FEEDBACK NULL CONTROLLABILITY OF THE SEMILINEAR HEAT EQUATION

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1 Differenial and Inegral Equaions Volume 5, Number, January 2002, Pages 5 28 FEEDBACK NULL CONTROLLABILITY OF THE SEMILINEAR HEAT EQUATION Mihai Sîrbu Deparmen of Mahemaical Sciences, Carnegie Mellon Universiy Pisburgh, PA 523, USA (Submied by: Viorel Barbu) Absrac. In his paper we sudy he null conrollabiliy of he semilinear hea equaion sudying he properies of he minimum energy ha one needs o seer an iniial sae x in 0. We prove his is locally Lipschiz, and consequenly we obain he expeced opimal feedback law. We also characerize he value funcion as he unique posiive viscosiy soluion (of he corresponding Hamilon Jacobi equaion wih singular final daa) which ends o 0 on admissible rajecories, or as he minimal posiive viscosiy supersoluion.. Inroducion This work is concerned wih he feedback null conrollabiliy of he sae sysem y (x, ) y(x, )+f(y(x, )) = m(x)u(x, ) for(x, ) Q =Ω (0,T) y(x, ) = 0 for (x, ) Σ= Ω (0,T) (.) y(x, 0) = y 0 (x) for x Ω where Ω is an open and bounded subse of R n wih a smooh boundary Ω and m is he characerisic funcion of an open subse ω Ω. Here is he Laplace operaor wih respec o x. We assume ha f C (R), f(0) = 0 and f (r) L( )r R. Sof is globally Lipschiz. In he sequel we wan o rewrie he sae sysem (.) in a semigroup form. For his purpose we define H := L 2 (Ω) and A : D(A) H H o be Au = u, where D(A) = H0 (Ω) H2 (Ω). We also define he Nemyskii operaor associaed o f, F : H H by (Fu)(x) =f(u(x)) for almos every x Ω and he bounded linear conrol operaor B : H H by (Bu)(x) =m(x)u(x) for almos every x Ω. Acceped for publicaion: Ocober AMS Subjec Classificaions: 49N35, 93B05, 49L25. 5

2 6 Mihai Sîrbu Taking ino accoun all his noaion, he sae sysem becomes { y + Ay + Fy = Bu on (0,T) y(0) = y 0. (.2) We know ha A generaes a C 0 -semigroup, F is globally Lipschiz on H and B is bounded. So, for each u L 2 (0,T; H) and y 0 H, (.2) has a unique mild soluion y C([0,T]; H). For he definiion of mild soluion, see [7]. Wih hese hypoheses we know from [4] ha (.) (or (.2)) is globally null conrollable; i.e., for each x H and 0 < T here exiss u L 2 (, T ; H) such ha { y + Ay + Fy = Bu on (, T ) y() =x and y(t )=0. (.3) For fixed x H we wan o find a feedback law for u L 2 (, T ; H) which fulfills (.3) and minimizes he energy J(u) := u 2 dτ. (.4) 2 Throughou he paper we will denoe by he norm in any Hilber space, and by X he norm of a Banach space in case we wan o emphasize he space X. We wan o sudy he minimizaion problem minj(u) subjec o (.3) by means of dynamic programming argumens. For his reason we define he value funcion ϕ :[0,T) H R by { } ϕ(, x) = inf 2 u 2 dτ : y +Ay +Fy = Bu, y() =x, y(t )=0. (.5) We expec ha ϕ is he soluion (unique if possible), in a weak sense, of he Hamilon Jacobi equaion ϕ (Ax + Fx), ϕ 2 B ϕ 2 =0 (.6) subjec o he formal final condiion { + for x 0 ϕ(t,x)= (.7) 0 for x =0. In Secion 2 we prove ha for fixed 0 <T, ϕ(, ) is locally Lipschiz, and consequenly we have he opimal feedback law, u (s) B ϕ(s, y (s)), (.8)

