The uniform controllability property of semidiscrete approximations for the parabolic distributed parameter systems in Banach spaces

The unif orm c on tr oll abil ity prop er t y o f se midis cr et e a pproximat ions fo r t he par ab olic dis tri bute d para met er sys tems in Ba nac h spa ces Th uy NGUYEN ∗ Abstract The probl em we co nsider in this w ork is to minimize the L q -norm ( q > 2) of the semidiscrete co n trols. As sho wn in [L T06], under the main appro ximation assumption s that the discret ized semigroup is uniformly analytic and th at the degree of un b ound edness of con trol operator is lo w er than 1/2, the uniform con trollabil it y property of semidiscrete ap pro ximations for th e parabolic systems is ac hiev ed in L 2 . In the presen t paper, w e sho w that the uniform con trollabil it y property still con tin ue to b e asserted in L q ( q > 2) ev en with the con- dition th at th e de gree of un b ounded ness of con trol operator is greater than 1/2. Moreo v er, the minimizati on pro cedu re to compute the ap- pro ximation con trols is p rovi ded. An e xample of app lication is imple- men ted fo r the one dimensiona l heat equat ion with Diric hlet b oundary con trol. 1 In tro du ction Consider an in finite dimensional linear con trol sys tem ˙ y ( t ) = Ay ( t ) + B u ( t ) , y ( 0 ) = y 0 , (1) where the state y(t) b elongs to a reflexiv e Banac h space X, the con trol u(t) b elongs to a reflexiv e Banac h sp ace U, A : D ( A ) → X is a n op erator, and ∗ Univ ersité d’Orléa ns, Labor atoire Mathématiques et Applications, Ph ysique Mathé- matiques d’O rléans (MAPMO), B âtimen t de Mathématiques, B.P . 6 759, 450 67 Orléans cedex 2, FRANCE. Email address : nthu yc8@ya hoo.com 1 B is a con trol op erator (in general, unbounded) on U . Discretizing this par- tial differen tial equation by us ing, fo r instance, a finite difference or a finite elemen t sc heme, leads to a family of finite dime nsional linear con tro l sy stems . y h ( t ) = A h y h ( t ) + B h u h ( t ) , y h (0) = y 0 h , (2) where y h ( t ) ∈ X h and u h ( t ) ∈ U h , for 0 < h < h 0 . Let y 1 ∈ X . If the control system (1) is con trollable in time T th en there exists a solution y ( . ) of (1) associated with a con trol u suc h that y ( T ) = y 1 . As kno wn, w e hav e man y a v ailable methods in order to iden tify the con t rolla- bilit y . W e refer to J.-L. Lions [L88] for a w ell-kno wn metho d in attainning the con trol of the min imal L 2 norm- the so called HUM (HUM stands for Hilbert uniquenes s method). In this w ork, how ev er, w e in v estigate a metho d whic h w e can ac hiev e the minimiz ation pro cedure in L q norm ( q > 1) . Namely , w e will establis h some conditions ensuring the e xistence and con ve rgence o f the discretize d con trol of the mi nimal L q norm min 1 q Z T 0 k u h ( t ) k q dt ( q > 1) . (3) Necessary conditions for optimal con trol in finite dimensional state space w ere deriv ed b y P on try agin et al [PS62] (se e also [E], [T05]). The Maxim um Principle a s a set of nece ssary conditions for optimal con trol in infinite di- mensional sp ace has b een studied b y man y authors. Since it is w ell kn ow n that the Maxim um Principle ma y b e false in infinite dimensional space, there are still man y papers that giv e some conditions to ensure that the Maxim um Principle remains true. It w as Li and Y a o [L Y85] who used the Eidelhe it sep- aration theorem a nd of the Uhl’s theorem in order to exten d the Maxim um Principle to a large class of problems in infinite dimensional space s wh en the target set is con v ex and the final time T is fix ed. Add itionally , the authors of [F87], [FF 91], [L Y91], by making use of Ek eland’s v ariational principle, giv e some conditions on the reac hable set and on the target set in order to get an extension of Maxim um Principle. Nev ertheless, the problem is that when w e applied the result of [F87], [L Y91] for the system (1) in the case the final state and final time are fixed, the finite-codimensional condition in [F87], [L Y9 1] do es not satisfy for the system (1) in general. Hence w e c annot use Maxim um Principle in our problem. F ortunately , thank to the F enc hel- Ro c k afellar dualit y theorem whic h is used in the same manner in [CGL94], [GLH08], the constrain ted minimization of the function can be replaced b y the unc onstrain ted minimization problem of corresponding conjugate func- tion. Therefore, we will consider the ab o v e minimization pro cedu re in the same framew ork with [GLH08]. 2 The uniform controllabilit y is an imp ortan t area of con trol theory re- searc h and it has b een the sub j ect of many pap ers in recen t y ears. The main goal of this article is to establish conditions ensuring a uniform con trolla- bilit y prop ert y of the f amily of discretized con trol systems (2) in L q and to establish a computationally feasible approximation metho d for iden tifying con trollabilit y . It is we ll known that con trollabilty and observ abilit y are dual asp ects of the same problem. W e therefore will fo cus on the uniform observ abilit y whic h is shown to hold when the observ abilit y constan t of the finite dimen- sional appro ximation syste ms do es not dep end on h . Some relev an t reference s concerning this prop ert y has b een inv estigated by man y authors in series of articles [IZ 99], [LZ98], [LZ02], [NZ03], [Zua99], [Zua02], [Zua04], [Zua05], [Zua06], [BHL10a], [BHL10b] and [L T06]. F or finite difference sc hemes, a uni- form observ abilit y prop ert y holds for one-dimensional heat equation [LZ02], b eam eq uation [LZ98], Sc hro dinger equations [Zua05], but does not h old for 1-D w av e equations [IZ 99]. This is due to the fact that the discrete dynam- ics generates high frequency spurious solutions for whic h the g roup v elo cit y v anishes that do not exist a