Null controllability of neutral system with infinite delays

Transcription

1 Null conrollabiliy of neural sysem wih infinie delays Davies, I. and Haas, O.C.L. Pre-prin deposied in Curve Sepember 215 Original ciaion: Davies, I. and Haas, O.C.L. (215) Null conrollabiliy of neural sysem wih infinie delays. European Journal of Conrol, volume In press. DOI: 1.116/j.ejcon hp://dx.doi.org/1.116/j.ejcon Lavoisier Copyrigh and Moral Righs are reained by he auhor(s) and/ or oher copyrigh owners. A copy can be downloaded for personal non-commercial research or sudy, wihou prior permission or charge. This iem canno be reproduced or quoed exensively from wihou firs obaining permission in wriing from he copyrigh holder(s). The conen mus no be changed in any way or sold commercially in any forma or medium wihou he formal permission of he copyrigh holders. CURVE is he Insiuional Reposiory for Covenry Universiy hp://curve.covenry.ac.uk/open

2 Null conrollabiliy of neural sysem wih infinie delay I. Davies* and O. C. L. Haas* * Conrol Theory and Applicaion Cenre (CTAC), The Fuures Insiue, Faculy of Engineering and Compuing, Covenry Universiy, 1 Covenry Innovaion village, Covenry Universiy Technology Park, Cheeah Road, Covenry, CV1 2TL, Unied Kingdom (Auhors daviesi6@uni.covenry.ac.uk; o.haas@covenry.ac.uk) Absrac Sufficien condiions are developed for he null conrollabiliy of neural conrol sysems wih infinie delay when he values of he conrol lie in an m-dimensional uni cube. Condiions are placed on he perurbaion funcion which guaranee ha; if he unconrolled sysem is uniformly asympoically sable and he conrol sysem saisfies a full rank condiion so ha K(λ)ξ(exp( λh)), for every complex λ, where K(λ) is an n n polynomial marix in λ consruced from he coefficien marices of he conrol sysem and ξ(exp( λh)) is he ranspose of [1, exp( λh),, exp( (n 1)λh)], hen he conrol sysem is null conrollable wih consrain. Keywords: Neural sysems, infinie delay, null conrollabiliy, rank condiion, sable Subjec classificaion codes : 93Cxx, 93C23, 34k4, 93Bxx, 93B5 1. Inroducion Neural funcional differenial sysems have applicaions in many areas of sudy because of is imporance as mahemaical models for phenomena in boh science and engineering; see Corduneanu (22), Kharovskii and Pavlovskaya (213) and references herein. The conrollabiliy of neural funcional differenial sysems, in paricular null conrollabiliy of neural funcional differenial sysems have been sudied, see for example Onwuau (1984) and Underwood and Chukwu (1988). These sudies have been exended o neural funcional differenial sysems wih infinie delays in (Onwuau, 1993; Balachandran and Leelamani, 26; Umana, 28, 211; Dauer e al. 1998; Davies, 26). In Dauer e al. (1998), null conrollabiliy of neural funcional differenial sysems having infinie delay has been sudied using he Schauder fixed poin heorem. In Umana (28), he resul was based on he uniform asympoical sabiliy of he unconrolled sysem and he conrollabiliy of he linear conrol sysem. In Davies (26), i was obained by exploiing he sabiliy of he free sysem and he well known rank crierion for properness. The approach used in his paper is differen. I is shown ha when he conrols of a neural funcional differenial sysem wih infinie delays are resrained o lie in he uni cube, he requiremens for null conrollabiliy are ha he sysem has full rank wih he condiion ha K(λ)ξ(exp( λh)) and an addiional condiion ha he rivial soluion of he unconrolled sysem be uniformly asympoically sable. The resuls obained by his mehod will exend ha of (Underwood and Chukwu, 1988; Jacobs and Langenhop, 1976; Rivera Rodas and Langenhop, 1978) o neural funcional differenial sysem having infinie delays by using he Schauder fixed poin heorem. The res of he paper is organised as follows. Secion 2 conains he mahemaical noaions, preliminaries and problem definiion. In Secion 3 sabiliy heorems for he sysem are saed. Secion 4 develops and proves he conrollabiliy heorems for he sysem; he main resul of he paper is also developed and proved in his secion. Finally, Secion 5 conains numerical examples of he heoreical resuls prior o he conclusions. Corresponding auhor daviesi6@uni.covenry.ac.uk 1