3 feedback conrollabiliy 7 where by ϕ we denoe he generalized gradien of ϕ wih respec o x in he sense of F. Clarke. In Secion 3 we characerize he value funcion as he minimal posiive viscosiy supersoluion of (.6) subjec o (.7) or as he unique posiive viscosiy soluion which ends o 0 on admissible rajecories. In paricular, he exisence of a posiive viscosiy supersoluion is equivalen o he null conrollabiliy of he sae sysem. 2. Properies of he value funcion. Feedback laws Before saing he main resul of his secion, we need some preliminaries. Firs, we would like o remind he reader of some properies of he operaors A and F : A is self-adjoin and A generaes a compac C 0 -semigroup, F is Gaeaux differeniable and (F (u)v)(x) =f (u(x))v(x) for almos every x Ω. In general, F is no Fréche differeniable. Then we define, for each ε>0, ϕ ε :[0,T] H R, by { ϕ ε (, x) = inf 2 u 2 dτ + 2ε y(t ) 2 : y + Ay + Fy = Bu,y() =x (2.) For fixed ε, since he funcion x 2ε x 2 is locally Lipschiz, we have ha ϕ ε (, ) is locally Lipschiz for each 0 T. In fac, he Lipschiz consan on bounded ses is independen of. For he proof, see []. Since A generaes a compac semigroup, by usual compacness argumens, we can conclude ha for fixed ε he infimum is aained in (2.) and ϕ ε (, x) ϕ(, x) asε 0 for each (, x) [0,T) H. (2.2) We also need he following lemma of independen ineres: Lemma 2.. Le A, F and B be defined as before. We consider h, g, l 0,l : H R such ha h, g and l are convex and C, and l 0 is merely coninuous. If (u,y ) is an opimal pair for he Bolza conrol problem, { } inf (h(u)+g(y))d + l 0 (y(0)) + l (y(t )) : y + Ay + Fy = Bu, (2.3) 0 hen here exiss p C([0,T]; H) saisfying p A p (F (y )) p = g(y ), B p = h(u )a.e. on (0,T) p(t )= l (y (T )) and p(0),h liminf ε 0 ε (l 0(y (0) + εh) l 0 (y (0))) ( ) h H. }.

4 8 Mihai Sîrbu Proof of Lemma 2.. Le (u,y ) be an opimal pair. If we denoe y 0 = y (0), we can see ha (u,y ) is also an opimal pair for he problem { inf (h(u)+g(y))d + l (y(t )) : y + Ay + Fy = Bu, y(0) = y 0 }. 0 By [2], here exiss p C([0,T]; H) saisfying p A p (F (y )) p = g(y ) B p = h(u )a.e. on (0,T) p(t )= l (y (T )). Now, we can fix u and ake he iniial daa y 0 + εh. We denoe by y y0 +εh he soluion of he sae sysem y + Ay + Fy = Bu, y(0) = y 0 + εh. Since (u,y ) is an opimal pair for he minimizaion problem (2.3), we see ha ε (g(y y 0 +εh) g(y ))d + ε (l 0(y 0 + εh) l 0 (y 0 )) 0 + ε (l (y y0 +εh(t )) l (y (T ))) 0. (2.4) Using compacness argumens (an infinie dimensional version of Arzelà Ascoli), and he specific form of F, we obain ha ε (y y 0 +εh y ) z srongly in C([0,T]; H), where z is he soluion of he variaion sysem z + Az + F (y )z =0, z(0) = h. (2.5) Since g and l are C on H, we can conclude from (2.4) ha g(y ),z d + liminf ε 0 0 ε (l 0(y 0 + εh) l 0 (y 0 )) + l (y (T )),z(t) 0. (2.6) We now follow he usual compuaion z,p = z,p + z,p = Az F (y )z,p + z,a p +(F (y )) p + g(y ) = z, g(y ). By inegraion we obain 0 z, g(y ) d = z(t ),p(t ) h, p(0).