t the con tin uous lev el. T o o v ercome these high frequency spurious fo r w a v e equations , [Zua05] sho w ed s ome remedies suc h that T yc honoff’s regularization, m ultigrid m etho d, mixed finite elemen t and filtering of high freq uency , etc. T o our k now ledge, in 1-D heat equ ation case, due to the fact that the dis- sipativ e effect of the 1-D heat equation acts as a filtering mec hanicsm by itself and it is strong enough to exc lude high frequency spurious osc illations[LZ98]. Ho w ev er, the situation is more complex in m ulti-dimensional. The coun ter- example is sho wn in [Zua06] for the sim plest finite difference semi-discretization sc heme for the heat eq uation in the square. In recen t w orks in L 2 -norm, b y means of discrete Carleman inequali- ties, the authors in [BHL10a], [BHL10b] obtain the w eak uniform observ- abilit y inequalit y fo r parab olic case b y adding reminder terms of the form e − C h − 2 k ψ h ( T ) k 2 L 2 (Ω) whic h v anishes asymptotically as h → 0 . Moreov er, as in [L T06], the appro ximate con trollability is deriv ed from using semigroup argumen ts and in tro ducing a v anishing term of the form h β k ψ h ( T ) k 2 L 2 (Ω) for some β > 0 . In fact, an efficien t computing the n ull con trol for a n umerical approx - imation sc heme of the heat equation is itself a difficult problem. Acc o ding to the pioneering w ork of Carthel, G lo winski and Lions in [CGL94], the n ull con trol problem i s reduced to the minimiz ation of a dual conjugate function with respect to final condition of the adjoint state. Ho w ev er, as a consequence of high regularizing prop erty of the heat k enel, this final condition do es not b elong to L 2 , but a muc h large space that can hardly b e appro ximated by 3 standard tec hniques in n umerical analys ic. Recen tly , A. Munc h and collab o- rators ha v e dev elop ed some feasible n umeric als suc h that the transm utation metho d, v ariational approac h, dual and primal algorithms allo w to more ef- ficien tly compute the n ull con trol (see in series [CM09], [CM10], [MZ10], [PM10]). The discretiz ation framew ork in this pap er is the same spirit as [L T06], [L T00]. In [L T06], under standard ass umptions on the discretization pro cess and for an exactly n ull con trollable parab olic system (1), if the degree of un b oundedness of the con trol op erator is lo w er than 1/2 then the semidisc rete appro ximation mo dels are uniformly con trollable and they also sho w ed that for the (2), the minimiz ing of the cost function of discretized con trol with pow er q = 2 is obtained . In this article, w e pro v e the exis tence of the minim um of the cost function of discre tized c ontrol p o w er q ( q > 2) for t yp e (2), in the case the op erator A generates an analytic semigroup. Of course, due to regularization prop erties, the control sys tem (1) is not ex actly c ontrollable in general. Hence, w e fo cus on exact n ull controllabilit y . Our main result, theorem 3.1, states that for exactly null con trollable parab olic system (1) and unde r standard approx i- mation assumptions, if the discretized semigroup is uniformly analytic, a nd if the degree of un b oundedness of the con trol operator B with re- sp ect to A is greater than 1/ 2 , then a uniform observ abilit y ineq uality in ( L p ) is pro v ed. W e stress that w e do not p rov e uniform exact n ull con trolla- bilit y property for the approxim ating system (2). Moreov er, a minimization pro cedure to compute the approx imation controls is prov ided. The outline of the paper is as follow s. In Section 2, w e briefly rev iew some w ell-kno wn f acts on con trollabilit y of linear partial differen tial equation in reflexiv e Banac h spaces. In Section 3, w e consider the existence and unique solution of the minimiz ation problem in contin uous case . By making use of the F enc hel-Ro c k afellar dualit y theorem, w e giv es a constructiv e w a y t o build the con trol of minimal L q norm. The main result is stated in Section 4 and pro v ed in Section 5. An example of application and n umerical sim ulations are pro vided in Section 6, for the one-dimensional heat equation with Diric hlet b oundary con trol. An Appendix is dev oted to the proof of a lemm a. 4 2 A short review on con trollabilit y of linear partial differen tial equations in reflexiv e Ba- nac h s paces In this section, it is con v enien t to first ha ve a quic k lo ok to con trollabilit y of infin ite dimensional linear con trol s ystems in reflexiv e Banac h sp aces (see more [CT06], [P83], [ TW07]). The notation L ( E , F ) stands for the set of linear con t in uous mappings from E to F , where E and F are reflexiv e Banac h spaces. Let X be a reflexiv e Banac h space. Denote b y <, > X the inner pro duct on X , and b y k . k X the asso ciated norm. Let S ( t ) denote a strongly contin uous semigroup on X , of generator ( A, D ( A )) . Let X − 1 denote the completion o f X for norm k x k − 1 = k ( β I − A ) − 1 x k , where β ∈ ρ ( A ) is fixed. Note t hat X − 1 do es not depend on the sp ecific v alue of β ∈ ρ ( A ) . The space X − 1 is isomor- phic to ( D ( A ∗ )) ′ , the dual sp ace of D ( A ∗ ) with respect to the piv ot space X, and X ⊂ X − 1 , with a con tin uous and dense em b edding. The semigroup S ( t ) extends to a semigroup on X − 1 , still denoted S ( t ) , whose generator is an extension of the op erator A , still den oted A . With these notations, A is a linear op erator from X to X − 1 . Let U be a reflexiv e Banac h space. De note b y < , > U the inner pro duct on U , and by k . k U the associated norm. A linear contin uous op erator B : U → X − 1 is admissible for the semigroup S ( t ) if ev ery solution of y ′ = Ay ( t ) + B u ( t ) , (4) with y (0) = y 0 ∈ X and u ( . ) ∈ L q (0 , + ∞ ; U ) , satisfies y ( t ) ∈ X , for ev ery t ≥ 0 . The solution of e quation (1) is understoo d in the mild sense , i.e, y ( t ) = S ( t ) y (0) + Z T 0 S ( t − s ) B u ( s ) ds, (5) for ev ery t ≥ 0 . F or T > 0 , define L T : L q (0 , T ; U ) → X − 1 b y L T u = Z T 0 S ( T − s ) B u ( s ) ds. (6) A control op erator B ∈ L ( U, X − 1 ) is admiss ible, if and only if I mL T ⊂ X , for some (and hence for ev ery) T > 0 . 