3 2. Basic noaions, preliminaries and definiions Suppose h > is a given number, E = (, ), E n is a real n dimensional Euclidean space wih norm. Le J be any inerval in E, he convenion W 2 () ( J, E n ) will represens he Lebesgue space of square-inegrable funcions from J o E n, and W 2 (1) ([ h, ], E n ) is he Sobolev space of all absoluely coninuous funcions x: [ h, ] E n whose derivaives are square inegrable. C = C([ h, ], E n ) is he space of coninuous funcion mapping he inerval [ h, ] ino E n wih he norm, where φ = sup h<s φ(s). Define he symbol x C by x (s) = x( + s), h s. This paper will consider neural funcional differenial sysem wih infinie delay of he form d d D()x = L(, x, u) + A(θ)x( + θ)dθ }, (1) x = φ, and is perurbaion d d D()x = L(, x, u) + A(θ)x( + θ)dθ 2 + f(, x, u()) (2) hrough is linear base conrol sysem d D()x d = L(, x, u), (3) and is free sysem d d D()x = L(, x, ) + A(θ)x( + θ)dθ, (4) where, D()x = x() A ()x( h), L(, x, u) = A 1 ()x() + A 2 ()x( h) + B()u(), wih he following assumpions: (i) A (), A 1 () and A 2 () are coninuous n n marices (ii) B(), is a coninuous n m consan marix (iii) A(θ) is an n n marix whose elemens are square inegrable on (, ] (1) (iv) f: [, ) W 2 E n E n is a nonlinear coninuous marix funcion. I is assumed ha f saisfy enough smoohness condiions o ensure ha a soluion of (2) exiss hrough each (, φ), is unique, and depends coninuously upon (, φ) and can be exended o he righ as long as he rajecory remains in a bounded se [, ) C. These condiions are given in Cruz and Hale, (197). Le x(, φ) be a soluion of (3) wih u = and se T(, )φ = x (, φ), φ W 2 (1). Then (1) (1) T(, ) is a coninuous linear operaor from W 2 W2. There is an n n marix funcion X(, s) which is defined on s 1, J = [, ), coninuous in s from he righ of bounded variaion in s; X(, s) =, < s 1, such ha X(, s) saisfies D()X(, s) = L(, X s (, s), ), s. Now, define he n n marix funcion X as, h s <, X (s) = { I, s =,

4 where I is he ideniy marix. Wrie T(, s)x (s) = X( +, s) = X (, s)(s), so ha T(, s)i = X(, s). A soluion of (3) hrough (, φ) saisfies he equaion or x (, φ, u) = T(, )φ + T(, )X B(s)u(s)ds, x (, φ, u) = x (, φ, ) + X(, s)b(s)u(s)ds. (5) In a similar manner, any soluion of sysem (2) will be given by x (, φ, u, f) = x (, φ, ) + X(, s)b(s)u(s)ds + X(, s) A(θ)x( + θ)dθds + X(, s) f(s, x s, u(s))ds, (6) Define he marix funcions Z by Z(, s) = X(, s)b(s), (7) for s, i follows hen from (6) ha x (, φ, u, f) = x (, φ, ) + Z(, s)u(s)ds + X(, s) A(θ)x( + θ)dθds + X(, s) f(s, x s, u(s))ds. (8) The conrols of ineres will be funcions u: [, ) C m which are square inegrable on finie inervals wih values in C m, and such conrols will be denoed by U, where C m = {u: u E m, u j 1, j = 1, 2,, m}. Definiion 1. The sysem (3) is proper on [, 1 ] if η T Z( 1, s) = almos everywhere s [, 1 ] implies η = for η E n, where η T is he ranspose of η. If (3) is proper on each inerval [, 1 ], hen he sysem is said o be proper in E n Definiion 2. Sysem (3) is said o be compleely conrollable on [, 1 ], if for each funcion φ W 2 (1), x1 E n here is an admissible conrol u L 2 ([, 1 ], E m ) such ha he soluion x(,, φ, u) of (3) saisfies x (,, φ, u) = φ, x 1 (,, φ, u) = x 1. I is compleely conrollable on [, 1 ] wih consrains, if he above holds wih u U. Definiion 3. The sysem (2) is null-conrollable on [, 1 ] if for each φ W 2 (1) ([ h, ], E n ), here exiss a u L 2 ([, 1 ], E m ) such ha he soluion of (2) saisfies x (,, φ, u, f) = φ, x 1 (,, φ, u, f) =. The sysem (2) is null-conrollable wih consrains if he above holds wih conrol u U Definiion 4. The domain U of null-conrollabiliy of (3) wih consrains is he se of all iniial funcions φ W 2 (1) for which he soluion x(, φ, u) of (3) wih x (,, φ, u) = φ, x 1 (,, φ, u) = a some 1, u U 3

