , y = 0 on Σ T (cid:27) endowed with the scalar product(( y, f ) , ( y, f )) H = ( y, y ) + (cid:0) ( y ( · , , ∂ t y ( · , , ( y ( · , , ∂ t y ( · , (cid:1) V + ( ∂ tt y − ∆ y, ∂ tt y − ∆ y ) + ( f, f ) ,q T and the norm k ( y, f ) k H := p (( y, f ) , ( y, f )) H . Then, for any ( u , u ) , ( z , z ) ∈ V , we define the subspacesof H A = (cid:26) ( y, f ) ∈ H | ( y ( · , , ∂ t y ( · , u , u ) , ( y ( · , T ) , ∂ t y ( · , T )) = ( z , z ) (cid:27) , A = (cid:26) ( y, f ) ∈ H | ( y ( · , , ∂ t y ( · , , , ( y ( · , T ) , ∂ t y ( · , T )) = (0 , (cid:27) . We consider the following non convex extremal problem :inf ( y,f ) ∈A E ( y, f ) , E ( y, f ) := 12 (cid:13)(cid:13) ∂ tt y − ∆ y + g ( y ) − f ω (cid:13)(cid:13) (6)justifying the least-squares terminology we have used. Remark that we can write A = ( y, f ) + A for anyelement ( y, f ) ∈ A . The problem is therefore equivalent to the minimization of E ( y + y, f + f ) over A for any ( y, f ) ∈ A .The functional E is well-defined in A . Precisely, Lemma 1.
There exists a positive constant
C > such that E ( y, f ) ≤ C k ( y, f ) k H for any ( y, f ) ∈ A .Proof. A priori estimate for the linear wave equation reads as k ( y, ∂ t y ) k L ∞ (0 ,T ; V ) ≤ C (cid:18) k ∂ tt y − ∆ y k + k u , u k V (cid:19) for any y such that ( y, f ) ∈ A . Using that | g ( r ) | ≤ C (1 + | r | ) log(2 + | r | ) for every r ∈ R and some C > k g ( y ) k ≤ C Z Q T (cid:18) (1 + | y | ) log(2 + | y | ) (cid:19) ≤ C Z Q T (1 + | y | ) ≤ C ( | Q T | + k y k L ( Q T ) ) ≤ C (cid:0) | Q T | + k y k L ∞ (0 ,T ; H (Ω)) (cid:1) E ( y, f ) ≤ C (cid:0) k ∂ tt y − ∆ y k + k f k ,q T + | Q T | + k y k L ∞ (0 ,T ; H (Ω)) (cid:1) leading to the result.Within the hypotheses of Theorem 1, the infimum of the functional of E is zero and is reached byat least one pair ( y, f ) ∈ A , solution of (1) and satisfying ( y ( · , T ) , ∂ t y ( · , T )) = ( z , z ). Conversely,any pair ( y, f ) ∈ A for which E ( y, f ) vanishes is solution of (1). In this sense, the functional E is an error functional which measures the deviation of ( y, f ) from being a solution of the underlying nonlinearequation. A practical way of taking a functional to its minimum is through the use of gradient descentdirections. In doing so, the presence of local minima is always something that may dramatically spoilthe whole scheme. The unique structural property that discards this possibility is the convexity of thefunctional E . However, for nonlinear equation like (1), one cannot expect this property to hold for thefunctional E . Nevertheless, we are going to construct a minimizing sequence which always converges toa zero of E .In order to construct such minimizing sequence, we formally look, for any ( y, f ) ∈ A , for a pair( Y , F ) ∈ A solution of the following formulation ∂ tt Y − ∆ Y + g ′ ( y ) · Y = F ω + (cid:0) ∂ tt y − ∆ y + g ( y ) − f ω (cid:1) , in Q T ,Y = 0 , on Σ T , ( Y ( · , , ∂ t Y ( · , , , in Ω . (7)Since ( Y , F ) belongs to A , F is a null control for Y . Among the controls of this linear equation,we select the control of minimal L ( q T ) norm. In the sequel, we shall call the corresponding solution( Y , F ) ∈ A the solution of minimal control norm. We have the following property. Proposition 1.
For any ( y, f ) ∈ A , there exists a pair ( Y , F ) ∈ A solution of (7). Moreover, thepair ( Y , F ) of minimal control norm satisfies the following estimates : k ( Y , ∂ t Y ) k L ∞ (0 ,T ; V ) + k F k ,q T ≤ Ce C k g ′ ( y ) k L ∞ (0 ,T ; Ld (Ω)) p E ( y, f ) , (8) and k ( Y , F ) k H ≤ C (cid:0) k g ′ ( y ) k L ∞ (0 ,T ; L (Ω)) (cid:1) e C k g ′ ( y ) k L ∞ (0 ,T ; Ld (Ω)) p E ( y, f ) (9) for some positive constant C > .Proof. The first estimate is a consequence of Proposition 7 using the equality k ∂ tt y − ∆ y + g ( y ) − f ω k = p E ( y, f ). The second one follows from k ( Y , F ) k H ≤ k ∂ tt Y − ∆ Y k + k Y k + k F k ,q T + k Y ( · , , ∂ t Y ( · , k V ≤ k Y k + k g ′ ( y ) Y k + 2 k F k ,q T + √ p E ( y, f ) ≤ C (cid:0) k g ′ ( y ) k L ∞ (0 ,T ; L (Ω)) (cid:1) e C k g ′ ( y ) k L ∞ (0 ,T ; Ld (Ω)) p E ( y, f )using that k g ′ ( y ) Y k ≤ Z T k g ′ ( y ) k L (Ω) k Y k L (Ω) ≤ k g ′ ( y ) k L ∞ (0 ,T ; L (Ω)) k Y k L ∞ (0 ,T ; L (Ω)) ≤ C k g ′ ( y ) k L ∞ (0 ,T ; L (Ω)) k Y k L ∞ (0 ,T ; H (Ω)) . .2 Main properties of the functional E The interest of the pair ( Y , F ) ∈ A lies in the following result. Proposition 2.
Assume that g satisfies ( H s ) for some s ∈ [0 , . Let ( y, f ) ∈ A and let ( Y , F ) ∈ A be a solution of (7). Then the derivative of E at the point ( y, f ) ∈ A along the direction ( Y , F ) satisfies E ′ ( y, f ) · ( Y , F ) = 2 E ( y, f ) . (10) Proof.