5 The adjoin t L ∗ T of L T satisfies L ∗ T : X ∗ → ( L q (0 , T ; U )) ∗ = L p (0 , T ; U ∗ ) L ∗ T ψ ( t ) = B ∗ S ( T − t ) ∗ ψ (7) , a.e on [0,T ] for ev ery ψ ∈ D ( A ∗ ) . Moreo v er, w e ha v e k L ∗ T ψ k = sup k u k q ≤ 1 Z T 0 < B ∗ S ( T − s ) ∗ ψ , u ( s ) > ds, (8) for ev ery ψ ∈ X ∗ . Let B ∈ L ( U, X − 1 ) denote an admis sible con trol op erator. W e use t w o follo wing lemmas (for the proofs w e refer to [T W07 ]) Lemma 1. Z , X ar e r eflexive Banac h sp ac es. G ∈ L ( Z, X ) then the fol lowing statements a r e e quiva lent: • G i s onto. • G ∗ b ound e d fr o m b elow i.e ther e exists C > 0 such that k G ∗ x k Z ≥ C k x k X every x ∈ X. Lemma 2. Z 1 , Z 2 , Z 3 ar e r eflexive Banach sp ac es. And f ∈ L ( Z 1 , Z 3 ) , g ∈ L ( Z 2 , Z 3 ) . Then the fol lowing statements ar e e quivalent: • I mf ⊂ I mg . • Ther e exists a c onstant C > 0 such that : k f ∗ z k Z 1 ≤ C k g ∗ z k Z 2 for every z ∈ Z 3 . • Ther e exists an op er ator h ∈ L ( Z 1 , Z 2 ) such that f = g h . W e state some concepts as follo ws F or y 0 ∈ X , and T > 0 , the sys tem (4) is exac tly controllable from y 0 in time T if for ev ery y 1 ∈ X , there exists u ( . ) ∈ L q (0 , T ; U ) so that the correspo nding solution (4), w ith y (0) = y 0 satisfies y ( T ) = y 1 . In fact that the sys tem (4) is exactly controllable from y 0 in time T if and only if L T is on to, that is I mL T = X . Making use of Lemma 1 , there exists C > 0 suc h that C k ψ k X ≤ k L ∗ T ψ k = sup k u k q ≤ 1 Z T 0 < B ∗ S ( T − s ) ∗ ψ , u ( s ) > ds ≤ sup k u k q ≤ 1 Z T 0 k B ∗ S ( T − t ) ∗ ψ kk u ( t ) k dt ≤ ( Z T 0 k B ∗ S ( t ) ∗ ψ k p dt ) 1 p . 6 Therefore, the system (4) is exactly con trollable in time T if and only if Z T 0 k B ∗ S ( t ) ∗ ψ k p dt ≥ C k ψ k p X . (9) F or T > 0 , the system (4) is s aid to b e exactly n ull con trollable in time T if for e v ery y 0 ∈ X , there ex ists u ( . ) ∈ L q (0 , T ; U ) so that the corres p onding solution of ( 4), with y (0) = y 0 satisfies y ( T ) = 0 . This means that the system (4) is exactly n ull controllable in time T if and only if I mS ( T ) ⊂ I mL T . Making use o f Lemma 2 and the same arg umen t as ab o v e, there exists C > 0 such that C k S ( T ) ∗ ψ k X ≤ k L ∗ T ψ k ≤ ( Z T 0 k B ∗ S ( t ) ∗ ψ k p dt ) 1 p . Th us, the sys tem (4) is exactly n ull con trollable in time T if and only if Z T 0 k B ∗ S ( t ) ∗ ψ k p dt ≥ C k S ( T ) ∗ ψ k p X . (10) 3 Dualit y The goal of this section is to sho w that, with using dualit y argumen ts and F enc hel- Ro ck afellar theorem w e can ac hiev e the con trol of minimal L q norm ( q > 1) for the con tin uous framew ork. Consider the sy stem : ( . y ( t ) = Ay ( t ) + B u ( t ) on Q T = (0 , T ) × Ω y ( 0 ) = y 0 (11) where B is admissible and A generates an analytic semigroup S ( t ) in the reflexiv e Banac h space X . Our aim is to mimimize the follo wing functional: ( Minimize J ( u ) = 1 q R T 0 k u k q dt ( q > 1) Sub j ect to u ∈ E (12) where E = { u ∈ U : u steerin g the system from y 0 at time z ero to y(T)=0 } . Theorem 1. The pr oblem (12) has a unique so lution u . 7 Pr o of. First of all, w e sho w the exis tence of the solution of the optimal prob- lem. Consider a min imizing sequence ( u n ) n ∈ N of con trols on [0 , T ] , i.e, Z T 0 k u n k q dt con v erges to inf J ( u ) as n → + ∞ . (13) Hence, ( u n ) n ∈ N b ounded in L q (0 , T ; U ) . Since U is reflexiv e space and q < + ∞ , the n L q (0 , T ; U ) is also reflex iv e. Th us, up to a sequence , u n con v erges we akly to u in L q . Note that the tra- jectory y n (resp. y ) asso ciated w ith the con trol u n (resp. u ) on [0,T] through the sys tem ˙ y n = Ay n + B u n , y n (0) = y 0 , and the solution of the ab ov e system is express ed in form y n ( t ) = S ( t ) y 0 + Z T 0 S ( T − s ) B u n ds. A pass age to the limit imples that ˙ y = Ay + B u, y (0) = y 0 , and the solution y asso ciated with con trol u in the form y ( t ) = S ( t ) y 0 + Z T 0 S ( T − s ) B uds. As u n con v erges w eakly to u in L q , w e get the in equalit y Z T 0 k u ( t ) k q dt ≤ lim inf n → + ∞ Z T 0 k u n ( t ) k q dt = inf Z T 0 k u ( t ) k q dt. It follo ws easily that Z T 0 k u ( t ) k q dt = inf Z T 0 k u ( t ) k q dt. Hence u is optimal of (1 2). This ensures the existence of a o ptimal con trol. Moreo v er, the cost function is strictly con v ex then the solution is ob vious uniquenes s. 8 By making use of conv ex dualit y , the problem of con trol to tra jecto- ries is red uced to the mi nimization of the corres p onding conjugate function. Roughly speaking, it is stated through the fo llo wing theorem: Theorem 2. (i) W e have the identity: inf u ∈ E 1 q Z T 0 k u k q dt = − inf ψ T ( 1 p Z T 0 k B ∗ ψ k p dt + < ψ (0) , y 0 > ) , (14) wher e ψ b e s olution of : − ˙ ψ = A ∗ ψ (15) ψ ( T ) = ψ T . (16) Or, we have in the form inf u ∈ E 1 q Z T 0 k u k q dt = − inf ψ ∈ X ∗ ( 1 p Z T 0 k B ∗ S ( T − t ) ∗ ψ k p dt + < S ( T ) ∗ ψ , y 0 > ) . (17) (ii) If u op is o ptimal of the pr oblem (12) then u op ( t ) = k B ∗ S ( T − t ) ∗ ϕ k p − 2 B ∗ S ( T − t ) ∗ ϕ, wher e ϕ b e o ptimal of the function: J ∗ ( ψ ) = 1 p Z T 0 k B ∗ S ( T − t ) ∗ ψ k p dt + < S ( T ) ∗ ψ , y 0 > . Pr o of. (i) Let ¯ y b e solution of (1) with u = 0 and we in tro duce the op erator N ∈ L ( L q ( Q T ) , X ) with N u = z u ( ., T ) fo r all u ∈ L q ( Q T ) , where z u is solution to ˙ z = Az + B u (18) z ( x, 0) = 0 . (19) A ccordingly , the solution y of (11) can b e decompo sed in the form y = z u + ¯ y . (20) The adjoin t N ∗ is giv en