5 feedback conrollabiliy 9 Subsiuing his in (2.6) we obain z(t ),p(t) h, p(0) + l (y (T )),z(t ) + liminf ε 0 ε (l 0(y 0 + εh) l 0 (y 0 )) 0. Bu p(t )= l (y (T )), so we obain p(0),h liminf ε 0 ε (l 0(y 0 + εh) l 0 (y 0 )). Since h H was arbirarily chosen, he proof of he lemma is complee. We are now ready o sae he main resul of his secion: Theorem 2.. For each 0 <T, ϕ(, ) is locally Lipschiz. Moreover, if (u,y ) is a pair for which he infimum is aained in (.5), hen u (s) B ϕ(s, y (s)), a.e. s [, T ). Remark 2.. In Theorem 2. we denoed by ϕ he generalized gradien in he sense of F. Clarke wih respec o he sae variable x. For he definiion of he generalized gradien, see [2]. Proof of Theorem 2.. From Carleman esimaes in [4] we ge ha any soluion of he adjoin equaion { p + p + ap = 0 on (, T ) Ω p(τ,x) =0on(, T ) Ω saisfies he observabiliy inequaliy: p() 2 L 2 (Ω) C(T, a L (Q)) p(τ,x) 2 dx dτ = C(T, a L (Q)) ω B p(τ) 2 L 2 (Ω) dτ. (2.7) Since f (r) L we can obain by Schauder s fixed-poin heorem ha he sae sysem (.2) is null conrollable and ϕ ε (, x) ϕ(, x) 2 C(T, L) x 2. (2.8) For fixed 0 <T, ε>0 and x H, le us consider he minimizaion problem (2.). Since he semigroup generaed by A is compac, here exiss an opimal pair (u,y ) such ha ϕ ε (, x) = 2 u 2 dτ + 2ε y (T ) 2 and y + Ay + Fy = Bu,y() =x.

6 20 Mihai Sîrbu By he definiion of ϕ ε,(u,y ) is also an opimal pair for he problem { inf 2 u 2 dτ + } 2ε y(t ) 2 ϕ ε (, y()) : y + Ay + Fy = Bu. Considering l (y) = 2ε y 2, and l 0 (y) = ϕ ε (, y), by Lemma 2., we conclude here exiss p,x C([, T ]; H) such ha (p,x ) A p,x (F (y )) p,x = 0 on [, T ] p,x (T )= ε y (T ) B p,x = u a.e on[, T ] (2.9) and also p,x (),h liminf λ 0 λ (ϕ ε(, x + λh) ϕ ε (, x)) for each h H. (2.0) Replacing h by h in (2.0), we can see ha whenever he limi lim λ 0 λ (ϕ ε(, x + λh) ϕ ε (, x)) = d dλ (ϕ ε(, x + λh)) λ=0 exiss, hen d dλ (ϕ ε(, x + λh)) λ=0 = p,x (),h, where p,x saisfies (2.9). Using (2.7) for p = p,x and a = f (y ), and hen aking ino accoun (2.8), we obain p,x () 2 C(T, L) B p,x (τ) 2 dτ = C(T, L) u (τ) 2 dτ C(T, L)2ϕ ε (, x) C 2 (T, L) x 2. So, we conclude ha anyime d dλ (ϕ ε(, x + λh)) exiss, hen d dλ (ϕ ε(, x + λh)) p,x+λh () h C(T, L) x + λh h. (2.) Now, for fixed ε and, ϕ ε (, ) is locally Lipschiz. We fix x, y R, so he funcion ψ defined by ψ(λ) =ϕ ε (, x + λ(y x)) is Lipschiz on [0, ]. This means ha ψ is almos-everywhere differeniable and ψ() ψ(0) = 0 ψ (λ)dλ. So d ϕ ε (, y) ϕ ε (, x) = 0 dλ (ϕ ε(, x + λ(y x))) dλ. Since x + λ(y x) R for each 0 λ, using (2.) we can conclude ha ϕ ε (, y) ϕ ε (, x) C(T, L)R y x, for x, y R.