We preliminary check that for all (
Y, F ) ∈ A the functional E is differentiable at the point( y, f ) ∈ A along the direction ( Y, F ) ∈ A . For any λ ∈ R , simple computations lead to the equality E ( y + λY, f + λF ) = E ( y, f ) + λE ′ ( y, f ) · ( Y, F ) + h (( y, f ) , λ ( Y, F ))with E ′ ( y, f ) · ( Y, F ) := (cid:0) ∂ tt y − ∆ y + g ( y ) − f ω , ∂ tt Y − ∆ y + g ′ ( y ) Y − F ω (cid:1) (11)and h (( y, f ) , λ ( Y, F )) := λ (cid:0) ∂ tt Y − ∆ Y + g ′ ( y ) Y − F ω , ∂ tt Y − ∆ Y + g ′ ( y ) Y − F ω (cid:1) + λ (cid:0) ∂ tt Y − ∆ Y + g ′ ( y ) Y − F ω , l ( y, λY ) (cid:1) + (cid:0) ∂ tt y − ∆ y + g ( y ) − f ω , l ( y, λY ) (cid:1) + 12 ( l ( y, λY ) , l ( y, λY ))where l ( y, λY ) := g ( y + λY ) − g ( y ) − λg ′ ( y ) Y . The application ( Y, F ) → E ′ ( y, f ) · ( Y, F ) is linear andcontinuous from A to R as it satisfies | E ′ ( y, f ) · ( Y, F ) | ≤ k ∂ tt y − ∆ y + g ( y ) − f ω k k ∂ tt Y − ∆ Y + g ′ ( y ) Y − F ω k ≤ p E ( y, f ) (cid:18) k ( ∂ tt Y − ∆ Y ) k + k g ′ ( y ) k L ∞ (0 ,T ; L (Ω)) k Y k L ∞ (0 ,T ; H (Ω)) + k F k ,q T (cid:19) ≤ p E ( y, f ) max (cid:0) , k g ′ ( y ) k L ∞ (0 ,T ; L (Ω)) (cid:1) k ( Y, F ) k H . (12)Similarly, for all λ ∈ R ⋆ , (cid:12)(cid:12)(cid:12)(cid:12) λ h (( y, f ) , λ ( Y, F )) (cid:12)(cid:12)(cid:12)(cid:12) ≤ λ k ∂ tt Y − ∆ Y + g ′ ( y ) Y − F ω k + (cid:18) λ k ∂ tt Y − ∆ Y + g ′ ( y ) Y − F ω k + p E ( y, f ) + 12 k l ( y, λY ) k (cid:19) λ k l ( y, λY ) k . For any ( x, y ) ∈ R and λ ∈ R , we then write g ( x + λy ) − g ( x ) = R λ yg ′ ( x + ξy ) dξ leading to | g ( x + λy ) − g ( x ) − λg ′ ( x ) y | ≤ Z λ | y || g ′ ( x + ξy ) − g ′ ( x ) | dξ ≤ Z λ | y | s | ξ | s | g ′ ( x + ξy ) − g ′ ( x ) || ξy | s dξ ≤ [ g ′ ] s | y | s | λ | s s . It follows that | l ( y, λY ) | = | g ( y + λY ) − g ( y ) − λg ′ ( y ) Y | ≤ [ g ′ ] s | λ | s s | Y | s and 1 | λ | (cid:13)(cid:13) l ( y, λY ) (cid:13)(cid:13) ≤ [ g ′ ] s | λ | s s (cid:13)(cid:13) | Y | s (cid:13)(cid:13) . (13)But (cid:13)(cid:13) | Y | s (cid:13)(cid:13) = k Y k s +1)2( s +1) ≤ C k Y k s +1) L ∞ (0 ,T ; L (Ω)) . Consequently, for s > | λ |k l ( y, λY ) k → λ → | h (( y, f ) , λ ( Y, F )) | = o ( λ ). In the case s = 0 leading to g ′ ∈ L ∞ ( R ), the result follows from theLebesgue dominated convergence theorem: we have (cid:12)(cid:12)(cid:12) λ ℓ ( y, λY ) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) g ( y + λY ) − g ( y ) λ − g ′ ( y ) Y (cid:12)(cid:12)(cid:12) ≤ k g ′ k ∞ | Y | , a.e. in Q T (cid:12)(cid:12) λ ℓ ( y, λY ) (cid:12)(cid:12) = (cid:12)(cid:12) g ( y + λY ) − g ( y ) λ − g ′ ( y ) Y (cid:12)(cid:12) → λ → Q T . It follows that | λ |k ℓ ( y, λY ) k → λ → E is differentiable at the point ( y, f ) ∈ A along thedirection ( Y, F ) ∈ A .Eventually, the equality (10) follows from the definition of the pair ( Y , F ) given in (7).Remark that from the equality (11), the derivative E ′ ( y, f ) is independent of ( Y, F ). We can thendefine the norm k E ′ ( y, f ) k A ′ := sup ( Y,F ) ∈A \{ } E ′ ( y,f ) · ( Y,F ) k ( Y,F ) k H associated to A ′ , the topological dual of A .Combining the equality (10) and the inequality (8), we deduce the following estimate of E ( y, f ) interm of the norm of E ′ ( y, f ). Proposition 3.
For any ( y, f ) ∈ A , the following inequalities hold true: √ (cid:0) , k g ′ ( y ) k L ∞ (0 ,T ; L (Ω)) (cid:1) k E ′ ( y, f ) k A ′ ≤ p E ( y, f ) ≤ √ C (cid:18) k g ′ ( y ) k L ∞ (0 ,T ; L (Ω)) (cid:19) e C k g ′ ( y ) k L ∞ (0 ,T ; Ld (Ω)) k E ′ ( y, f ) k A ′ (14) where C is the positive constant from Proposition 1.Proof. (10) rewrites E ( y, f ) = E ′ ( y, f ) · ( Y , F ) where ( Y , F ) ∈ A is solution of (7) and therefore,with (9) E ( y, f ) ≤ k E ′ ( y, f ) k A ′ k ( Y , F ) k A ≤ C (cid:0) k g ′ ( y ) k L ∞ (0 ,T ; L (Ω)) (cid:1) e C k g ′ ( y ) k L ∞ (0 ,T ; Ld (Ω)) k E ′ ( y, f ) k A ′ p E ( y, f ) . On the other hand, for all (
Y, F ) ∈ A , the inequality (12), i.e. | E ′ ( y, f ) · ( Y, F ) | ≤ p E ( y, f ) max (cid:0) , k g ′ ( y ) k L ∞ (0 ,T ; L (Ω)) (cid:1) k ( Y, F ) k A leads to the left inequality.Consequently, any critical point ( y, f ) ∈ A of E (i.e., E ′ ( y, f ) vanishes) such that k g ′ ( y ) k L ∞ (0 ,T ; L (Ω)) is finite is a zero for E , a pair solution of the controllability problem. In other words, any sequence( y k , f k ) k> satisfying k E ′ ( y k , f k ) k A ′ → k → ∞ and for which k g ′ ( y k ) k L ∞ (0 ,T ; L (Ω)) is uniformlybounded is such that E ( y k , f k ) → k → ∞ . We insist that this property does not imply the convexityof the functional E (and a fortiori the strict convexity of E , which actually does not hold here in view ofthe multiple zeros for E ) but show that a minimizing sequence for E can not be stuck in a local minimum.On the other hand, the left inequality indicates the functional E is flat around its zero set. As aconsequence, gradient-based minimizing sequences may achieve a low speed of convergence (we refer to[23] and also [18] devoted to the Navier-Stokes equation where this phenomenon is observed).We end this section with the following estimate. Lemma 2.
Assume that g satisfies ( H s ) for some s ∈ [0 , . For any ( y, f ) ∈ A , let ( Y , F ) ∈ A bedefined by (7) . For any λ ∈ R the following estimate holds E (cid:0) ( y, f ) − λ ( Y , F ) (cid:1) ≤ E ( y, f ) (cid:18) | − λ | + λ s c ( y ) E ( y, f ) s/ (cid:19) (15) with c ( y ) := C (1 + s ) √ g ′ ] s d ( y ) s , d ( y ) := Ce C k g ′ ( y ) k L ∞ (0 ,T ; Ld (Ω)) . roof. Estimate (13) applied with Y = Y reads (cid:13)(cid:13) l ( y, λY ) (cid:13)(cid:13) ≤ [ g ′ ] s | λ | s s (cid:13)(cid:13) | Y | s (cid:13)(cid:13) . (16)But k| Y | s (cid:13)(cid:13) = k Y k s +1)2( s +1) ≤ C k Y k s +1) L ∞ (0 ,T ; H (Ω)) which together with (8) lead to (cid:13)(cid:13) | Y | s (cid:13)(cid:13) ≤ C (cid:18) Ce C k g ′ ( y ) k L ∞ (0 ,T ; Ld (Ω) (cid:19) s E ( y, f ) s . (17)Eventually, we write2 E (cid:0) ( y, f ) − λ ( Y , F ) (cid:1) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:0) ∂ tt y − ∆ y + g ( y ) − f ω (cid:1) − λ (cid:0) ∂ tt Y − ∆ Y + g ′ ( y ) Y − F ω (cid:1) + l ( y, − λY ) (cid:13)(cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13)(cid:13) (1 − λ ) (cid:0) ∂ tt y − ∆ y + g ( y ) − f ω (cid:1) + l ( y, − λY ) (cid:13)(cid:13)(cid:13)(cid:13) ≤ (cid:18)(cid:13)(cid:13) (1 − λ ) (cid:0) ∂ tt y − ∆ y + g ( y ) − f ω (cid:1)(cid:13)(cid:13) + (cid:13)(cid:13) l ( y, − λY ) (cid:13)(cid:13) (cid:19) ≤ (cid:18) | − λ | p E ( y, f ) + [ g ′ ] s | λ | s s (cid:13)(cid:13) | Y | s (cid:13)(cid:13) (cid:19) ≤ (cid:18) | − λ | p E ( y, f ) + [ g ′ ] s | λ | s s C (cid:18) Ce C k g ′ ( y ) k L ∞ (0 ,T ; Ld (Ω) (cid:19) s E ( y, f ) s (cid:19) (18)and we get the result. E We now examine the convergence of an appropriate sequence ( y k , f k ) ∈ A . In this respect, we observethat equality (10) shows that − ( Y , F ) given by the solution of (7) is a descent direction for E . Therefore,we can define, for any fixed m ≥
1, a minimizing sequence ( y k , f k ) k> ∈ A as follows: ( y , f ) ∈ A , ( y k +1 , f k +1 ) = ( y k , f k ) − λ k ( Y k , F k ) , k ∈ N ,λ k = argmin λ ∈ [0 ,m ] E (cid:0) ( y k , f k ) − λ ( Y k , F k ) (cid:1) , (19)where ( Y k , F k ) ∈ A is the solution of minimal control norm of ∂ tt Y k − ∆ Y k + g ′ ( y k ) · Y k = F k ω + ( ∂ tt y k − ∆ y k + g ( y k ) − f k ω ) , in Q T ,Y k = 0 , on Σ T , ( Y k ( · , , ∂ t Y k ( · , , , in Ω . (20)The real number m ≥ λ k ) k ∈ N bounded.Given any s ∈ [0 , β ⋆ ( s ) := r s C (2 s + 1) (21)where C >
0, only depending on Ω and T , is the constant appearing in Proposition 7. In this section, weprove our main result. Theorem 2.