5 Definiion 5. The reachable se of (3) is a subse of E m given by P(, ) = { Z(, s)u(s)ds: u L 2 ([, ], E m ) }. If he conrols are in L 2 ([, ], C m ), we define he consrain reachable se by R(, ) = { Z(, s)u(s)ds: u L 2 ([, ], C m ) }. Noe ha P(, ) is a subse of E m which is symmeric abou zero. Definiion 6. The conrollabiliy marix of (3) will be given by where Z T is he ranspose of Z. 3. STABILITY RESULTS W(, ) = Z(, s)z T (, s)ds, Here, some definiions, lemmas and heorem which are fundamenal o he developmen of he sabiliy resuls for he sysem (4) are given. Consider he sysems (3) defined by d d D()x = L(, x, ), (9) Definiion 7. The soluion x = of (9) is called sable a if and here exiss a b = b() > such ha if φ b, hen he soluion x(, φ) exiss for. Also, for each ε > here exiss a δ = δ(, ε) > such ha if φ δ, hen he soluion x(, φ) of (9) saisfies x (, φ) ε for. The rivial soluion of (9) is called sable if i is sable for each. I is called uniformly sable if i is sable and δ does no depend on. I is uniformly asympoically sable if i is uniformly sable for every η > and every > here exiss T(η) independen of and H > independen of η, such ha φ H implies x (, φ) η, for all + T(η). Definiion 8. The soluion x = of (9) is uniformly asympoically sable if and only if here exiss consan c >, k > such ha x (, φ) k exp[ c( )] φ, for all. Definiion 9. Le D(, φ) = Dφ, and consider he homogeneous difference equaion { D()x =, x = φ, Dφ = (1) (1) D() is uniformly sable if here are consans α, β > such ha for J, φ W 2 soluion of (1) saisfies x (, φ) β exp[ α( )] φ, for all. he The nex wo lemmas are due o Cruz and Hale (197), hey are very imporan for he analysis and developmen of he properies for operaor D( ), and for he overall sabiliy resul in his secion. 4

6 Lemma 1. Le A be an n n consan marix. The operaor Dφ = φ() A φ( h) is uniformly sable if all he roos of he equaion de[i A r h ] = have moduli less han 1. This holds if A < 1. Lemma 2. Dφ is uniformly sable if here are consans α, β > such ha for any φ W 2 (1), [τ, ), he soluion x(, φ) of (1) saisfies x (, φ) β φ e α( ),. The nex heorem is developed following Theorem 1 of Sinha (1985) and Corollary 2 of Hale (1974) for funcional differenial equaions wih infinie delay; see also Corollary 3.8 of Davies (26) and references herein for neural funcional differenial sysems wih infinie delays. Theorem 1. In sysem (4), assume here is a v >, and a consan M such ha A(θ) Mexp(vθ) M for θ (, ] and if B(λ) = {λ C: Re λ, de (λ) = } =, where (λ) = λ(i A exp( λh)) A 1 A 2 exp( λh) exp(λθ) A(θ)dθ, Then he soluion of (4) is uniformly asympoically sable if X(, s) k exp( α( )), s, k >, α > 4. Conrollabiliy Resuls This secion develops and proves necessary and sufficien conrollabiliy condiions for sysems (3) by exploiing he mehod in (Jacobs and Langenhop, 1976; Rivera Rodas and Langenhop, 1978). Some relevan conrollabiliy resuls which are relevan o his sudy are also given. Lemma 3. The sysem (3) is compleely conrollable if and only if W(, 1 ) is non-singular Proof. The proof can be observed from Proposiion 3.1 of Dauer and Gahl (1977). Proposiion 1. The sysem (3) is conrollable on [, 1 ] if and only if in R(, ) Proof. This is Lemma 3.2 of Davies (26). Proposiion 2. The sysem (3) is compleely conrollable on [, 1 ] if and only if R(, ). Proof. This is Theorem 3.3 of Davies (26). Proposiion 3. The following are equivalen. (i) (ii) (iii) W(, 1 ) is non-singular sysem (3) is compleely conrollable on [, 1 ], 1 > sysem (3) is proper on [, 1 ], 1 > Proof. This is Proposiion 3 of Onwuau (1993). Lemma 4. The sysem (3) is compleely conrollable on [, 1 ] if and only if i is conrollable on [, 1 ]. Proof. The proof follows immediaely from Proposiion 4.1 and