as follo ws F or eac h ψ T ∈ X ∗ , N ∗ ψ T = B ∗ ψ where ψ is solu tion of (15), (16). Let us in tro duce the fo llo wing functions F and G 9 F ( z T ) = ( 0 for z ( T ) = − ¯ y ( T ) + ∞ otherwise , G ( u ) = 1 q Z T 0 k u k q dt. Then, the problem (12 ), where the infimium is tak en o v er all u satis fying E, is eq uiv a len t to the follo wing minimization proble m inf u ∈ L q ( Q T ) ( F ( N u ) + G ( u )) . (21) W e can apply no w duality theorem of W.F ench el and T.R.Ro ck afellar (see Theorem 4.2 p.60 in [ET99]). It giv es inf u ∈ L q ( Q T ) ( F ( N u ) + G ( u )) = − inf ψ T ∈ X ∗ ( G ∗ ( N ∗ ψ T ) + F ∗ ( − ψ T )) , (22) where F ∗ and G ∗ are the con v ex conjugate of F and G, resp ectiv ely . Denote that ψ T = ψ ( T ) , z T = z ( T ) . Note that F ∗ ( ψ T ) = sup z T = − ¯ y T < z T , ψ T > = − < ψ T , ¯ y T >, for all ψ T ∈ X ∗ . A dditionally , G ∗ ( ω ) = 1 p Z T 0 k ω k p dt. Therefore, G ∗ ( N ∗ ψ T ) + F ∗ ( − ψ T ) = 1 p Z T 0 k B ∗ ψ k p dt + < ψ T ( x ) , ¯ y T ( x ) > . (23) Finally , mu ltiplying the state equation (15 ) b y ¯ y and due to (11), w e obtain < ψ T , ¯ y T > = < ψ ( 0 ) , y 0 > . Rewrite (23) as fo llo ws G ∗ ( N ∗ ψ T ) + F ∗ ( − ψ T ) = 1 p Z T 0 k B ∗ ψ k p dt + < ψ (0) , y 0 > = 1 p Z T 0 k B ∗ S ( T − t ) ∗ ψ T k p dt + < S ( T ) ∗ ψ T , y 0 >, since ψ is the solution of (15) , (16 ). 10 F rom (21 ) and (22), w e ha v e the iden tit y inf u ∈ E 1 q Z T 0 k u k q dt = − inf ψ T ( 1 p Z T 0 k B ∗ ψ k p dt + < ψ (0) , y 0 > ) , where ψ b e s olution of (15), (16). Or, w e ha v e in the form inf u ∈ E 1 q Z T 0 k u k q dt = − inf ψ ∈ X ∗ ( 1 p Z T 0 k B ∗ S ( T − t ) ∗ ψ k p dt + < S ( T ) ∗ ψ , y 0 > ) . (ii) If w e denote b y ( u op ) , ( ϕ T ) the unique solution to "LHS of (14)" and "RHS of (14)" resp ectiv ely , then one has 0 = 1 q Z T 0 k u op k q dt + 1 p Z T 0 k B ∗ ϕ T k p dt + < ϕ T (0) , y 0 > . (24) W e apply Y o ung’s ineq ualit y for the first t w o terms of RHS (24 ) 1 q Z Q T k u op k q dt + 1 p Z Q T k B ∗ ϕ T k p dt ≥ Z Q T u op .B ∗ ϕ T . (25) Then, "RHS of (24 )" ≥ R Q T u op .B ∗ ϕ T + < ϕ T (0) , y 0 > . F uthermore, b y m ultiplying tw o sides o f (15) b y y and applying G reen’s form ula, w e obtain < B ∗ ϕ T , u > + < ϕ T (0) , y 0 > = 0 . (26) On the one hand, "RHS of (24)" ≥ 0 ( due to (26 )). On the othe r hand, "RHS of (24)" = 0 ( due to (24 )). This is equiv a len t to that the sign "= " in ine qualit y (14) happ ens, i.e, k u op k q = k B ∗ ϕ T k p . It is als o rewritten as follo ws u op ( t ) = k B ∗ S ( T − t ) ∗ ϕ k p − 2 B ∗ S ( T − t ) ∗ ϕ, where ϕ b e optim al of the function J ∗ is giv en as abov e. 11 Remark : It is easily seen that th e functional J ∗ is con ve x, and from th e inequalit y (10), is co erciv e. Then , it follows that J ∗ attains a unique minim um in some p oin t ϕ ∈ D ( A ∗ ) . As ab o v e explanation, the control ¯ u is c hosen b y ¯ u ( t ) = k B ∗ S ( T − t ) ∗ ϕ k p − 2 B ∗ S ( T − t ) ∗ ϕ, (27) for ev ery t ∈ [0 , T ] a nd let y(.) b e the solution of (11), suc h that y (0) = y 0 , asso ciated with the con tro l ¯ u , th en w e ha v e y(T)=0. Therefore, ¯ u is the con trol of minimal of L q norm, among all con trols whose associated tra jectory satisfied y ( T ) = 0 . W e emphasize th at observ abilit y in L p norm ( 1 < p < 2 ) implies con trol- labilit y and gives a constructiv e w ay to build the con trol of min imal L q norm ( q > 2 ). A similar result w as kno wn in L 2 norm through using HUM ( see more [CT06], [L88], [Zua04], [Zua05]). 4 The main result W e a re concerned in this w ork with the uniform con trollabilit y prop ert y for the parab olic systems. As show n in [L T06], this prop ert y is kno wn to hold with the degree of unboundedness of control op erator γ ∈ [0 , 1 / 2) . In this section, w e also establish some appropriate assumptions and conditions ensuring that the unform controllabil ity stil l holds in the case γ ∈ h 1 / 2 , 1 p  . Let X and U b e Hilb ert spaces, and let A : D ( A ) → X b e a linear op erator and self-adjoin t, generating a strongly con tin uous semigroup S ( t ) on X . Let B ∈ L ( U, D ( A ∗ ) ′ ) b e a con trol op erator. W e mak e the following assumptions that will b e us ed along this article (also refer to [L T06]) (H1) The semigroup S(t) is analytic. Therefore, (see [P83]) there exist p ositive real n um b er C 1 and Ω suc h that k S ( t ) k X 6 C 1 e ω t k y k X , k AS ( t ) y k X 6 C 1 e ω t t k y k X , (28) for all t > 0 and y ∈ D ( A ) , and suc h that, if w e set ˆ A = A − ω I , for θ ∈ [0 , 1] and there holds    ( − ˆ A θ ) S ( t ) y    X ≤ C 1 e ω t t θ k y k X , (29) for all t > 0 and y ∈ D ( A ) . Of course, inequalities (28) hold as w ell if one replaces A b y A ∗ , S ( t ) b y S ( t ) ∗ , for y ∈ D ( A ∗ ) . 12 Moreo v er, if ρ ( A ) denotes the resolv en t set of A, then there exists δ ∈  0 , π 2  suc h that ρ ( A ) ⊃ ∆ δ =  ω + ρe iθ | θ > 0 , | θ | ≤ π 2 + δ  . F or λ ∈ ρ ( A ) , denote by R ( λ, A ) = ( λI − A ) − 1 the resolv en t of A . It follo ws from the previ ous estimates that exists C 2 > 0 s uc h that k R ( λ, A ) k L ( X ) ≤ C 2 | λ − ω | , k AR ( λ, A ) k L ( X )) ≤ C 2 , (30) for ev ery λ ∈ ∆ δ , and    R ( λ, ˆ A ))    L ( X ) ≤ C 2 | λ | ,    ˆ AR ( λ, ˆ A )    L ( X ) ≤ C 2 , (31) for ev ery λ ∈ ∆ δ + ω . Similarly , inequalities (30) and (31) hold as w ell with A ∗ and ˆ A ∗ . (H2) The degree of un b ounded ness of B is γ . Assume that γ ∈ h 1 / 2 , 1 p  ( where p,q are conjugate, i.e 1 p + 1 q = 1 and 1 ≤ p < 2 ). This means that B ∈ L ( U, D (( − ˆ A ∗ ) γ ) ′ ) . (32) In these conditions, the domain of B ∗ is D ( B ∗ ) = D (( − ˆ A ∗ ) γ ) , and there exists C 3 > 0 suc h that k B ∗ ψ k ≤ C 3    (( − ˆ A ∗ ) γ ) ψ    X , (33) for ev ery ψ ∈ D (( − ˆ A ∗ ) γ ) . (H3) W e consider t w o f amilies ( X h ) 0 0 and C 4 > 0 suc h t hat there holds, for ev ery h ∈ (0 , h 0 ) ,      I − ˆ ( A ∗ ) − γ + 1 2 ˜ P h P h  ψ     X ≤ C 4 h s k A ∗ ψ k X , (35) 13     (( − ˆ A ∗ ) γ )  I − ˆ ( A ∗ ) − γ + 1 2 ˜ P h P h  ψ     X ≤ C 4 h s (1 − γ ) k A ∗ ψ k X , (36) for ev ery ψ ∈ D ( A ∗ ) and    ( I − ˜ Q h Q h ) u    U → 0 , (37) for ev ery u ∈ U , and    ( I − ˜ Q h Q h ) B ∗ ψ    U ≤ C 4 h s (1 − γ ) k A ∗ ψ k X , (38) for ev ery ψ ∈ D ( A ∗ ) F or ev ery h ∈ (0 , h 0 ) , the v ector space X h (resp. U h ) is e ndow ed with the norm k . k X h (resp. k . k U h ) defined b y: k y h k X h =    ˜ P h y h    X for y h ∈ X h (resp. , k u h k U h =    ˜ Q h u h    U ). Therefore, w e ha v e the properties    ˜ P h    L ( X h ,X ) =    ˜ Q h    L ( U h ,U ) = 1 and     ˆ ( A ∗ ) − γ + 1 2 x     X ≤ C k x k X , (39) k P h k L ( X,X h ) ≤ C 5 and k Q h k L ( U,U h ) ≤ C 5 . (40) ( H 3 . 