Assume that g ′ satisfies ( H s ) for some s ∈ [0 , and H ) There exists α ≥ and β ∈ [0 , β ⋆ ( s )) such that | g ′ ( r ) | ≤ α + β ln / (1 + | r | ) for every r in R if s ∈ (0 , and ( H ) √ C k g ′ k ∞ e C k g ′ k ∞ | Ω | /d < if s = 0 .Then, for any ( y , f ) ∈ A , the sequence ( y k , f k ) k ∈ N defined by (19) strongly converges to a pair ( y, f ) ∈ A satisfying (1) and the condition ( y ( · , T ) , y t ( · , T )) = ( z , z ) , for all ( u , u ) , ( z , z ) ∈ V .Moreover, the convergence is at least linear and is at least of order s after a finite number of iterations. Consequently, the algorithm (19) provides a constructive way to approximate a control for the non-linear wave equation (19).The proof consists in showing that the decreasing sequence ( E ( y k , f k )) k ∈ N converges to zero. In view of(14), this property is related to the uniform property of the observability constant e C k g ′ ( y k ) k L ∞ (0 ,T ; Ld (Ω)) with respect to k . In order to fix some notations and the main ideas of the proof of Theorem 2, wefirst prove in Section 3.1 the convergence of the sequence ( y k , f k ) k ∈ N under the stronger condition that g ′ ∈ L ∞ ( R ), sufficient to ensure the boundedness of the sequence (cid:0) e C k g ′ ( y k ) k L ∞ (0 ,T ; Ld (Ω)) (cid:1) k ∈ N . Then, inSection 3.2, we prove Theorem 2 by showing that under the assumption ( H ), the sequence ( y k , f k ) k ∈ N isactually bounded in A . This implies the same property for the real sequence e C k g ′ ( y k ) k L ∞ (0 ,T ; Ld (Ω)) , andthen the announced convergence. g ′ ∈ L ∞ ( R ) We establish in this section the following preliminary result, which coincides with Theorem 2 in thesimpler case β = 0. Proposition 4.
Assume that g ′ satisfies ( H s ) for some s ∈ [0 , and that g ′ ∈ L ∞ ( R ) . If s = 0 , assumemoreover ( H ) . For any ( y , f ) ∈ A , the sequence ( y k , f k ) k ∈ N defined by (19) strongly converges to apair ( y, f ) ∈ A satisfying (1) and the condition ( y ( · , T ) , y t ( · , T )) = ( z , z ) , for all ( u , u ) , ( z , z ) ∈ V .Moreover, the convergence is at least linear and is at least of order s after a finite number of iterations. Proceeding as in [17, 24], Proposition 4 follows from the following lemma.
Lemma 3.
Under the hypotheses of Proposition 4, for any ( y , f ) ∈ A , there exists a k ∈ N such thatthe sequence ( E ( y k , f k )) k ≥ k tends to as k → ∞ with at least a rate s + 1 .Proof. Since g ′ ∈ L ∞ ( R ), the nonnegative constant c ( y k ) in (15) is uniformly bounded w.r.t. k : weintroduce the real c > c ( y k ) ≤ c := C (1 + s ) √ g ′ ] s (cid:18) Ce C k g ′ k ∞ | Ω | /d (cid:19) s , ∀ k ∈ N . (22) | Ω | denotes the measure of the domain Ω. For any ( y k , f k ) ∈ A , let us then denote the real function p k by p k ( λ ) := | − λ | + λ s c E ( y k , f k ) s/ , λ ∈ [0 , m ] . Lemma 2 with ( y, f ) = ( y k , f k ) then allows to write that p E ( y k +1 , f k +1 ) = min λ ∈ [0 ,m ] q E (( y k , f k ) − λ ( Y k , F k )) ≤ p k ( f λ k ) p E ( y k , f k ) (23) We recall that a sequence ( u k ) k ∈ N of real numbers converges to 0 with order α ≥ M > | u k +1 | ≤ M | u k | α for every k ∈ N . A sequence ( v k ) k ∈ N of real numbers converges to 0 at least with order α ≥ u k ) k ∈ N of nonnegative real numbers converging to 0 with order α ≥ | v k | ≤ u k for every k ∈ N . p k ( f λ k ) := min λ ∈ [0 ,m ] p k ( λ ). Assume first that s >
0. We then easily check that the optimal f λ k isgiven by f λ k := s ) /s c /s p E ( y k , f k ) , if (1 + s ) /s c /s p E ( y k , f k ) ≥ , , if (1 + s ) /s c /s p E ( y k , f k ) < p k ( f λ k ) := − s (1 + s ) s +1 c /s p E ( y k , f k ) , if (1 + s ) /s c /s p E ( y k , f k ) ≥ ,c E ( y k , f k ) s/ , if (1 + s ) /s c /s p E ( y k , f k ) < . (25)Accordingly, we may distinguish two cases : • If (1 + s ) /s c /s p E ( y , f ) <
1, then c /s p E ( y , f ) <
1, and thus c /s p E ( y k , f k ) < k ∈ N since the sequence ( E ( y k , f k )) k ∈ N is decreasing. Hence (23) implies that c /s p E ( y k +1 , f k +1 ) ≤ (cid:0) c /s p E ( y k , f k ) (cid:1) s ∀ k ∈ N . It follows that c /s p E ( y k , f k ) → k → ∞ with a rate equal to 1 + s . • If (1 + s ) /s c /s p E ( y , f ) ≥ I := { k ∈ N , (1 + s ) /s c /s p E ( y k , f k ) ≥ } is a finite subset of N ; indeed, for all k ∈ I , (23) implies that c /s p E ( y k +1 , f k +1 ) ≤ (cid:16) − s (1 + s ) s +1 c /s p E ( y k , f k ) (cid:17) c /s p E ( y k , f k ) = c /s p E ( y k , f k ) − s (1 + s ) s +1 (26)and the strict decrease of the sequence ( c /s p E ( y k , f k )) k ∈ I . Thus there exists k ∈ N such that for all k ≥ k , (1 + s ) /s c /s p E ( y k , f k ) <
1, that is I is a finite subset of N . Arguing as in the first case, itfollows that p E ( y k , f k ) → k → ∞ .It follows in particular from (25) that the sequence ( p k ( f λ k )) k ∈ N decreases as well.If now s = 0, then p k ( λ ) = | − λ | + λc with c = [ g ′ ] Ce C k g ′ k ∞ | Ω | /d and (23) with f λ k = 1 leads to p E ( y k +1 , f k +1 ) ≤ c p E ( y k , f k ). The convergence of ( E ( y k , f k )) k ∈ N to 0 holds if c <
1, i.e. ( H ). Proof. (of Proposition 4) In view of (9), we write (cid:0) k g ′ ( y ) k L ∞ (0 ,T ; L (Ω)) (cid:1) e C k g ′ ( y ) k ∞ (0 ,T ; Ld (Ω)) ≤ (1 + k g ′ k ∞ | Ω | / ) e C k g ′ k ∞ | Ω | /d ≤ e C k g ′ k ∞ | Ω | /d using that (1 + u ) e u ≤ e u for all u ∈ R + . It follows that k X n =0 | λ n |k ( Y n , F n ) k H ≤ m Ce C k g ′ k ∞ | Ω | /d k X n =0 p E ( y n , f n ) . (27)Using that p n ( e λ n ) ≤ p ( e λ ) for all n ≥
0, we can write for n > p E ( y n , f n ) ≤ p n − ( e λ n − ) p E ( y n − , f n − ) ≤ p ( e λ ) p E ( y n − , f n − ) ≤ ( p ( e λ )) n p E ( y , f ) . (28)Then, using that p ( e λ ) = min λ ∈ [0 ,m ] p ( λ ) < p (0) = 1 and p ′ (0) < k X n =0 | λ n |k ( Y n , F n ) k H ≤ m Ce C k g ′ k ∞ | Ω | /d p E ( y , f )1 − p ( e λ )for which we deduce (since H is a complete space) that the serie P n ≥ λ n ( Y n , F n ) converges in A .Writing from (19) that ( y k +1 , f k +1 ) = ( y , f ) − P kn =0 λ n ( Y n , F n ), we conclude that ( y k , f k ) stronglyconverges in A to ( y, f ) := ( y , f ) + P n ≥ λ n ( Y n , F n ).10hen, using that ( Y k , F k ) goes to zero as k → ∞ in A , we pass to the limit in (20) and get that( y, f ) ∈ A solves ∂ tt y − ∆ y + g ( y ) = f ω , in Q T ,y = 0 , on Σ T , ( y ( · , , ∂ t y ( · , y , y ) , in Ω . (29)Since the limit ( y, f ) belongs to A , we have that ( y ( · , T ) , y t ( · , T )) = ( z , z ) in Ω. Moreover, for all k > k ( y, f ) − ( y k , f k ) k H = (cid:13)(cid:13)(cid:13)(cid:13) ∞ X p = k +1 λ p ( Y p , F p ) (cid:13)(cid:13)(cid:13)(cid:13) H ≤ m ∞ X p = k +1 k ( Y p , F p k H ≤ m C ∞ X p = k +1 q E ( y p , f p ) ≤ m C ∞ X p = k +1 p ( e λ ) p − k p E ( y k , f k ) ≤ m C p ( e λ )1 − p ( e λ ) p E ( y k , f k ) (30)and conclude from Lemma 3 the convergence of order at least 1 + s after a finite number of iterates. Remark 1.