7 feedback conrollabiliy 2 So, on B R, ϕ ε (, ) has he Lipschiz consan C(T, L)R, independen of ε. Passing o limi in (2.2) we conclude ha ϕ(, ) is Lipschiz on B R wih Lipschiz consan C(T, L)R. We urn nex o he exisence of a feedback law for he minimum energy conrol. Firs we see ha ϕ saisfies he dynamic programming principle { s } ϕ(, x) = inf 2 u 2 dτ+ϕ(s, y(s)); y +Ay+Fy = Bu, y() =x (2.2) for each 0 <s<t and each x H. The proof of (2.2) is sandard, so we skip i. For fixed 0 <T we consider an opimal pair (u,y )on[, T ]; i.e., ϕ(, x) = 2 u 2 dτ and y + Ay + Fy = Bu ; y () =x, y (T )=0. We know such a pair exiss, since he semigroup generaed by A is compac. By usual dynamic programming argumens, we obain ha for a fixed s (, T ), (u,y ) is also an opimal pair on [, s] for he problem in (2.2). Since ϕ(s, ) is locally Lipschiz, by [2], we conclude ha here exiss p s C([, s]; H) such ha (p s ) A p s (F (y )) p s =0 B p s = u a.e. on [, s] p s (s) ϕ(s, y (s)). Here, by ϕ(s, ) we denoe he generalized gradien of ϕ(s, ). For < s s 2 < T, we consider he same argumens, obaining he dual arcs p s C([, s ]; H) and p s 2 C([, s 2 ]; H). Since B p s = u = B p s 2 almos everywhere on [, s ], and boh p s and p s 2 saisfy he adjoin sysem p A p (F (y )) p =0, we conclude by observabiliy inequaliy (2.7) (considered on subinervals of [, s ] for p = p s p s 2 and a = f (y )) ha p s = p s 2 on [, s ]. Using his device for each s [, T ), we obain ha here exiss a unique p C([, T ); H) such ha { p A p (F (y )) p =0 B p = u (2.3) a.e. on [, T ) and, furhermore, p(s) ϕ(s, y (s)) everywhere on (, T ). So we obained he feedback law u (s) B ϕ(s, y (s)) a.e. on [, T ). The proof of Theorem 2. is now complee.

8 22 Mihai Sîrbu Remark 2.2. By dynamic programming argumens we obained also opimaliy condiions for he singular problem, namely (2.3). I remains o sudy he behaviour of p for T. In he linear case (F = 0), his is lim T y (),p() =0. 3. Viscosiy approach This secion is devoed o he sudy of he Hamilon Jacobi equaion (.6) subjec o (.7). In fac we wan o characerize he value funcion ϕ as he viscosiy soluion (unique under exra assumpions) of (.6) and (.7). Since we have o deal wih quadraic growh soluions for he dynamic programming equaion associaed o a conrol problem, we will use he resuls in [6] (especially Secion VII). Firs of all, for echnical reasons, we rewrie (.6) in he form ϕ + Ax, ϕ + Fx, ϕ + 2 B ϕ 2 =0. (3.) The definiion of he viscosiy soluion for (3.) is he usual definiion for he unbounded linear case in [6], since A is a linear maximal monoone operaor on H: Definiion 3.. Le u C([0,T) H). Then u is a subsoluion (respecively, supersoluion) of (3.) if for every ψ :[0,T) H R, weakly sequenially lower semiconinuous such ha ψ and A ψ are coninuous and each g radial, nonincreasing and coninuously differeniable on H and a local maximum (respecively, minimum) (, z) ofu ψ g (respecively, u + ψ + g), we have ψ (, z)+ z,a ψ(, z) + Fz, ψ(, z)+ g(, z) + 2 B ( ψ(, z)+ g(, z)) 2 0 (3.2) (respecively ψ (, z) z,a ψ(, z) Fz, ψ(, z)+ g(, z) + 2 B ( ψ(, z)+ g(, z)) 2 0). (3.3) Since he paricular A we use is self-adjoin and A generaes a compac semigroup, we can choose D := (I + A) self-adjoin, posiive and compac o saisfy (A D + D)x, x x 2. (3.4) We denoed by D he operaor B in [6] since we already denoed by B he conrol operaor.