3 ) F or ev ery h ∈ (0 , h 0 ) , there holds P h = ˜ P h ∗ and Q h = ˜ Q h ∗ , (41) where the adjoin t op erators a re considered with resp ect to the piv ot spaces X , U , X h , U h . ( H 3 . 4 ) The re exists C 6 suc h that     B ∗ ˆ ( A ∗ ) − γ + 1 2 ˜ P h ψ h     U ≤ C 6 h − γ s k ψ h k X h , (42) for all h ∈ (0 , h 0 ) and ψ h ∈ X h . F or ev ery h ∈ (0 , h 0 ) , w e define the approx imation op erators A ∗ h : X h → X h of A ∗ and B ∗ h : X h → U h of B ∗ , b y A ∗ h = P h A ∗ ˜ P h and B ∗ h = Q h B ∗ ˆ ( A ∗ ) − γ + 1 2 ˜ P h . (43) ( H 4 ) The follo wing properties hold: ( H 4 . 1 ) The family of op erators e tA h is uniformly analytic, in sense that there exists C 7 > 0 suc h that   e tA h   L ( X h ) ≤ C 7 e ω t , (44) 14   A h e tA h   L ( X h ) ≤ C 7 e ω t t , (45) for all t > 0 and h ∈ (0 , h 0 ) . ( H 4 . 2 ) There exists C 8 > 0 suc h that, for ev ery f ∈ X and ev ery h ∈ (0 , h 0 ) , the respectiv e solutions of ˆ A ∗ ψ = f and ˆ A ∗ h ψ h = P h f satisfy k P h ψ − ψ h k X h ≤ C 8 h s k f k X . (46) In other w ords, there holds    P h ˆ A ∗− 1 − ˆ A ∗− 1 P h    L ( X,X h ) ≤ C 8 h s . Remark 4.1 Compare to [L T06], the impor tan t p oin t to note here is the app earance of the f unction ( ˆ A ∗ ) − γ + 1 2 in (35), (36) and (42). A ccording to [L T06], the inequalit y (22) mak e sense since γ < 1 2 and th us im ˜ P h ⊂ D (( − ˆ A ∗ ) 1 / 2 ) ⊂ D (( − ˆ A ∗ ) γ ) . In our con text, on accoun t of γ ≥ 1 2 , the inequalit y (36), whic h is sim - ilar to inequalit y (22) in [L T06], only mak e sense if w e add the functional ( ˆ A ∗ ) − γ + 1 2 in order that im ( ˆ A ∗ − γ + 1 2 ˜ P h ) ⊂ D (( − ˆ A ∗ ) γ ) . The c hoice of the func- tion ( ˆ A ∗ ) − γ + 1 2 seems to be the b est adapted to our theory . Namely , we giv e he re for instance ab out the functional ( ˆ A ∗ ) − γ + 1 2 through the heat equ ation with Diric hlet b oundary con trol as follo ws . y = ∆ y + c 2 y in (0 , T ) × Ω y ( 0 , . ) = y 0 in Ω y = u in (0 , T ) × Γ = Σ . Set X = L 2 (Ω) and U = L 2 (Γ) . It can b e written in the form (4), where the self-adjoin t op erator A : D ( A ) → X is defined b y Ay = ∆ y + c 2 y : D ( A ) = H 2 \ H 1 0 → L 2 . In this case, the degree of un b ounded of B is γ = 3 4 + ǫ ( ǫ > 0 ) (see [L T00], section 3.1). W e may take ˆ A as follo ws ˆ Ah = − ∆ h, D ( ˆ A ) = H 2 \ H 1 0 . Therefore, ( ˆ A ∗ ) − γ + 1 2 = ( ˆ A ∗ ) − 1 4 + ǫ : H 1 (Ω) → H 3 2 + ǫ (Ω) . 15 Remark 4.2 By means of the condition of the degree of unbounded of op erator B and (3 3), w e imply that B is admissible. Indeed, w e ha v e k L ∗ T ψ k = sup k u k q ≤ 1 Z T 0 < B ∗ S ∗ ( T − s ) x, u ( s ) > ds ≤ ( Z T 0 k B ∗ S ( t ) ∗ ψ k p dt ) 1 p ≤ C 3 ( Z T 0    ( − ˆ A ∗ ) γ S ( t ) ∗ ψ    p dt ) 1 p ≤ C 3 ( Z T 0 e ω t t pγ k ψ k p dt ) 1 p dt ( pγ < 1) ≤ C T k ψ k . Remark 4.3 . It is easily seen that assumptions ( H 3 ) (exce pt for the inequalities (3 5), (36), (42)) and ( H 4 . 2 ) hold f or most of the classical nu - merical appro ximation sc hemes, suc h as Ga lerkin metho ds, cen tered finite difference sc hemes ,...A dditionally , b y using some appro ximation prop erties and the properties o f the functional ( ˆ A ∗ ) − γ +1 / 2 , we pro v e that the inequal- ities (35), (36), (4 2) also hold fo r most of the ab ov e classic al sche mes (see the pro of in Section 5). As noted in [L T00], the assumption H 4 . 1 of uniform analyticit y is not standard, and has to be ch ec k ed in eac h specific case. The main res ult of the article is the follow ing : Theorem 3. Under the pr evious ass umptions, if the c ontr ol system ˙ y = Ay + B u is exactly nul l c ontr ol lable in time T > 0 , then ther e exist β > 0 , h 1 > 0 , and p ositive r e al numb ers C, C’ satisfying C   e T A ∗ h ψ h   p X h ≤ Z T 0   B ∗ h e tA ∗ h ψ h   p U dt + h β k ψ h k p X h ≤ C ′ k ψ h k p X h , (47) for e very h ∈ (0 , h 1 ) and every ψ h ∈ X h , (1 ≤ p < 2) . In these c onditions, for every y 0 ∈ X , and every h ∈ (0 , h 1 ) , ther e exists a s olution ϕ h ∈ X h minimizing the functional J h ( ψ h ) = 1 p Z T 0   B ∗ h e tA ∗ h ψ h   p U dt + 1 p h β k ψ h k p X h + < e T A ∗ h ψ h , P h y 0 > X h , (1 ≤ p < 2) (48) and the se q uenc e ( ˜ Q h u h ) 0 0 such that Z T 0 k u ( t ) | p U ≤ M p/ ( p − 1) k y 0 | p/ ( p − 1) X , and, for every h ∈ (0 , h 1 ) , Z T 0 k u h ( t ) k p U h ≤ M p/ ( p − 1) k y 0 k p/ ( p − 1) X , h β k ϕ h k p X h ≤ M p/ ( p − 1) k y 0 k p/ ( p − 1) X , k y h ( T ) k X h ≤ M 1 / ( p − 1) h β /p k y 0 k 1 / ( p − 1) X . (49) Remark 4.4 The left hand side of (47) is uniform observ ability ty p e inequalit y for con trol system (2). This inequalit y is w eaker than the uniform exact null con trollability . No attempt has been made here to prov e uniform exact n ull con trol for the app roxi mation systems (2). Remark 4.5 A similar res ult holds if the con trol system (1) is exactly con trollable in time T. How ev er, due to assumption ( H 1 ), the semigroup S(t) enjoys in g eneral regularit y prop erties. Therefore, the solution y(.) of the con trol system ma y b elong to a subspace of X, whatev er the con trol u is. F or instance, in