7 Le x: [α, β] E p, p a posiive ineger, be absoluely coninuous and define he differenial operaor for neural sysems D by (Dx)() = x () = dx() d, almos everywhere on [α, β]. Higher powers of he operaor D are defined inducively by D k+1 = DD k, wih domain equal o all x: [α, β] E p, such ha D k x is absoluely coninuous on [α, β]. Noe ha, by D he ideniy (D x)() = x(), [α, β]. Define W υ 2, (τ, E μ ), μ a posiive ineger and υ a nonnegaive ineger o be he collecion of all x: (, τ] E μ such ha x() = for and he resricion x [,τ] is in W υ 2 ([, τ], E μ ), and adop he convenion W 2 ([, τ], E μ ) = L 2 ([, τ], E μ ). For f W υ 2, (τ, E μ ) define he shif operaor S by (Sf)() = f( h), τ. (11) Define S o be he ideniy operaor on W υ 2, (τ, E μ ) and ake S k+1 = SS k, k =, 1, 2, by inducively using (11). Observe from he definiion of he differenial operaor D, and he shif operaor S ha for υ 1, if he funcion space W υ 2, (τ, E μ ) is aken as a common domain for he operaors S and D, hen S and D commue in his seing and each commues wih muliplicaion by a scalar (elemen in E). For any n n marix A and n m marix B, one can define n υm marix by P υ [A, B] = [B,, A υ 1 B], for inegers υ 1. Consider he neural sysem wih delay of he form (d d)(x A x( h)) = A 1 x() + A 2 x( h) + Bu(), (12) where A i, i =, 1, 2 are n n consan marices, B is chosen o be n 1 consan real marix. The soluion x(,, u) of (12) is he resricion o [ h, ] of he soluion x W (1) 2, (τ, E n ) of he equaion (ID A SD A 1 A 2 S)x = Bu. Now define he marix Q(D, S) by he equaion Q(D, S) = ID A SD A 1 A 2 S, and le P(D, S) = adj Q(D, S), where adj denoes he ransposed marix of cofacors. Some basic relaionship exiss beween hese wo operaors which by Jacobs and Langenhop (1976) can be expressed as n 1 P(D, S) = P i (D)S i = P i(s)d i, (13) i= n 1 i= where he n n marix polynomials P i (D), P i(s) are a mos of degree n 1 in heir argumen. Using he polynomial P i (D) in (13) define a unique marix operaor by K(D) = [P (D)B, P 1 (D)B,, P n 1 (D)B], Now, he operaor K(D) can be wrien in he form of a polynomial o ge n 1 K(D) = K i D n 1 i, i= (14) where, he K i, i =, 1,, n 1 are n n consan real marices, and le ξ(exp( λh)) be he ranspose of [1, exp( λh),, exp( (n 1)λh)] for all complex numbers λ, h >. Theorem 2. Le τ > nh, hen for (12) o be conrollable on [, τ] i is necessary and sufficien for rank P n [A, B] = n and K(λ)ξ(exp( λh)) for every complex λ. 6