9 feedback conrollabiliy 23 In order o inerpre condiion (.7), we firs spli i in wo pars: ϕ(t,x)= + for x 0, and (3.5) ϕ(t,0)=0. (3.6) We replace hen he formal condiion (3.5) by lim T y x ϕ(, y) =+ for x 0. (3.7) Since we wan he soluions of (3.) subjec o (3.6) and (3.7) resriced o [0,s] (for 0 s<t) o fi in he framework of [6] (Secion VII), we have o look for soluions u which saisfy he following uniform coninuiy condiions: { for each ε>0 here exiss a modulus mε such ha u(, x) u(, y) m ε ( x y ) for x, y ε, 0 T ε (3.8) and { for each ε>0 here exiss a modulus ρε such ha u(τ,x) u(, e A( τ) x) ρ ε ( τ) for x ε, 0 τ T ε. (3.9) By a modulus, we mean here, as usual, a funcion m : [0, ) [0, ) coninuous, nonincreasing and subaddiive such ha m(0) = 0. Now we can sae he main resul of his secion: Theorem 3.. (i) If here exiss ψ C([0,T) H) a posiive viscosiy supersoluion of (3.)saisfying he final condiion (3.7) and also (3.8) and (3.9), hen he sae sysem (.2) is null conrollable and ψ ϕ (where ϕ is defined in (.5)). (ii) If he sae sysem (.2) is null conrollable and ϕ(, ) is locally Lipschiz for each <T, hen ϕ is he unique posiive viscosiy soluion of (3.) saisfying (3.7), (3.8), (3.9) and ϕ(s, y(s)) 0 for s T (3.0) whenever { y + Ay + Fy = Bu on [, T ] y() =x, y(t )=0 for an u L 2 (, T ; H). Remark 3.. Condiion (3.0) means ha ϕ goes o zero along he admissible rajecories of he sae sysem (.3). We can regard i as a subsiue for (3.6).

10 24 Mihai Sîrbu In order o prove Theorem 3. we need he following lemma, which is in fac a resul in [6] (Secion VII). Lemma 3.. Le g : H R uniformly coninuously on bounded ses and g 0. (i) Then v defined by { } v(, x) = inf 2 u 2 dτ + g(y(t )) : y + Ay + Fy = Bu, y() =x is posiive, coninuous on [0,T] H and is also a soluion of (3.), subjec o v(t,x)=g(x) ( ) x H. Moreover, for each R>0 here exis moduli m R and ρ R such ha v(, x) v(, y) m R ( x y ) for all 0 T, x, y R and (3.) v(τ,x) v(, e A( τ) x) ρ R ( τ) for all 0 τ T, x R. (3.2) (ii) If u C([0,T] H) is a posiive subsoluion and w C([0,T] H) is a posiive supersoluion of (3.), boh saisfying (3.), (3.2) and u(t,x) = w(t,x)( ) x H, hen u w on [0,T] H. Consequenly, he value funcion v is he unique posiive viscosiy soluion of (3.) subjec o v(t,x)= g(x) ( )x H, under he exra assumpions (3.) and (3.2). Proof of Lemma 3.. Since his is a resul in [6](Secion VII) we jus need o verify he hypoheses. Because u 2 u 2 is convex coercive and F : H H is globally Lipschiz, he assumpions concerning he Hamilonian in equaion (3.) are fulfilled. Regarding he comparison par (ii), we jus have o see ha condiions (3.) and (3.2) ogeher imply D-coninuiy (in fac weak sequenial coninuiy, since D is compac) on [0,T) H. This is indeed rue, since we have he inequaliy (see [5], page 259) e A x 2 D +2 e A x 2 e 2 x 2 D, and consequenly e A :(H, D ) (H, ) is bounded for each >0. We consider here x 2 D = Dx, x. Sou and w are D-coninuous, and saisfy (3.) and (3.2). We can conclude ha u w. The exisence par (i) is saed in [6]. Proof of Theorem 3.. (i) Le ψ 0 be a viscosiy supersoluion of (3.), saisfying (3.7), (3.8) and (3.9). Le (, x) [0,T) H and suppose < < 2 < < n < <T saisfy n T for n. We define