In particular, along the sequence ( y k , f k ) k ∈ N defined by (19), (30) is a kind of coercivityproperty for the functional E . We emphasize, in view of the non uniqueness of the zeros of E , that anestimate (similar to (30) ) of the form k ( y, f ) − ( y, f ) k H ≤ C p E ( y, f ) does not hold for all ( y, f ) ∈ A .We also insist in the fact the sequence ( y k , f k ) k ∈ N and its limits ( y, f ) are uniquely determined from theinitialization ( y , f ) ∈ A and from our selection criterion for the control F . Remark 2.
Estimate (27) implies the uniform estimate on the sequence ( k ( y k , f k ) k H ) k ∈ N : k ( y k , f k ) k H ≤ k ( y , f ) k H + m Ce C k g ′ k ∞ | Ω | /d k − X n =0 p E ( y n , f n ) ≤ k ( y , f ) k H + m Ce C k g ′ k ∞ | Ω | /d p E ( y , f )1 − p ( e λ ) . In particular, for s > and the less favorable case for which (1 + s ) /s c /s p E ( y , f ) ≥ , we get √ E ( y ,f )1 − p ( e λ ) = (1+ s ) s +1 s c /s E ( y , f ) , (see (25) ) leading to k ( y k , f k ) k H ≤ k ( y , f ) k H + m Ce C k g ′ k ∞ | Ω | /d (1 + s ) s +1 s c /s E ( y , f ) , and then, in view of (22) to the explicit estimate in term of the data k ( y k , f k ) k H ≤ k ( y , f ) k H + m (1 + s ) s (cid:18) C [ g ′ ] s √ (cid:19) /s (cid:18) Ce C k g ′ k ∞ | Ω | /d (cid:19) s +1 s E ( y , f ) . (31) The case s = 0 under the hypothesis c < leads to k ( y k , f k ) k H ≤ k ( y , f ) k H + m c √ E ( y ,f )1 − c . Remark 3.
For s > , recalling that the constant c is defined in (22) , if (1 + s ) /s c /s p E ( y , f ) ≥ ,inequality (26) implies that c /s p E ( y k , f k ) ≤ c /s p E ( y , f ) − k s (1 + s ) s +1 , ∀ k ∈ I. (32) Hence, the number of iteration k to achieve a rate s is estimate as follows : k = (cid:22) (1 + s ) (cid:18) c /s (1 + s ) /s p E ( y , f ) (cid:19) − s (cid:23) + 111 here ⌊·⌋ denotes the integer part. As expected, this number increases with p E ( y , f ) and k g ′ k ∞ . If (1 + s ) /s c /s p E ( y , f ) < , then k = 0 . In particular, as s → + , k → ∞ if c > , i.e. if ( H ) doesnot hold.For s = 0 , the inequality p E ( y k +1 , f k +1 ) ≤ c p E ( y k , f k ) with c < leads to k = 0 . We also have the following convergence result for the optimal sequence ( λ k ) k> . Lemma 4.
Assume that g ′ satisfies ( H s ) for some s ∈ [0 , and that g ′ ∈ L ∞ ( R ) . The sequence ( λ k ) k>k defined in (19) converges to as k → ∞ at least with order s .Proof. In view of (18), we have, as long as E ( y k , f k ) >
0, since λ k ∈ [0 , m ](1 − λ k ) = E ( y k +1 , f k +1 ) E ( y k , f k ) − − λ k ) h (cid:0) y k,tt + ∆ y k + g ( y k ) − f k ω (cid:1) , l ( y k , λ k Y k ) i E ( y k , f k ) − (cid:13)(cid:13) l ( y k , λ k Y k ) (cid:13)(cid:13) E ( y k ) ≤ E ( y k +1 , f k +1 ) E ( y k , f k ) − − λ k ) h (cid:0) y k,tt + ∆ y k + g ( y k ) − f k ω (cid:1) , l ( y k , λ k Y k ) i E ( y k , f k ) ≤ E ( y k +1 , f k +1 ) E ( y k , f k ) + 2 √ m p E ( y k , f k ) k l ( y k , λ k Y k ) k L ( Q T ) E ( y k , f k ) ≤ E ( y k +1 , f k +1 ) E ( y k , f k ) + 2 √ m k l ( y k , λ k Y k ) k p E ( y k , f k ) . But, from (16) and (17), we have k l ( y k , λ k Y k ) k L ( Q T ) ≤ cλ sk E ( y k , f k ) s ≤ cm s E ( y k , f k ) s andthus (1 − λ k ) ≤ E ( y k +1 , f k +1 ) E ( y k , f k ) + 2 √ m s cE ( y k , f k ) s/ . Consequently, since E ( y k , f k ) → E ( y k +1 ,f k +1 ) E ( y k ,f k ) → s , we deduce the result. In this section, we relax the condition g ′ ∈ L ∞ ( R ) and prove Theorem 3.2, for s > H ). This assumption implies notably that | g ( r ) | ≤ C (1 + | r | ) ln(2 + | r | ) for every r ∈ R , mentioned inthe introduction to state the well-posedness of (1). The case β = 0 corresponds to the case developed inthe previous section, i.e. g ′ ∈ L ∞ ( R ).Within this more general framework, the difficulty is to have a uniform control with respect to k ofthe observability constant Ce C k g ′ ( y k ) k L ∞ (0 ,T ; Ld (Ω)) appearing in the estimates for ( Y k , F k ), see Proposition1. In other terms, we have to show that the sequence ( y k , f k ) k ∈ N uniquely defined in (19) is uniformlybounded in A , for any ( y , f ) ∈ A .We need the following intermediate result. Lemma 5.