8 Proof. This is Theorem 3.4 of Rivera Rodas and Langenhop, (1978). Corollary 1. Le τ > nh, [, τ] and assume ha sysem (12) saisfies he following (i) rank P n [A, B] = n ; (ii) K(λ)ξ(exp( λh)), for every complex λ. Then, sysem (12) is compleely conrollable on [, τ]. Proof. If condiion (i) and (ii) holds, hen by Theorem 2, he sysem (12) is conrollable on [, τ]. This by Lemma 4 implies ha sysem (12) is compleely conrollable on [, τ]. Conversely, if sysem (12) is compleely conrollable on [, τ], hen i is conrollable by Lemma 4, and by Theorem 2; rank P n [A, B] = n and K(λ)ξ(exp( λh)) for every complex λ, and he proof is complee. Theorem 3. In sysem (1), assume he following (i) A, A 1, A 2 are n n consan marices, B is n 1 consan real marix (ii) for τ > nh, rank P n [A, B] = n ; (iii) K(λ)ξ(exp( λh)), for every complex λ, (iv) sup{re(λ), de Δ( λ) = } <, (v) Δ(λ) = λ(i A exp( λh)) A 1 A 2 exp( λh) exp(λθ) A(θ)dθ Dφ = φ() A φ( h) is uniformly sable. Then, sysem (1) is null conrollable wih consrains on (, ), > τ. Proof. Because of (i), (ii) and (iii) sysem (1) is conrollable on [, τ] by Theorem 2. Hence, In R(, τ) by Proposiion 1. By condiion (iv), and (v) he sysem (1) wih u = saisfies x (, φ, ) as. Hence, a some 1 >, x 1 (, φ, ) In R(, τ) and hence In U, he domain of null conrollabiliy of (1). Suppose for he conrary ha In U. Since x = is a soluion of (1) wih u =, hen U. This implies ha, here exiss a sequence {φ i } W (1) 2 such ha φ i as i and φ i U, for any i, herefore φ i. I follows from he variaion of consan formula (5) ha; x i (, φ i, u) = x (, φ i, ) + 1 Z( 1, s)u(s)ds. Le z i = x 1 (, φ i, ). Then, since φ i U, x 1 (, φ i, u), for any i, and so z i R(, 1 ), for any 1 > and z i. Bu z i as i, and In R(, 1 ) which is a conradicion. Therefore, In U, and hence here exiss a ball S 2 around which is conained in U. Again, by (iv) here exiss some 2 <, x 2 (, φ, ) S 2. Therefore, using 2 as iniial poin and x 2 (, φ, ) ψ as iniial funcion, here exiss u U and 3 > 2 such ha, he soluion x( 2, x 1 (, φ, ), u) of (1) saisfies x 3 (, 1, x 1, u) =, and he proof is complee Main resul The main resul for he neural conrol sysem wih infinie delay will now be developed and proved in his secion. Theorem 4. Assume for sysem (2) ha (i) he consrain se U is an arbirary compac se of E n 7