11 v :[0, ] H R by { v(, y) = inf feedback conrollabiliy 25 2 u 2 dτ + ψ(,y( )) : y + Ay + Fy = Bu, y() =y (3.3) Since ψ saisfies (3.8) and (3.9), ψ saisfies (3.) and (3.2) for T =. By Lemma 3. (i), since ψ(, ) is uniformly coninuous on bounded ses and posiive, v saisfies also (3.) and (3.2), and v is a posiive viscosiy soluion for (3.) on [0, ] H. Since ψ is a posiive viscosiy supersoluion and ψ(,x)=v(,x)( ) x H, by Lemma 3. (ii) we can conclude ha ψ v on [0, ] H. This in paricular means ha ψ(, x) v(, x). Since A generaes a compac semigroup, here exiss an opimal pair (u,y )on[, ] for he problem (3.3) considered when (, y) =(, x). So we have ψ(, x) v(, x) = 2 u 2 dτ + ψ(,y ( )) and { y + Ay + Fy = Bu on [, ] y () =x. We apply he same device on [, 2 ] o find (u 2,y 2 ) such ha 2 ψ(,y( )) 2 u 2 2 dτ + ψ( 2,y2( 2 )) and { y 2 + Ay2 + Fy 2 = Bu 2 on [, 2 ] y2 ( )=y ( ). Using he same argumen on [ 2, 3 ] and so on o [ n, n ] and hen maching he soluions we find a pair (u,y ) defined on [, T )byy = yn on [ n, n ] and u = u n almos everywhere on [ n, n ], such ha n ψ(, x) 2 u 2 dτ + ψ( n,y ( n )) ( ) n, and (3.4) { y + Ay + Fy = Bu on [, T ) y (3.5) () =x. From (3.4), since ψ( n,y ( n )) 0, we conclude ha u L 2 (, T ; H), and consequenly (3.5) implies ha y C([, T ]; H) (y is defined in T also). By condiion (3.7) we have y (T ) = 0. (Oherwise ψ( n,y ( n )), which conradics ψ(, x) ψ( n,y ( n )).) Using (3.4) we also obain ψ(, x) 2 u 2 dτ, soψ(, x) ϕ(, x). }.

12 26 Mihai Sîrbu (ii) We know ha ϕ saisfies he dynamic programming principle (2.2). Since ϕ(s, ) is locally Lipschiz and posiive, we can use Lemma 3. (i) o conclude ha ϕ is a posiive viscosiy soluion on [0,s) H and saisfies (3.) and (3.2) on [0,s] for each 0 s<t. This implies ha ϕ is a viscosiy soluion on [0,T) H. Also, choosing s and R such ha s>t ε and R> ε, we can see ha ϕ saisfies (3.8) and (3.9) on [0,T). I is easy o prove ha ϕ fulfills condiion (3.0) since 0 ϕ(s, y(s)) s 2 u 2 dτ whenever y + Ay + Fy = Bu, y() =x, y(t )=0 for u L 2 (, T ; H). Passing o limi for s T, we obain (3.0). From Secion 2 we know ha ϕ ϕ ε for each ε>0. Since lim s T y x ϕ ε (s, y) = 2ε x 2 we easily obain ha ϕ mus saisfy (3.7). So, he exisence par is proved. Regarding uniqueness, le ψ be a posiive viscosiy soluion saisfying he assumpions. We already proved in (i) ha ψ ϕ. Now, using he same device in (i), for fixed (, x) [0,T) H and s [, T ), we use Lemma 3. o ge { s } ψ(, x) = inf 2 u 2 dτ + ψ(s, y(s)) : y + Ay + Fy = Bu, y() =x. (3.6) We fix (u, y) such ha y + Ay + Fy = Bu, y() =x, y(t )=0 (3.7) and u L 2 (, T ; H). From (3.6) we obain ha ψ(, x) s 2 u 2 dτ + ψ(s, y(s)). Taking ino accoun ha ψ(s, y(s)) 0 for s T (since ψ saisfies (3.0)), we also have ha ψ(, x) 2 u 2 dτ for each pair (u, y) which saisfies (3.7). So ψ(, x) ϕ(, x). We hus obained ha ψ = ϕ, and he proof of Theorem 3. is complee. Since we proved in Secion 2 ha ϕ is locally Lipschiz, we can use Theorem 3. o conclude ha ϕ is eiher he minimal posiive viscosiy supersoluion of (3.) saisfying (3.7), (3.8) and (3.9) or he unique posiive viscosiy soluion wih he same properies and (3.0) as well. Since (3.0) is a subsiue for (3.6), we can consider ha he value funcion ϕ is he unique