Let
C > , only depending on Ω and T be the constant appearing in Proposition 7. Assumethat g satisfies the growth condition ( H ) and Cβ ≤ . Then for any ( y, f ) ∈ A , e C k g ′ ( y ) k L ∞ (0 ,T ; Ld (Ω)) ≤ C max(1 , e Cα | Ω | ) (cid:18) k y k L ∞ (0 ,T ; L p⋆ (Ω)) | Ω | /p ⋆ (cid:19) Cβ for any p ⋆ ∈ N ⋆ with p ⋆ < ∞ if d = 2 and p ⋆ ≤ if d = 3 .Proof. We use the following inequality (direct consequence of the inequality (3.8) in [19]): e C k g ′ ( y ) k L ∞ (0 ,T ; Ld (Ω)) ≤ C (cid:18) t ∈ (0 ,T ) Z Ω e C | g ′ ( y ) | (cid:19) , ∀ ( y, f ) ∈ A . (33)12riting that | g ′ ( y ) | ≤ (cid:0) α + β ln(1 + | y | ) (cid:1) , we get that R Ω e C | g ′ ( y ) | ≤ e Cα R Ω (1 + | y | ) Cβ .Assuming 2 Cβ ≤ p ⋆ , Holder inequality leads to Z Ω e C | g ′ ( y ) | ≤ e Cα (cid:18)Z Ω (1 + | y | ) p ⋆ (cid:19) Cβ p⋆ | Ω | − Cβ p⋆ ≤ e Cα | Ω | (cid:18) k y k L p⋆ (Ω) | Ω | /p ⋆ (cid:19) Cβ . It follows, by (33), that for every ( y, f ) ∈ A , e C k g ′ ( y ) k L ∞ (0 ,T ; Ld (Ω)) ≤ C (cid:18) e Cα | Ω | (cid:18) k y k L ∞ (0 ,T ; L p⋆ (Ω)) | Ω | /p ⋆ (cid:19) Cβ (cid:19) ≤ C max(1 , e Cα | Ω | ) (cid:18) (cid:18) k y k L ∞ (0 ,T ; L p⋆ (Ω)) | Ω | /p ⋆ (cid:19) Cβ (cid:19) ≤ Cβ C max(1 , e Cα | Ω | ) (cid:18) k y k L ∞ (0 ,T ; L p⋆ (Ω)) | Ω | /p ⋆ (cid:19) Cβ and the result. Lemma 6.
Assume that g satisfies the growth condition ( H ) and Cβ ≤ . For any ( y, f ) ∈ A , theunique solution ( Y , F ) ∈ A of (7) satisfies k ( Y , Y t ) k L ∞ (0 ,T ; V ) + k F k ,q T ≤ d ( y ) p E ( y, f ) (34) with d ( y ) := C ( α ) (cid:18) k y k L ∞ (0 ,T ; L (Ω)) | Ω | (cid:19) Cβ , C ( α ) := 2 C max(1 , e Cα | Ω | ) . (35) Proof.
Lemma 5 with p ⋆ = 1 and (8) lead to the result.With these notations, the term c ( y ) in (15) rewrites as c ( y ) = C (1 + s ) √ g ′ ] s d ( y ) s , ∀ ( y, f ) ∈ A , ∀ s ∈ (0 , . (36) Proof. (of Theorem 2) If the initialization ( y , f ) ∈ A is such that E ( y , f ) = 0, then the sequence( y k , f k ) k ∈ N is constant equal to ( y , f ) and therefore converges. We assume in the sequel that E ( y , f ) >
0. We are going to prove that, for any β < β ⋆ ( s ), there exists a constant M > y k ) k ∈ N defined by (19) enjoys the uniform property k y k k L ∞ (0 ,T ; L (Ω)) ≤ M, ∀ k ∈ N . (37)The convergence of the sequence ( y k , f k ) k ∈ N in A will then follow by proceeding as in Section 3.1. Remarkpreliminary that the assumption β < β ⋆ ( s ) implies 2 Cβ < s s +1 ≤ s ∈ (0 , Proof of the uniform property (37) for some M large enough - As for n = 0, from any initialization ( y , f )chosen in A , it suffices to take M larger than M := k y k L ∞ (0 ,T ; L (Ω)) . We then proceed by inductionand assume that, for some n ∈ N , k y k k L ∞ (0 ,T ; L (Ω)) ≤ M for all k ≤ n . This implies in particular that, d ( y k ) ≤ d M ( β ) := C ( α ) (cid:18) M | Ω | (cid:19) Cβ , ∀ k ≤ n c ( y k ) ≤ c M ( β ) := C (1 + s ) √ g ′ ] s d sM ( β ) , ∀ k ≤ n. (38)Then, we write that k y n +1 k L ∞ (0 ,T ; L (Ω)) ≤ k y k L ∞ (0 ,T ; L (Ω)) + P nk =0 λ k k Y k k L ∞ (0 ,T ; L (Ω)) . But, Lemma6 implies that k Y k k L ∞ (0 ,T ; L (Ω)) ≤ d M ( β ) p E ( y k , f k ) for all k ≤ n leading to k y n +1 k L ∞ (0 ,T ; L (Ω)) ≤ k y k L ∞ (0 ,T ; L (Ω)) + m d M ( β ) n X k =0 p E ( y k , f k ) . (39)Moreover, inequality (28) implies that P nk =0 p E ( y k , f k ) ≤ − p ( f λ ) p E ( y , f ) where p ( f λ ) is given by(25) with c = c M ( β ).Now, we take M large enough so that (1 + s ) /s c /sM ( β ) p E ( y , f ) ≥ (cid:18) C √ g ′ ] s (cid:19) /s C ( α ) /s (cid:18) M | Ω | (cid:19) Cβ s p E ( y , f ) ≥ . (40)Such M exists since p E ( y , f ) > M and since the left hand side is of order O ( M Cβ s ) with Cβ s >
0. We denote by M the smallest value of M such that (40) hold true.Then, from (25), we get that p ( f λ ) = 1 − s (1+ s ) s +1 c /sM ( β ) √ E ( y ,f ) and therefore11 − p ( f λ ) = (1 + s ) s +1 s c /sM ( β ) p E ( y , f )so that P nk =0 p E ( y k , f k ) ≤ (1+ s ) s +1 s c /sM ( β ) E ( y , f ). It follows from (39) that k y n +1 k L ∞ (0 ,T ; L (Ω)) ≤ k y k L ∞ (0 ,T ; L (Ω)) + m d M ( β ) (1 + s ) s +1 s c /sM ( β ) E ( y , f ) . The definition of c M ( β ) (see (38)) then gives k y n +1 k L ∞ (0 ,T ; L (Ω)) ≤k y k L ∞ (0 ,T ; L (Ω)) + m (1 + s ) s (cid:18) C [ g ′ ] s √ (cid:19) /s (cid:18) C ( α ) (cid:19) s E ( y , f ) (cid:18) M | Ω | (cid:19) (2 Cβ s +1) s . Now, we take
M > M , i.e. k y k L ∞ (0 ,T ; L (Ω)) + m (1 + s ) s (cid:18) C [ g ′ ] s √ (cid:19) /s (cid:18) C ( α ) (cid:19) s E ( y , f ) (cid:18) M | Ω | (cid:19) (2 Cβ s +1) s ≤ M. (41)Such M exists under the assumption β < β ⋆ ( s ) i.e. (2 Cβ )(2 s +1) s <
1. We denote by M the small-est value of M such that (41) holds true. Eventually, taking M := max( M , M , M ), we get that k y n +1 k L ∞ (0 ,T ; L (Ω)) ≤ M as well. We have then proved by induction the uniform property (37) for some M large enough. Proof of the convergence of the sequence ( y k , f k ) k ∈ N - In view of Lemma 5 with p ⋆ = 1, the uniformproperty (37) implies that the observability constant Ce C k g ′ ( y k ) k L ∞ (0 ,T ; Ld (Ω)) appearing in the estimatesfor ( Y k , F k ) (see Proposition 1) is uniformly bounded with respect to the parameter k . As a consequence,the constant c ( y k ) appearing in the instrumental estimate (15) is bounded by c M ( β ) given by (38).Consequently, the developments of Section 3.1 apply with c = c M ( β ). Theorem 2 then follows from theproof of Proposition 4. 14 emark 4. Remark that M := max( M , M ) since M ≥ M . The constant M can be made explicitsince the constraint (40) implies that (cid:18) C [ g ′ ] s √ (cid:19) /s C ( α ) /s (cid:18) M | Ω | (cid:19) Cβ s p E ( y , f ) ≥ . i.e. (cid:18) M | Ω | (cid:19) Cβ ≥ C ( α ) − p E ( y , f ) − s/ (cid:18) C √ g ′ ] s (cid:19) − / . In particular, M is large for small values of p E ( y , f ) , for any s > . On the other hand, theconstant M is implicit, hence whether M > M or M > M depend on the values of p E ( y , f ) and k y k L ∞ (0 ,T ; L (Ω)) . Remark that p E ( y , f ) can be large and k y k L ∞ (0 ,T ; L (Ω)) small, and vice versa. Exact controllability of (1) has been established in [11], under a growth condition on g , by meansof a Leray-Schauder fixed point argument that is not constructive. In this paper, under a slightlystronger growth condition and under the additional assumption that g ′ is uniformly H¨older continuouswith exponent s ∈ [0 , s after afinite number of iterations.In turn, our approach gives a new and constructive proof of the exact controllability of (1). Moreover,we emphasize that the method is general and may be applied to any other equations or systems - notnecessarily of hyperbolic nature - for which a precise observability estimate for the linearized problem isavailable: we refer to [15] addressing the case of the heat equation. Among the open issues, we mentionthe extension of this constructive approach to the case of the boundary controllability (see for instance[28]).Several comments are in order. Asymptotic condition.