9 (ii) Dφ = φ() A φ( h) is uniformly sable (iii) he sysem (4) is uniformly asympoically sable; so ha he soluion of (4) saisfies x (, φ, ) k exp( α( )) φ, α >, k >. (iv) he sysem (3) is compleely conrollable (v) The coninuous funcion f saisfies f(, x( ), u( )) exp( b) π(x( ), u( )), for all (, x( ), u( )) [, ) W () 2 L 2, where and b α. 8 π(x( ), u( ))ds M <, Then, he sysem (2) is null conrollable. Proof. By (iv), W 1 exiss for each 1 >. Assuming he pair of funcions x, u forms a pair o he inegral equaions u() = Z( 1, s) T W 1 (, 1 ) [x 1 (, φ, ) + X( 1, s) A(θ)x( + θ)dθds X( 1, s) f(s, x( ), u( ))ds], (15) for some suiably chosen 1, u() = v(), [ h, ] x() = x (, φ, ) + Z(, s)u(s)ds + X(, s) A(θ)x( + θ)dθds + X(, s) f(s, x( ), u( ))ds, (16) x() = φ(), [ h, ]. Then u is square inegrable on [ h, 1 ] and x is a soluion of (2) corresponding o u wih iniial sae x() = φ. Also, x( 1 ) =. I is necessary o show now ha u: [, 1 ] U is in he arbirary compac consrain subse of E m, ha is u() a 1, for some consan a 1 >. By (ii) and (iii), and he coninuiy of B in compac inervals, i follows ha Z( 1, s) T W 1 (, 1 ) d 1, 1 x 1 (, φ, ) + X( 1, s) A(θ)x( + θ)dθds d 2 exp( α( 1 )), for some d 1 >, d 2 >. Hence, 1 u() d 1 [d 2 exp( α( 1 )) + k exp( α( s)) exp( bs) π(x( ), u( ))ds] and herefore, u() d 1 d 2 exp( α( 1 )) + d 1 kmexp( α( 1 )), (17) since b α and s. Hence, 1 from (17) can be chosen sufficienly large ha u() a 1, [, 1 ], showing ha u is admissible conrol. I remains o prove he exisence of a pair of he inegral equaions (15) and (16). Le W () 2 represen he Banach space of all funcions (x, u): [ h, 1 ] [ h, 1 ] E n E m, where x W () 2 ([ h, 1 ], E n ); u L 2 ([ h, 1 ], E m ) wih he norm defined by (x, u) = x 2 + u 2, where x 2 = { 1 h x(s) ds } 1 2 Define he operaor T: W 2 () W 2 () by T(x, u) = (y, w), where ; u 2 = { u(s) ds 1 h } 1 2.

10 w() = Z( 1, s) T W 1 (, 1 ) [x 1 (, φ, ) + X( 1, s) A(θ)x( + θ)dθds X( 1, s) f(s, x s, u(s))ds], (18) for J = [, 1 ] and v() = w() for [ h, ]. y() = x (, φ, ) + Z(, s)u(s)ds + X(, s) A(θ)x( + θ)dθds + X(, s) f(s, x s, u(s))ds, (19) for J and y() = φ() for [ h, ]. I is clear from (17) ha v() a 1 for J and also v [ h, ] U, so ha v() k. Hence, v 2 a 1 ( 1 + h ) 1 2 = b. Again, y() d 2 exp ( α( 1 )) + d 3 v(s) ds + kmexp( α( 1 )), where d 3 = sup Z(, s). Since α >,, i follows ha y() d 2 + d 3 a 1 ( ) + km = b 1, J and y() sup φ() = δ, [ τ, ]. Hence, if λ = max[b 1, δ], hen y 2 λ( 1 + h ) 1 2 = b 2 <. Le r = max[b 1, b 2 ]. Then leing Q(r) = {(x, u) W 2 () : x 2 r, u 2 r}, i follows ha T: Q(r) Q(r). Now, since Q(r) is closed, bounded and convex, by Riesz heorem (see Kanorovich and Akilov, 1982 ), i is relaively compac under he ransformaion T. Hence, he Schauder s fixed poin heorem implies ha T has a fixed poin. Hence, sysem (2) is null conrollable. 5. Example Consider he neural conrol sysem (d d)(x() A x( h)) = A 1 x() + A 2 x( h) + C exp (vθ)x( + θ)dθ + Bu(), (2) and is perurbaion (d d)(x() A x( h)) = A 1 x() + A 2 x( h) + C exp (vθ)x( + θ)dθ + Bu() + f(, x(), x( h), u()). (21) Is linear conrol base sysem is given by (d d)(x() A x( h)) = A 1 x() + A 2 x( h) + Bu(), (22) and is free sysem (d d)(x() A x( h)) = A 1 x() + A 2 x( h) + C exp (vθ)x( + θ)dθ, where (23) 9