the case of the heat equation with a Diric hlet o r Neumann 17 b oundary con trol, the solution is the smo oth f unction of the state v ariable x, as soon as t > 0 ,for ev ery con trol and initial condition y 0 ∈ L 2 . Hence, exact con trollabilit y do es not hold in this case L 2 . Moreo v er, one ma y w onder under w hic h assump tions the control u is the con trol, is defined b y (2 7), suc h that y(T)=0. As in [L T06], the f ollo wing prop osition giv e an answ er: Prop osition 1. With the notations of the or em, if the se quenc e of r e al num- b ers k ψ h k X h , 0 < h < h 1 , is mor e over b ounde d, then the c ontr ol u is the unique c on tr ol, is define d by (27), such t hat y(T)=0. Mor e over, the se quenc e ( ˜ Q h u h ) 0 0 such that the c on tr ol system . y = Ay + B u is exactly nul l c ontr o l lable in time t, for every t ∈ [ T − η , T + η ] , and the tr aje ctory t 7→ S ( t ) y 0 in X, for t ∈ [ T − η , T + η ] , is n ot c ontaine d in a hyp erpla ne of X. Other sufficient c ondition o n c ontr ol u, also e nsuring the b ounde dness of the se quenc e ( k ϕ h k X h ) 0 0 such that the c ontr ol system . y = Ay + B u is exa ctly nul l c ontr o l lable in time t, for every t ∈ [ T − η , T + η ] , and with the c ontr ol u is define d as (27), the tr aje ctory t 7→ S ( t − ξ ) B u ( ξ ) in X, fo r t ∈ [ T − η , T + η ] , every ξ ∈ (0 , t ) is not c ontaine d in a hyp erplane of X. 5 Pro of of the main results 1. The pro of of theorem : Pr o of. F or conv enience, we f irst state the follo wing useful appro xima- tion lemma, whose pro o f readily follow s that o f [L T06], [L T00]. The pro of of this le mma is pro vided in the Appendix. Lemma 3. Ther e exists C 9 > 0 such that, for al l t ∈ (0 , T ] and h ∈ (0 , h 0 ) , ther e holds   ( e tA ∗ h P h − P h S ( t ) ∗ ) ψ   X h ≤ C 9 h s t k ψ k X , (50)    ˜ Q h B ∗ h e tA ∗ h ψ h    U ≤ C 9 t γ k ψ h k X h , (51) 18 for e very θ ∈ [0 , 1] .    ˜ Q h B ∗ h e tA ∗ h ψ h − B ∗ S ( t ) ∗ ˜ P h ψ h    U ≤ C 9 h s (1 − γ ) θ t θ +(1 − θ ) γ k ψ h k X h every ψ h ∈ X h . (52) W e carry out pro ving the theorem as fo llo ws: The degree of un b oundedness γ of the control op erator B is lo w er than 1 p , there exi sts θ ∈ (0 , 1) suc h that 0 < θ + (1 − θ ) γ < 1 p . F or all h ∈ (0 , h 0 ) and ψ h ∈ X h w e ha v e Z T 0    ˜ Q h B ∗ h e tA ∗ h ψ h    p U dt = Z T 0 (    ˜ Q h B ∗ h e tA ∗ h ψ h    p U −    B ∗ S ( t ) ∗ ˜ P h ψ h    p U ) dt + Z T 0    B ∗ S ( t ) ∗ ˜ P h ψ h    p U dt. (53) W e estimate t w o terms of righ t hand side of (53 ). The con trol system is exactly n ull con trollable in time T, then there exists a positiv e real n um b er C > 0 such that Z T 0    B ∗ S ( t ) ∗ ˜ P h ψ h    p dt ≥ C    S ( T ) ∗ ˜ P h ψ h    p X . (54) W e hav e the followin g inequalit y | y p − z p | < p ( y p − 1 + z p − 1 ) | y − z | , (55) where y , z ∈ R + , p > 1 . Indeed, w e apply mean-v alue theorem for f ( x ) = x p ( p > 1 , x ∈ R + ) , there exis ts ξ ∈ ( y , z ) suc h that | y p − z p | = | f ′ ( ξ ) | | y − z | = p   ξ ( p − 1)   . | y − z | < p ( y p − 1 + z p − 1 ) . | y − z | . W e apply the ab o v e inequalit y and mak e use of (40), (28), (44), (50) 19 to obtain        P h S ( T ) ∗ ˜ P h ψ h    p X h −   e T A ∗ h ψ h   p X h     ≤ p (    P h S ( T ) ∗ ˜ P h ψ h    p − 1 X h +   e T A ∗ h ψ h   p − 1 X h ) ×        P h S ( T ) ∗ ˜ P h ψ h    X h −   e T A ∗ h ψ h   X h     ≤ p ( C 5 C 1 e ω t k ψ h k p − 1 X h + C 7 k ψ h k p − 1 X h ) .    P h S ( T ) ∗ ˜ P h ψ h − e T A ∗ h ψ h    X h ≤ C p k ψ h k p − 1 X h C 9 C 5 h s k ψ h k X h ≤ C 14 h s k ψ h k p X h . Therefore , from abov e estimate and (39), w e g et   e T A ∗ h ψ h   p X h − C 14 h s k ψ h k p X h ≤    P h S ( T ) ∗ ˜ P h ψ h    p X h ≤ C p 5    S ( T ) ∗ ˜ P h ψ h    p X . (56) Com bine (54) with (56) we ha v e: Z T 0    B ∗ S ( t ) ∗ ˜ P h ψ h    p U dt ≥ C 15   e T A ∗ h ψ h   p X h − C 14 h s k ψ h k p X h . (57) F or t he first term on the righ t hand side of (53 ), one has, using (3 3), (51), (52) and app lying the inequalit y (55)       ˜ Q h B ∗ h e tA ∗ h ψ h    p U −    B ∗ S ( t ) ∗ ˜ P h ψ h    p U    ≤ p (    ˜ Q h B ∗ h e tA ∗ h ψ h    p − 1 U +    B ∗ S ( t ) ∗ ˜ P h ψ h    p − 1 U ) ×       ˜ Q h B ∗ h e tA ∗ h ψ h    U −    B ∗ S ( t ) ∗ ˜ P h ψ h    U    ≤ p ( C p − 1 9 t γ ( p − 1) k ψ h | p − 1 X h + C p − 1 3 e ω t ( p − 1) t γ ( p − 1) k ψ h k p − 1 X h ) ×       ˜ Q h B ∗ h e tA ∗ h ψ h − B ∗ S ( t ) ∗ ˜ P h ψ h    U    ≤ C 16 t γ ( p − 1) k ψ h k p − 1 X h .C 9 h s (1 − γ ) θ t θ +(1 − θ ) γ k ψ h k X h ≤ C 17 h s (1 − γ ) θ t θ +(1 − θ ) γ + γ ( p − 1) k ψ h k p X h . 20 W e hav e γ < 1 p ( p ≥ 1) , therefore θ + (1 − θ ) γ + γ ( p − 1) < 1 and w e can get, b y in tegrat ion,    R T 0 (    ˜ Q h B ∗ h e tA ∗ h ˜ P h ψ h    p U − k B ∗ S ( t ) ∗ ψ h k p U ) dt    ≤ C 18 h s (1 − γ ) θ k ψ h k p X h . Therefore, Z T 0    ˜ Q h B ∗ h e tA ∗ h ψ h    p U dt ≥ Z T 0 k B ∗ S ( t ) ∗ ψ h k p U dt − C 18 h s (1 − γ ) θ k ψ h k p X h . (58) W e choose a real nu m b er β suc h that 0 ≤ β ≤ s (1 − γ ) θ . Comb ine results (53), (57), (58 ) w e hav e ine qualit y ( 47). F or h ∈ ( 0 , h 1 ) , the functional J h is conv ex, and inequalit y (47 ), is co erciv e. Therefore, it admits a solution minim um at ϕ h ∈ X h so that 0 = ▽ J h ( ϕ h ) = G h ( T ) ϕ h + h β k ϕ h k p − 2 ϕ h + e T A h P h y 0 , where G h ( T ) = R T 0   B ∗ h e tA ∗ h ϕ h   p − 2 e tA h B h B ∗ h e tA ∗ h dt is t he Gramian of the semidi screte syste m. With u h ( t ) =   B ∗ h e ( T − t ) A ∗ h ϕ h   p − 2 B ∗ h e ( T − t ) A ∗ h ϕ h is c hosen then, the so- lution y h ( . ) satisfies y h ( T ) = e T A h y h (0) + Z T 0 e ( T − t ) A h B h u h ( t ) dt = e T A h P h y 0 + G h ( T ) ϕ h = − h β k ϕ h k p − 2 ϕ h . Note that, since J h (0) = 0 , there m ust hold, at the minim um, J h ( ϕ h ) ≤ 0 . Hence, using the obse rv abilit y inequalit y (47) and the Cauc h