The asymptotic condition ( H ) on g ′ is slightly stronger than the asymptoticcondition ( H ) made in [11]: this is due to our linearization of (1) which involves r → g ′ ( r ) while thelinearization (2) in [11] involves r → ( g ( r ) − g (0)) /r . There exist cases covered by Theorem 1 in whichexact controllability for (1) is true but that are not covered by Theorem 2. Note however that the example g ( r ) = a + br + cr ln / (1 + | r | ), for any a, b ∈ R and for any c > H ) as well as ( H s ) for any s ∈ [0 , s . Minimization functional.
Among all possible admissible controlled pair ( y, v ) ∈ A , we have selectedthe solution ( Y , F ) of (7) that minimizes the functional J ( v ) = k v k ,q T . This choice has led to theestimate (8) which is one of the key points of the convergence analysis. The analysis remains true whenone considers the quadratic functional J ( y, v ) = k w v k ,q T + k w y k for some positive weight functions w and w (see for instance [4]). Link with Newton method.
Defining F : A → L ( Q T ) by F ( y, f ) := ( ∂ tt y − ∆ y + g ( y ) − f ω ),we have E ( y, f ) = k F ( y, f ) k and we observe that, for λ k = 1, the algorithm (19) coincides with theNewton algorithm associated to the mapping F (see (5)). This explains the super-linear convergence15roperty in Theorem 2, in particular the quadratic convergence when s = 1. The optimization of theparameter λ k gives to a global convergence property of the algorithm and leads to the so-called dampedNewton method applied to F . For this method, global convergence is usually achieved with linear orderunder general assumptions (see for instance [8, Theorem 8.7]). As far as we know, the analysis of dampedtype Newton methods for partial differential equations has deserved very few attention in the literature.We mention [16, 25] in the context of fluids mechanics. A variant.
To simplify, let us take λ k = 1, as in the standard Newton method. Then, for each k ∈ N ,the optimal pair ( Y k , F k ) ∈ A is such that the element ( y k +1 , f k +1 ) minimizes over A the functional( z, v ) → J ( z − y k , v − f k ) with J ( z, v ) := k v k ,q T (control of minimal L ( q T ) norm). Alternatively, wemay select the pair ( Y k , F k ) so that the element ( y k +1 , f k +1 ) minimizes the functional ( z, v ) → J ( z, v ).This leads to the sequence ( y k , f k ) k ∈ N defined by ∂ tt y k +1 − ∆ y k +1 + g ′ ( y k ) y k +1 = f k +1 ω + g ′ ( y k ) y k − g ( y k ) in Q T ,y k = 0 , on Σ T , ( y k +1 ( · , , ∂ t y k +1 ( · , u , u ) in Ω . (42)In this case, for every k ∈ N , ( y k , f k ) is a controlled pair for a linearized wave equation, while, in the caseof the algorithm (19), ( y k , f k ) is a sum of controlled pairs ( Y j , F j ) for 0 ≤ j ≤ k . This formulation usedin [10] is different and the convergence analysis (at least in the least-squares setting) does not seem to bestraightforward because the term g ′ ( y k ) y k − g ( y k ) is not easily bounded in terms of p E ( y k , f k ). Initialization with the controlled pair of the linear equation.
The number of iterates to achieveconvergence (notably to enter in a super-linear regime) depends on the size of the value E ( y , f ). Anatural example of an initialization ( y , f ) ∈ A is to take ( y , f ) = ( y ⋆ , f ⋆ ), the unique solution ofminimal control norm of (1) with g = 0 (i.e., in the linear case). Under the assumption ( H ), this leadsto the estimate E ( y , f ) = 12 k g ( y ) k ≤ | g (0) | | Q T | + 2 Z Q T | y | (cid:0) α + β ln(1 + | y | ) (cid:1) . Local controllability when removing the growth condition ( H ) . If the real E ( y , f ) is smallenough, then we may remove the growth condition ( H ) on g ′ . Proposition 5.
Assume g ′ satisfies ( H s ) for some s ∈ [0 , . Let ( y k , f k ) k> be the sequence of A definedin (19) . There exists a constant C ([ g ′ ] s ) such that if E ( y , f ) ≤ C ([ g ′ ] s ) , then ( y k , f k ) k ∈ N → ( y, f ) in A where f is a null control for y solution of (1) . Moreover, the convergence is at least linear and is atleast of order s after a finite number of iterations.Proof. In this proof, the notation k · k ∞ ,d stands for k · k L ∞ (0 ,T ; L d (Ω)) . We note D := C (1+ s ) √ [ g ′ ] s and e k = c ( y k ) E ( y k , f k ) s/ with c ( y ) := Dd ( y ) s and d ( y ) := Ce C k g ′ ( y ) k ∞ ,d . (23) then reads p E ( y k +1 , f k +1 ) ≤ min λ ∈ [0 ,m ] (cid:0) | − λ | + λ s e k (cid:1)p E ( y k , f k ) . (43)We write | g ′ ( y k ) − g ′ ( y k − λ k Y k ) | ≤ [ g ′ ] s | λ k Y k | s so that k g ′ ( y k +1 ) k L ∞ (0 ,T ; L d (Ω)) ≤ k g ′ ( y k ) k ∞ ,d + (cid:0) [ g ′ ] s λ sk k ( Y k ) s k ∞ ,d (cid:1) + 2 k g ′ ( y k ) k ∞ ,d [ g ′ ] s λ sk k ( Y k ) s k ∞ ,d and e C k g ′ ( y k +1 ) k ∞ ,d ≤ e C k g ′ ( y k ) k ∞ ,d e C (cid:0) [ g ′ ] s λ sk k ( Y k ) s k ∞ ,d (cid:1) e C k g ′ ( y k ) k ∞ ,d (cid:0) [ g ′ ] s λ sk k ( Y k ) s k ∞ ,d (cid:1) c ( y k +1 ) c ( y k ) ≤ (cid:18) e C (cid:0) [ g ′ ] s λ sk k ( Y k ) s k ∞ ,d (cid:1) e C k g ′ ( y k ) k ∞ ,d (cid:0) [ g ′ ] s λ sk k ( Y k ) s k ∞ ,d (cid:1)(cid:19) s . We infer that k ( Y k ) s k ∞ ,d = k Y k k s ∞ ,sd . Moreover, (8) leads to k Y k k s ∞ ,sd ≤ d s ( y k ) E ( y k , f k ) s/ = c ( y k ) s s D s s E ( y k , f k ) s/ ≤ D − s s c ( y k ) E ( y k , f k ) s/ using that c ( y k ) ≥ C is necessary). Consequently, e C (cid:0) [ g ′ ] s λ s k ( Y k ) s k ∞ ,d (cid:1) ≤ e C (cid:0) [ g ′ ] s λ s D − s s e k (cid:1) := e C e k . Similarly, k g ′ ( y k ) k ∞ ,d k ( Y k ) s k ∞ ,d ≤ k g ′ ( y k ) k ∞ ,d d s ( y k ) E ( y k , f k ) s/ ≤ k g ′ ( y k ) k ∞ ,d (cid:18) Ce C k g ′ ( y ) k L ∞ (0 ,T ; Ld (Ω)) (cid:19) s E ( y k , f k ) s/ ≤ (cid:18) Ce C k g ′ ( y ) k L ∞ (0 ,T ; Ld (Ω)) (cid:19) s +1 E ( y k , f k ) s/ ≤ c ( y k ) D E ( y k , f k ) s/ = e k D using that a ≤ Ce Ca for all a ≥ C > e C k g ′ ( y k ) k ∞ ,d (cid:0) [ g ′ ] s λ sk k ( Y k ) s k ∞ ,d (cid:1) ≤ e C [ g ′ ] s λ sk ekD := e C e k and then c ( y k +1 ) c ( y k ) ≤ ( e C e k + C e k ) s . By multiplying (43) by c ( y k +1 ), we obtain the inequality e k +1 ≤ min λ ∈ [0 ,m ] (cid:0) | − λ | + e k λ s (cid:1) ( e C e k + C e k ) s e k . If 2 e k <