13 feedback conrollabiliy 27 posiive viscosiy soluion of (3.) subjec o he formal final condiion (.7), under he exra assumpions (3.8) and (3.9). 4. Final remarks Remark 4.. Alhough in he framework here considered he operaors A, B and F (u) are self-adjoin, we wroe he opimaliy condiions in erms of A,B and (F (u)), in order o be consisen wih he usual noaion for he adjoin sysem. Remark 4.2. All saemens and proofs in he previous secions can be exended o more general cases. Thus we can consider insead of f C (R) a funcion f depending on he space variable also, f :Ω R R which is measurable in x, C in r, saisfies f(x, 0) = 0 ( )x Ω and r f(x, r) L ( ) (x, r) Ω R. We sill can wrie he sae sysem in he form (.2) if we denoe by F : H H he operaor F (u)(x) =f(x, u(x)) for almos every x Ω. Insead of he energy (.4), we can consider he generalized energy J(u) = ( 2 u Cy 2 )dτ (4.) where C is a linear bounded operaor from H o a differen Hilber space Y. There are choices of Y and C such ha he energy (4.) has a significan meaning. In his case, as we poined ou, he sae sysem has he absrac form (.2), bu he Hamilon Jacobi equaion becomes ϕ (Ax + Fx), ϕ 2 B ϕ Cx 2 =0, (4.2) wih he same final condiion (.7). Since Lemma 3. is sill valid in his case, he only hing ha remains o check is wheher he value funcion ϕ, defined in (.5), replacing he energy by he generalized energy, is sill locally Lipschiz. This is indeed rue, since from [4] we have he sronger observabiliy inequaliy p() 2 L 2 (Ω) C (T, a L (Q)) ( B p(τ) 2 L 2 (Ω) dτ + ) Cy(τ) 2 Y dτ (4.3)

14 28 Mihai Sîrbu if p saisfies he adjoin sysem { p + p + ap = C Cy on (, T ) Ω p(τ,x)=0 on(, T ) Ω. From here, we can deduce ha ϕ(, x) 2 C (T ) x 2, and using again Lemma 2. and he observabiliy inequaliy (4.3) we obain ha ϕ(, ) has he Lipschiz consan C (T, L)R on B R. Remark 4.3. The resuls presened are rue for differen kinds of homogenous boundary condiions of he ype y ν + αy = 0 on Σ, where by y ν we denoed he normal derivaive and α 0 is a consan. In his case we can define A : D(A) A by D(A) ={u H 2 (Ω) : u ν + αu =0on Ω} and Au = u, and he sae sysem sill has he same semigroup form (.2). Remark 4.4. Since ϕ and ϕ ε are weakly sequenially coninuous on [0,T) H and ϕ ε (, x) ϕ(, x) asε 0 for each (, x) [0,T) H, we can use Dini s crierion o conclude ha ϕ ε (, x) ϕ(, x) uniformly for (, x) in he weak (sequenially) compac se [0,s] B R, for each 0 s<t. Acknowledgmen. The auhor express his warmes hanks o Professor V. Barbu for proposing his subjec of sudy and for helpful suggesions. References [] V. Barbu and G. Da Prao, Hamilon Jacobi Equaions in Hilber Spaces, Research Noes in Mahemaics, Piman Advanced Publishing Program, Boson London Melbourne, 983. [2] V. Barbu, Opimal Conrol of Variaional Inequaliies, Research Noes in Mahemaics, Piman Advanced Publishing Program, Boson London Melbourne, 984. [3] F. Clarke and R. Viner, The relaionship beween he maximum principle and dynamic programming, SIAM J. Conrol Opim., 25 (987), [4] A.V. Fursikov and O. Yu. Imanuvilov, Conrollabiliy of Evoluion Equaions, Lecure Noes Series 34 (996), RIM Seoul Universiy, Korea. [5] M.G. Crandall and P.L. Lions, Hamilon Jacobi equaions in infinie dimensions, Par IV, Unbounded linear erms, J. Func. Anal., 90 (990), [6] M.G. Crandall and P.L. Lions, Hamilon Jacobi equaions in infinie dimensions, Par V, Unbounded linear erms and B-coninuous soluions, J. Func. Anal., 97 (99), [7] A. Pazy, Semigroups of Linear Operaors and Applicaions o Parial Differenial Equaions, Springer Verlag, 983. [8] M. Sirbu, A Riccai equaion approach o he null conrollabiliy of linear sysems, Comm. Appl. Anal., o appear.