y-Sc h w arz inequalit y , o ne ge ts c   e T A ∗ h ϕ h   p X h ≤ Z T 0   B ∗ h e tA ∗ h ϕ h   p U h + h β k ϕ h k p X h ≤ 2   e T A ∗ h ϕ h   X h k P h y 0 k X h , and th us,   e T A ∗ h ϕ h   X h ≤ ( 2 c ) 1 / ( p − 1) ( k P h y 0 k X h ) 1 / ( p − 1) . (59) 21 As a conseq uence, Z T 0   B ∗ h e tA ∗ h ϕ h   p U h ≤ ( 2 p c ) 1 / ( p − 1) ( k P h y 0 k p/ ( p − 1) X h ) , (60) and h β k ϕ h k p X h ≤ ( 2 p c ) 1 / ( p − 1) ( k P h y 0 k p/ ( p − 1) X h ) , and the estim ates (49) fol- lo w. 2. Pro of of proposition Pr o of. If the sequence (    ˜ P h ϕ h    X ) 0 U h ds is b o unded, uniformly for h ∈ (0 , h 1 ) . Th us, passing to the l imit, one gets Z t 0 < Φ , S ( t − s ) B u ( s ) > X ds = 0 . This equalit y is equiv alen t to the fact that : there exists ξ ∈ (0 , t ) suc h that < Φ , S ( t − ξ ) B u ( ξ ) > X = 0 . This con tradicts the fact that the tra jectory t 7→ S ( t − ξ ) B u ( ξ ) , t ∈ [ T − η , T + η ] and ev ery ξ ∈ (0 , t ) , is not con tained in a hyperplane of X. 6 Numerical sim ulation for the heat equation with Diric hlet b oundary con trol In this section, w e give an example of a situation where the theorem 3.1 are satisfied. Let Ω ⊂ R n b e an op en bo unded domain with sufficien tly smo oth bound- ary Γ . W e consider the D iric hlet mixe d problem for the heat e quation: . y = ∆ y + c 2 y in (0 , T ) × Ω y ( 0 , . ) = y 0 in Ω y = u in (0 , T ) × Γ = Σ , with b oundary con tro l u ∈ L 6 (0 , T ; L 2 (Γ)) and y 0 ∈ L 2 (Ω) . Set X = L 2 (Ω) and U = L 2 (Γ) . W e in tro duce the self-adjoin t op erator: Ah = ∆ h + c 2 h : D ( A ) = H 2 ∩ H 1 0 → L 2 (Ω) . 23 The adjoin t B ∗ ∈ L ( D ( A ∗ ) , U ) of B is giv en b y B ∗ ψ = − ∂ ψ ∂ ν , ψ ∈ D ( A ∗ ) . Moreo v er, the degree of un b oundedness of B is γ = 3 4 + ǫ ( ǫ > 0 ) (see [L T00], section 3.1). 1. One-dimens ional Finite-Difference semi-discretized model: W e next in tro duce a semi-discretiz ed mo del of the ab ov e heat equation, using 1D Finite-Differe nce. F or s implicit y , w e set Ω = (0 , 1) , Γ = { 0 , 1 } , c=1 and T=1. Giv en n ∈ N we de fine h = 1 n +1 > 0 . W e consider the follo wing simplex mesh: Ω h = { x 0 = 0 ; x i = ih, i = 1 , ..., N ; x n +1 = 1 } , whic h divides [0,1] in to n+1 subin terv als I j = [ x j , x j + 1 ] j=0 ,...,n+1. Set X h = { y ∈ C 0 (Ω h ) } , U h = { y ∈ C 0 (Γ) } . Define ˜ P h (resp., ˜ Q h ) as the canonical em b edd ing from X h in to D (( − A ) 1 / 2 ) (resp., f rom U h to U). F or x h ∈ X h and u h ∈ U h , set, ˜ P h ( x h ) = x h and ˜ Q h ( u h ) = u h . F or y ∈ D (( − A ) 1 / 2 ) ′ = H 1 (Ω) ′ , set P h y = ( y 1 , .., y i , .., y n +2 ) where y i = y (( i − 1) h ) and, for u ∈ U , set, Q h u = ( u 1 , .., u i , .., u n +2 ) whe re u i = u (( i − 1) h ) . It is clear that the assu mptions ( H 3 . 1 ) and ( H 3 . 3 ) are here satisfied. Our aim is nex t to v erify the inequalitie s in ( H 3 . 2 ) and ( H 3 . 4 ) . In order to get thes e ineq ualities, it will necessary to making use of the follo wing usual appro ximation properties (see [L T00], section 5): (i) k Π h y − y k H l (Ω) ≤ ch s − l k y k H s (Ω) , s ≤ r + 1 , s − l ≥ 0 , 0 ≤ l ≤ 1 , and the in v erse appro ximation properties (ii) k y h k H α (Ω) ≤ ch − α k y h k 2 L (Ω) , 0 ≤ α ≤ 1 . (iii) h − 1 k y − Π h y k L 2 (Γ) +   ( I − Π h ) ∂ y ∂ ν   L 2 (Γ) ≤ ch s − 3 2 k y k H s (Ω) , 3 2 < s < r + 1 , y ∈ H s (Ω) . 24 (iv) k y h k L 2 (Γ) + h   ∂ y h ∂ ν   L 2 (Γ) ≤ C h − 1 2 k y h k L 2 (Ω) , y h ∈ V h . where r is the order of appro ximation (degree of p olynomials) and Π h is the orthogonal p ro jection of L 2 (Ω) on to V h . First, b y applying (i) we easily get the inequalit y (36)     ( I − ˆ A ∗ − γ + 1 2 ˜ P h P h ) ψ     L 2 (Ω) ≤ C h 2 k ψ k H 2 (Ω) ≤ C h 2 k ψ k D ( A ∗ ) ≤ C h 2 k A ∗ ψ k X , in this case s=2. W e next v erify the inequalit y (37) as follo ws     (( − ˆ A ∗ ) γ )  I − ˆ ( A ∗ ) − γ + 1 2 ˜ P h P h  ψ     X ≤ C     ( I − ˆ ( A ∗ ) − γ + 1 2 ˜ P h P h ) ψ     D (( − ˆ A ∗ ) γ ) ≤ C h s − l k ψ k D ( A ∗ ) ≤ C h s (1 − γ ) k A ∗ ψ k X , where we ha v e used (i) with s=2, D ( A ∗ ) = H s (Ω) and D (( − ˆ A ∗ ) γ ) = H l (Ω) . F or th e inequalit y (39), w e apply (iii) wi th s=2 as    ( I − ˜ Q h Q h ) B ∗ ψ    L 2 (Γ) =     ( I − ˜ Q h Q h ) ∂ ψ ∂ ν     L 2 (Γ) ≤ C h 1 / 2 k ψ k H 2 (Ω) ≤ C h s (1 − γ ) k ψ k D ( A ∗ ) ≤ C h s (1 − γ ) k A ∗ ψ k X . 25 F or th e inequalit y (43), b y using (iv) and (40 ) we get     B ∗ ˆ ( A ∗ ) − γ + 1 2 ˜ P h ψ h     U =       ∂ ( ˆ ( A ∗ ) − γ + 1 2 ˜ P h ψ h ) ∂ ν       L 2 (Γ) ≤ C h − 3 2     ˆ ( A ∗ ) − γ + 1 2 ˜ P h ψ h     L 2 (Ω) ≤ C h − 3 2 k ψ h k X h . Therefore, the ineq uality (43) is satisfied for s=2, γ = 3 4 + ǫ . Moreo v er, the assumption ( H 4 . 2 ) is satisfied with s=2 (see [L T00]). Hence, theorem 3.1 applie s, with β = 0 . 1 6 , for in stance. W e then conside r the follo wing finite difference appro ximation of the ab o v e heat equation as follws ˙ y j = 1 h 2 [ y j + 1 + y j − 1 − 2 y j ] + c 2 y j 0 < t < T , j = 1 , ..., n y j (0) = y j 0 , j = 1 , ..., n y 0 ( t ) = y n +1 ( t ) = u h , 0 < t < T , where y ∈ R n +2 , y 0 ∈ R n +2 , u h ∈ R and A h = 1 h 2            0 0 0 . . . 0 0 ( c 2 h 2 − 2) 1 . . . 0 0 1 ( c 2 h 2 − 2) . . . 0 . . . . . . . . . . . . . . . 0 . . . ( c 2 h 2 − 2) 1 0 0 . . . 1 ( c 2 h 2 − 2) 0 0 . . . 0 0 0            , B h =        1 0 . . . 0 1        . 26 2. Numerical sim ulation The minimization pro cedure described in Theorem 3.1 has b een imple- men t for d=1, b y using a simpl e gra dien t method that has the adv and- tages not to require the complex computations and this metho d can applied with any p o w er p. Ho wev er, the computation of gradien t of J h is v ery expensiv e since the gradie n t is related to the Gramian matrix. name S h h y 0 1D-10 10 10 − 1 y 0 ( x ) = e − x 2 1D-100 100 10 − 2 y 0 ( x ) = e − x 2 1D-500 500 2 . 