1, the minimum is reached for λ = 1 leading e k +1 e k ≤ e k ( e C e k + C e k ) s . Consequently, if the ini-tial guess ( y , f ) belongs to the set { ( y , f ) ∈ A , e < / , e ( e C e + C e ) s < } , the sequence ( e k ) k> goes to zero as k → ∞ . Since c ( y k ) ≥ k ∈ N , this implies that the sequence ( E ( y k , f k )) k> goesto zero as well. Moreover, from (8), we get D k ( Y k , F k ) k H ≤ e k p E ( y k , f k ) and repeating the argumentsof the proof of Proposition 4, we conclude that the sequence ( y k , f k ) k> converges to a controlled pairfor (1).These computations does not use the assumption ( H ) on the nonlinearity g . However, the smallnessassumption on e requires a smallness assumption on E ( y , f ) (since c ( y ) > g (0) = 0, the smallness assumption on E ( y , f ) is achieved as soon as k ( u , u ) k V is small enough. Therefore, the convergence result statedin Proposition 5 is equivalent to the local controllability property for (1). Proposition 5 can actually beseen as a consequence of the usual convergence of the Newton method: when E ( y , f ) is small enough,i.e., when the initialization is close enough to the solution, then λ k = 1 for every k ∈ N and we recoverthe standard Newton method. Weakening of the condition ( H s ) . Given any s ∈ [0 , g ∈ C ( R ) the followinghypothesis :( H ′ s ) There exist α, β, γ ∈ R + such that | g ′ ( a ) − g ′ ( b ) | ≤ | a − b | s (cid:0) α + β ( | a | γ + | b | γ ) (cid:1) , ∀ a, b ∈ R which coincides with ( H s ) if γ = 0 for α + β = [ g ′ ] s . If γ ∈ (0 ,
1) is small enough and related to theconstant β appearing in the growth condition ( H ), Theorem 2 still holds if ( H s ) is replaced by the17eaker hypothesis ( H ′ s ). Precisely, if g satisfies ( H ) and ( H ′ s ) for some s ∈ (0 , y k , f k ) k ∈ N defined by (19) fulfills the estimate E ( y k +1 , f k +1 ) ≤ E ( y k , f k ) min λ ∈ [0 ,m ] (cid:18) | − λ | + λ s c ( y k ) E ( y k , f k ) s/ (cid:19) with c ( y ) := s ) √ (cid:18)(cid:0) α +2 β k y k k γ ∞ , γ )+ βm γ d ( y ) γ E ( y , f ) γ (cid:19) d ( y ) s and d ( y ) := Ce C k g ′ ( y ) k L ∞ (0 ,T ; Ld (Ω)) .Using Lemma 5 with p ⋆ = 6 γ ≤ k y k k L ∞ (0 ,T ; L (Ω)) ) k ∈ N is uniformly bounded under the condition γ +(2 Cβ )(1+2 s ) s < y k , f k ) k ∈ N . A Appendix: controllability results for the linearized wave equa-tion
We recall in this section some a priori estimates for the linearized wave equation with potential in L ∞ (0 , T ; L d (Ω)) and right hand side in L ( Q T ). We first recall the crucial observability type estimateproved in [11, Theorem 2.2] (see also [19, Theorem 2.1]). Proposition 6. [11] Assume that ω and T satisfy the assumptions of Theorem 1. For any A ∈ L ∞ (0 , T ; L d (Ω)) , and ( φ , φ ) ∈ H := L (Ω) × H − (Ω) , the weak solution φ of ∂ tt φ − ∆ φ + Aφ = 0 , in Q T ,φ = 0 , on Σ T , ( φ ( · , , φ t ( · , φ , φ ) , in Ω , (44) satisfies the observability inequality k φ , φ k H ≤ Ce C k A k L ∞ (0 ,T ; Ld (Ω)) k φ k ,q T for some C > only depend-ing on Ω and T . Classical arguments then lead to following controllability result.
Proposition 7. [11] Let A ∈ L ∞ (0 , T ; L d (Ω)) , B ∈ L ( Q T ) and ( z , z ) ∈ V . Assume that ω and T satisfy the assumptions of Theorem 1. There exists a control function u ∈ L ( q T ) such that the solutionof ∂ tt z − ∆ z + Az = u ω + B, in Q T ,z = 0 , on Σ T , ( z ( · , , z t ( · , z , z ) , in Ω , (45) satisfies ( z ( · , T ) , z t ( · , T )) = (0 , in Ω . Moreover, the unique pair ( u, z ) of minimal control norm satisfies k u k ,q T + k ( z, ∂ t z ) k L ∞ (0 ,T ; V ) ≤ C (cid:18) k B k + k z , z k V (cid:19) e C k A k L ∞ (0 ,T ; Ld (Ω)) (46) for some constant C > only depending on Ω and T . Let p ⋆ ∈ N ⋆ such that p ⋆ < ∞ if d = 2 and p ⋆ < d = 3. We next discuss some properties of theoperator K : L ∞ (0 , T ; L p ⋆ (Ω)) → L ∞ (0 , T ; L p ⋆ (Ω)) defined by K ( ξ ) = y ξ , a null controlled solution ofthe linear boundary value problem (2) with the control f ξ of minimal L ( q T ) norm. Proposition 7 with B = − g (0) gives k ( y ξ , ∂ t y ξ ) k L ∞ (0 ,T ; V ) ≤ C (cid:16) k u , u k V + k g (0) k (cid:17) e C k b g ( ξ ) k L ∞ (0 ,T ; Ld (Ω)) (47)where the function ˆ g is defined in (2). We assume that g ∈ C ( R ) satisfies the following asymptoticcondition (slightly weaker than ( H )): there exists a β small enough such that lim sup | r |→∞ | g ( r ) || r | ln / | r | ≤ β ,i.e. 18 H ′ ) There exist α ≥ β ≥ | g ( r ) | ≤ α + β (1 + | r | )) ln / (1 + | r | ) forevery r in R .This implies that ˆ g satisfies | b g ( r ) | ≤ α + β ln / (1 + | r | ) for every r ∈ R and some constant α > b g ( ξ ) ∈ L ∞ (0 , T ; L d (Ω)) for any ξ ∈ L ∞ (0 , T ; L p ⋆ (Ω)). Assuming 2 Cβ ≤ e C k b g ( ξ ) k L ∞ (0 ,T ; Ld (Ω)) ≤ C (cid:18) k ξ k L ∞ (0 ,T ; L p⋆ (Ω)) | Ω | /p ⋆ (cid:19) Cβ , ∀ ξ ∈ L ∞ (0 , T ; L p ⋆ (Ω))for some C = C ( α ). Using (47), we then infer that k y ξ k L ∞ (0 ,T ; L p⋆ (Ω)) ≤ C (cid:16) k u , u k V + k g (0) k (cid:17) C (cid:18) k ξ k L ∞ (0 ,T ; L p⋆ (Ω)) | Ω | /p ⋆ (cid:19) Cβ , ∀ ξ ∈ L ∞ (0 , T ; L p ⋆ (Ω)) . Taking β small enough so that 2 Cβ <
1, we conclude that there exists
M > k ξ k L ∞ (0 ,T ; L p⋆ (Ω)) ≤ M implies k K ( ξ ) k L ∞ (0 ,T ; L p⋆ (Ω)) ≤ M . This is the argument (introduced in [29] for the one dimensionalcase and) implicitly used in [19] to prove the controllability of (1). Note that, in contrast to β , M dependson k u , u k V (and increases with k u , u k V ).The following result gives an estimate of the difference of two controlled solutions. Lemma 7.
Let A ∈ L ∞ (0 , T ; L d (Ω)) , a ∈ L ∞ (0 , T ; L d + ǫ (Ω)) for any ǫ > , B ∈ L ( Q T ) and ( u , u ) ∈ V . Let u and v be the null controls of minimal L ( q T ) norm for y and z respectively solutions of ∂ tt y − ∆ y + Ay = u ω + B in Q T ,y = 0 on Σ T , ( y ( · , , ∂ t y ( · , u , u ) in Ω , (48) and ∂ tt z − ∆ z + ( A + a ) z = v ω + B in Q T ,z = 0 on Σ T , ( z ( · , , ∂ t z ( · , u , u ) in Ω . (49) Then, k y − z k L ∞ (0 ,T ; H (Ω)) ≤ C k a k L ∞ (0 ,T ; L d + ǫ (Ω)) (cid:0) k B k + k u , u k V (cid:1) e C k A + a k L ∞ (0 ,T ; Ld (Ω)) e C k A k L ∞ (0 ,T ; Ld (Ω)) for some constant C > only depending on Ω and T .Proof. The controls of minimal L ( q T ) norm for y and z are given by u = φ ω and v = φ a ω where φ and φ a respectively solve the adjoint equations ∂ tt φ − ∆ φ + Aφ = 0 in Q T ,φ = 0 on Σ T , ( φ ( · , , ∂ t φ ( · , φ , φ ) in Ω , ∂ tt φ a − ∆ φ a + ( A + a ) φ a = 0 in Q T ,φ = 0 on Σ T , ( φ ( · , , ∂ t φ ( · , φ a, , φ a, ) in Ω , for some appropriate ( φ , φ ) , ( φ a, , φ a, ) ∈ H . In particular, φ, φ a ∈ C ([0 , T ]; L (Ω)) ∩ C ([0 , T ]; H − (Ω)).Hence Z := z − y solves ∂ tt Z − ∆ Z + ( A + a ) Z = Φ1 ω − ay in Q T ,Z = 0 on Σ T , ( Z ( · , , ∂ t z ( · , ,
0) in Ω , (50)19nd Φ := φ a − φ solves ∂ tt Φ − ∆Φ + ( A + a )Φ = − aφ in Q T , Φ = 0 on Σ T , (Φ( · , , ∂ t Φ( · , φ a, − φ , φ a, − φ ) in Ω . In particular (since a ∈ L ∞ (0 , T ; L d + ǫ (Ω)) and φ ∈ L ∞ (0 , T ; L (Ω))), we get that aφ ∈ L ∞ (0 , T ; H − (Ω))and therefore (Φ , Φ t ) ∈ C ([0 , T ]; H ), see [13, Theorem 2.3]. We decompose Φ := Ψ + ψ where Ψ and ψ solve respectively ∂ tt Ψ − ∆Ψ + ( A + a )Ψ = 0 in Q T , Ψ = 0 on Σ T , (Ψ( · , , ∂ t Ψ( · , φ a, − φ , φ a, − φ ) in Ω , ∂ tt ψ − ∆ ψ + ( A + a ) ψ = − aφ in Q T ,ψ = 0 on Σ T ,ψ ( · , , ∂ t ψ ( · , ,
0) in Ω , and we deduce that Ψ1 ω is the control of minimal L ( q T ) norm for Z solution of ∂ tt Z − ∆ Z + ( A + a ) Z = Ψ1 ω + (cid:16) ψ ω − ay (cid:17) in Q T ,Z = 0 on Σ T , ( Z ( · , , ∂ t Z ( · , ,
0) in Ω . Proposition 7 implies that k Ψ k ,q T + k ( Z, ∂ t Z ) k L ∞ (0 ,T ; V ) ≤ C k ψ ω − ay k e C k A + a k L ∞ (0 ,T ; Ld (Ω)) . Moreover, [19, Lemma 2.4] applied to ψ leads to k ( ψ, ψ t ) k L ∞ (0 ,T ; H ) ≤ k aϕ k L ∞ (0 ,T ; H − (Ω)) e C k A + a k L ∞ (0 ,T ; Ld (Ω)) and k ψ k L ( q T ) ≤ C k a k L ∞ (0 ,T ; L d + ǫ (Ω)) k φ k e C k A + a k L ∞ (0 ,T ; Ld (Ω)) . But, using again [19, Lemma 2.4], we inferthat k φ k ≤ C k φ , φ k H e C k A k L ∞ (0 ,T ; Ld (Ω)) while [19, Theorem 2.1] gives k φ , φ k H ≤ Ce C k A k L ∞ (0 ,T ; Ld (Ω)) k φ k ,q T .Since u = φ ω , we obtain k φ k ≤ C e C k A k L ∞ (0 ,T ; Ld (Ω)) k u k ,q T e C k A k L ∞ (0 ,T ; Ld (Ω)) and then k ψ k L ( q T ) ≤ C k a k L ∞ (0 ,T ; L d + ǫ (Ω)) e C k A k L ∞ (0 ,T ; Ld (Ω)) e C k A + a k L ∞ (0 ,T ; Ld (Ω)) k u k ,q T from which we deduce that k Z k L ∞ (0 ,T ; H (Ω)) ≤ C (cid:16) k ψ k ,q T + k a k L ∞ (0 ,T ; L d + ǫ (Ω)) k y k (cid:17) e C k A + a k L ∞ (0 ,T ; Ld (Ω)) ≤ C k a k L ∞ (0 ,T ; L d + ǫ (Ω)) (cid:18) k B k + k u , u k V (cid:19) e C k A + a k L ∞ (0 ,T ; Ld (Ω)) e C k A k L ∞ (0 ,T ; Ld (Ω)) leading to the result.This result allows to establish the following property for the operator K . Lemma 8.
Under the assumptions done in Theorem 1, let M = M ( k u , u k V , β ) be such that K maps B L ∞ (0 ,T ; L d + ǫ (Ω)) (0 , M ) into itself and assume that ˆ g ′ ∈ L ∞ ( R ) . For any ξ i ∈ B L ∞ (0 ,T ; L d + ǫ (Ω)) (0 , M ) , i = 1 , , there exists c ( M ) > such that k K ( ξ ) − K ( ξ ) k L ∞ (0 ,T ; H (Ω)) ≤ c ( M ) k ˆ g ′ k ∞ k ξ − ξ k L ∞ (0 ,T ; L d + ǫ (Ω)) . roof. For any ξ i ∈ B L ∞ (0 ,T ; L p⋆ (Ω)) (0 , M ), i = 1 ,
2, let y ξ i = K ( ξ i ) be the null controlled solution of ∂ tt y ξ i − ∆ y ξ i + y ξ i b g ( ξ i ) = − g (0) + f ξ i ω in Q T ,y ξ i = 0 on Σ T , ( y ξ i ( · , , ∂ t y ξ i ( · , u , u ) in Ω , with the control f ξ i ω of minimal L ( q T ) norm. We observe that y ξ is solution of ∂ tt y ξ − ∆ y ξ + y ξ b g ( ξ ) + y ξ ( b g ( ξ ) − b g ( ξ )) = − g (0) + f ξ ω in Q T ,y ξ = 0 on Σ T , ( y ξ ( · , , ∂ t y ξ ( · , u , u ) in Ω . It follows from Lemma 7 applied with B = − g (0), A = ˆ g ( ξ ), a = ˆ g ( ξ ) − ˆ g ( ξ ), that k y ξ − y ξ k L ∞ (0 ,T ; H (Ω)) ≤ A ( ξ , ξ ) k b g ( ξ ) − b g ( ξ ) k L ∞ (0 ,T ; L d + ǫ (Ω)) (51)where the positive constant A ( ξ , ξ ) := C (cid:18) k g (0) k + k u , u k V (cid:19) e C k ˆ g ( ξ ) k L ∞ (0 ,T ; Ld (Ω)) e C k ˆ g ( ξ ) k L ∞ (0 ,T ; Ld (Ω)) is bounded by some c ( M ) > ξ i ∈ B L ∞ (0 ,T ; L d + ǫ (Ω)) (0 , M ). The result follows from (51). Remark 5.
By Lemma 8, if k ˆ g ′ k ∞ < /c ( M ) then the operator K : L ∞ (0 , T ; L d + ǫ (Ω)) → L ∞ (0 , T ; L d + ǫ (Ω)) is contracting. Note however that the bound depends on the norm k u , u k V of the initial data to be con-trolled. References [1]
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