10 − 3 y 0 ( x ) = e − x 2 T able 1: Data for the one -dimensional heat equation. name k ϕ h k X h β ( β = 0 . 1 6 ) k y h ( T ) k 1D-10 0.1690 0 . 6814 0.4775 1D-100 0.7960 0 . 4779 0.4565 1D-500 2.0570 0 . 3699 0.4273 T able 2: Numerical results for one dimensional equation for β = 0 . 16 . name k ϕ h k X h β ( β = 2) k y h ( T ) k 1D-10 4.4266 10 − 2 0.0111 1D-100 4.8933 10 − 4 1.3467e-004 1D-500 5.0956 4 . 10 − 6 5.5178e-006 T able 3: Numerical results for one dimensional equation for β = 2 . Numerical sim ulation are carried out with a space-discretization step equal to 0.005, with the data of T a ble 1. The n umerical results a re pro vided in T able 2 for beta =0.16 and T able 3 for beta=2 . The conv ergence of the me tho d is v ery slow. F r om the result of The- orem 3, the final s tate y h ( T ) is equal to − h β k ϕ h k p − 2 ϕ h in wh ic h ϕ h is minimizer of J h . W e note that the maxim um v a lue for whic h the theorem asserts the con v ergence is v ery small. F or suc h a small v alue of β (for instance β = 0 . 16 ), h β con v erges to 0 v ery slo wly . It follo ws that y h ( T ) con v erges v ery extreme ly slow . In practic e, the unique minimizer ϕ h of J h is obtaine d through the sim- ple gradien t method in whic h the step is equal to 0.01 and the error ǫ = 10 − 2 is tak en. With the small v alue β = 0 . 16 , it to ok a long time 27 to ac hiev e the res ults in T able 2. Namely , for n=500, it to ok more one mon th to get the result after 1000 iterations. It is clearly seen fro m T a- ble 2 that the con v ergence of y h ( T ) is very slo w. Th ese resul ts illustrate the difficult in using our metho d to compute con trols. Although the case b eta=2 is not cov ered b y our main theorem, the method seems to con v erge for this v alue for b eta and w e prov ide hereafter the nume rical results in T able 3. Since the v alue of the b eta is greater, the con v ergence is quic ker. App endix: pro of of lemma Pr o of. • First of all, w e will pro v e (51) F or e v ery ψ ∈ D ( A ∗ ) , one has     ˜ Q h B ∗ h e tA ∗ h P h ψ − B ∗ ˆ ( A ∗ ) − γ + 1 2 ˜ P h P h S ( t ) ∗ ψ     U ≤    ˜ Q h B ∗ h e tA ∗ h P h ψ    U +     B ∗ ˆ ( A ∗ ) − γ + 1 2 ˜ P h P h S ( t ) ∗ ψ     U . (62) W e estimate eac h term of the righ t hand side of (62). Since B ∗ h = Q h B ∗ ˆ ( A ∗ ) − γ + 1 2 ˜ P h and th us, using (40) (42) (44) one gets    ˜ Q h B ∗ h e tA ∗ h P h ψ    U ≤ C 5     B ∗ ˆ ( A ∗ ) − γ + 1 2 ˜ P h e tA ∗ h P h ψ     U ≤ C 5 C 6 h − γ s   e tA ∗ h P h ψ   X h ≤ C 2 5 C 6 C 7 h − γ s e ω t k ψ k X . (63) On the other hand, from (28), (40), (42) one obtains     B ∗ ˆ ( A ∗ ) − γ + 1 2 ˜ P h P h S ( t ) ∗ ψ     U ≤ C 6 h − γ s k P h S ( t ) ∗ ψ k X h ≤ C 5 C 6 h − γ s k S ( t ) ∗ ψ k X ≤ C 1 C 5 C 6 h − γ s e ω t k ψ k X . (64) Hence, com bine (63), (64) with (62), there exis ts C 10 > 0 suc h that     ˜ Q h B ∗ h e tA ∗ h P h ψ − B ∗ ˆ ( A ∗ ) − γ + 1 2 ˜ P h P h S ( t ) ∗ ψ     U ≤ C 10 h − γ s k ψ k X . (65) 28 for ev ery ψ ∈ D ( A ∗ ) , ev ery t ∈ [0 , T ] , and ev ery h ∈ (0 , h 0 ) . Moreo v er, we get a nother estim ate o f this term. By using (33), (38), (40) , (42), (50) one gets     ˜ Q h B ∗ h e tA ∗ h P h ψ − B ∗ ˆ ( A ∗ ) − γ + 1 2 ˜ P h P h S ( t ) ∗ ψ     U =     ˜ Q h Q h B ∗ ˆ ( A ∗ ) − γ + 1 2 ˜ P h e tA ∗ h P h ψ − B ∗ ˆ ( A ∗ ) − γ + 1 2 ˜ P h P h S ( t ) ∗ ψ     U ≤     ˜ Q h Q h B ∗ ˆ ( A ∗ ) − γ + 1 2 ˜ P h ( e tA ∗ h P h ψ − P h S ( t ) ∗ ψ )     U +     ˜ Q h Q h B ∗ ( ˆ ( A ∗ ) − γ + 1 2 ˜ P h P h − I ) S ( t ) ∗ ψ     U +    ( ˜ Q h Q h − I ) B ∗ S ( t ) ∗ ψ    U +     B ∗ ( I − ˆ ( A ∗ ) − γ + 1 2 ˜ P h P h ) S ( t ) ∗ ψ     U ≤ C 5 C 6 h γ s   e tA ∗ h P h ψ − P h S ( t ) ∗ ψ   X h + C 5 C 3     ( − ˆ A ) γ ( ˆ ( A ∗ ) − γ + 1 2 ˜ P h P h − I ) S ( t ) ∗ ψ     X + C 4 h s (1 − γ ) k A ∗ S ( t ) ∗ ψ k X + C 3     ( − ˆ A ) γ ( ˆ ( A ∗ ) − γ + 1 2 ˜ P h P h − I ) S ( t ) ∗ ψ     X ≤ C 5 C 6 C 9 h s (1 − γ ) t k ψ k X + ( C 3 ( C 5 + 1) + 1) C 4 h s (1 − γ ) k A ∗ S ( t ) ∗ ψ k X ≤ C 11 h s (1 − γ ) t k ψ k X . (66) Then, raising (65) to the pow er 1 − γ , (6 6) to p ow er to γ and m ultiplying b oth result estim ates, we obtain     ˜ Q h B ∗ h e tA ∗ h P h ψ − B ∗ ˆ ( A ∗ ) − γ + 1 2 ˜ P h P h S ( t ) ∗ ψ     U ≤ C 12 t γ k ψ k X . Hence,    ˜ Q h B ∗ h e tA ∗ h P h ψ    U ≤ C 12 t γ k ψ k X +     B ∗ ˆ ( A ∗ ) − γ + 1 2 ˜ P h P h S ( t ) ∗ ψ     U . (67) 29 F rom (29 ), (33), (36) one yieds     B ∗ ˆ ( A ∗ ) − γ + 1 2 ˜ P h P h S ( t ) ∗ ψ     U ≤     B ∗ ( I − ˆ ( A ∗ ) − γ + 1 2 ˜ P h P h ) S ( t ) ∗ ψ     U + k B ∗ S ( t ) ∗ ψ k U ≤ C 13 e ω t t γ k ψ k X . (68) Com bine (67) with (68) and by setting ψ = ˜ P h ψ h w e get (51). • Finally , w e pro v e (52). On the one hand, r easoning as ab o v e for obtain- ing (66), w e get    ˜ Q h B ∗ h e tA ∗ h P h ψ − B ∗ S ( t ) ∗ ψ    U ≤ C h s (1 − γ ) t k ψ k X , (69) for ev ery ψ ∈ D ( A ∗ ) , ev ery t ∈ [0 , T ] and ev ery h ∈ (0 , h 0 ) . On the other hand, from (51) and settin g ψ = ˜ P h ψ h one obtains    ˜ Q h B ∗ h e tA ∗ h P h ψ − B ∗ S ( t ) ∗ ψ    U ≤    ˜ Q h B ∗ h e tA ∗ h P h ψ    U + k B ∗ S ( t ) ∗ ψ k U ≤ C 9 t γ k ψ h k + C 3    ( − ˆ A ∗ ) γ S ( t ) ∗ ψ    X ≤ C t γ k ψ h k X . (7 0) Raising (69) to the p o w er θ , (70) to the p o w er 1 − θ and m ultiplying b oth resulting estim ates, w e obtain (52). The pro of of the inequalit y (50) is found in [L T06], [L T00]. 7 Conclusio n W e ha v e sho wn that the appropriate dualit y tec hniques can b e applied to solv e (3), namely the F enc hel-Ro c k afellar theorem. A dditionally , it is also stated that under standard assumptions on the discretization pro cess, f or an exactly n ull controllable linear con trol system, if the semigroup of appro ximating syste m is uniformly analytic, and if the degree of un b oundedness of the con trol op erator is greater than 1 2 then the unform observ ability t yp e inequalit y is pro v ed. 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