11 A = ( ), A 1 = ( ), A 2 = ( ), C = ( 1 4 ), f(, x(), x( h), u()) = ( exp ( α)sin(x() + x( h)) cos u() ), α >, B = ( 1 ). Now, use Lemma 1 o check ha, he operaor D is uniformly sable as follows: The condiion de[i A r h ] = of Lemma 1 gives, 1 1 ( 2 r h 1 ) =, 2 r h 1 which implies 1 (1 4)r 2h =, and r = (1 2) 1 h. Hence, he operaor D is uniformly sable if h >. Nex, observe by Theorem 1 ha he characerisic roo of (23) is (4 exp( 2λh))λ 2 + (12 2 exp( λh) exp( 2λh))λ (λ + 1) exp (λ + v)θdθ =, (24) and all he roos of (23) have negaive real par. Hence by Theorem 1, sysem (23) is uniformly asympoically sable. Finally, check ha (22) is conrollable as follows: rank[b, A B] = rank ( ) = 2, P (λ) = adj(i λ A 1 ) λ = adj ( 1 λ + 2 ) = ( λ λ + 1 ), P 1 (λ) = adj( A λ A 2 ) λ 2 = adj [( λ 2 ) ( (λ + 1) )] = (1 2 ), 1 2 λ 2 K(λ) = [P (λ)b, P 1 (λ)b] 1 (λ + 1) 2 = ( 1 + λ ), 1 (λ + 1) 2 K(λ)ξ(exp( λh)) = ( 1 + λ ) ( 1 exp( λh) ). Observe ha for all complex λ, h >, K(λ)ξ(exp( λh)) ( ). 1 Therefore sysem (22) is conrollable on (, τ), τ > 2h. Moreover, f(, x(), x( h), u()) = exp ( α)sin(x() + x( h)) cos u() exp ( α) 1 (25) Hence, all he condiions of Theorem 4 are saisfied and sysem (21) is null conrollable. 5.1 Simulaion sudies The sabiliy and conrollabiliy of he open loop sysem (21) can be illusraed using Simulink and MATLAB based simulaion sudies. The simulaion model parameers are as given (21) wih he defaul parameer seing and a square wave inpu where α and v are chosen o be 2 and 1 respecively wih h =.25s. Fig. 5.1 depics he sabiliy and conrollabiliy of he saes when simulaion is done wih he linear conrol base sysem i.e. (22), and when he simulaion is done wih he perurbaion funcion (see (21)). The

12 ampliude of simulaion is observed o be slighly higher wih he perurbaion funcion and faser response when he simulaion is done wihou he perurbaion funcion. The seling imes for he sysems wihou he perurbaion funcion are also observed o be faser; his is as expeced and depends on he assumpions placed on he perurbaion funcion (see (25)). The simulaion showed ha, he sysem (23) is sable and he overall conrol sysem (21) is conrollable. Figure 5.1. Simulaion of conrol inpu and sysem saes wih perurbaion funcion and linear conrol base sysem Conclusion Null conrollabiliy for neural funcional differenial sysem wih infinie delay have been developed and proved. The resuls were obained wih respec o he sabiliy of he free sysem and he linear conrol base sysem having full rank wih he condiion ha K(λ)ξ(exp( λh)), wih he assumpion ha he perurbaion funcion fsaisfies he smoohness and growh condiion placed on i. Acknowledgemen The auhors are hankful o he Rivers Sae Scholarship Board, Rivers Sae Secrearia Complex, Por Harcour, Rivers Sae, Nigeria for he supporing his research 11

13 References Balachandran, K., Leelamani, A., 26. Null conrollabiliy of neural evoluion inegrodifferenial sysems wih infinie delay. Mahemaical Problems in Engineering 26, Corduneanu, C., 22. Absolue sabiliy for neural differenial sysems. European Journal of Conrol 8, Cruz, M. A., Hale, J. K., 197. Sabiliy of funcional differenial equaions of neural ype. Journal of Differenial Equaions 7, Dauer, J., Balachandran, K., Anhoni, S., Null conrollabiliy of nonlinear infinie neural sysems wih delays in conrol. Compuers & Mahemaics wih Applicaions 36, Dauer, J., Gahl, R., Conrollabiliy of nonlinear delay sysems. Journal of Opimizaion Theory and Applicaions 21, Davies, I., 26. Euclidean null conrollabiliy of infinie neural differenial sysems. Anziam Journal 48, Hale, J. K., Funcional differenial equaions wih infinie delays. Journal of Mahemaical Analysis and Applicaions 48, Jacobs, M. Q. Langenhop, C., Crieria for funcion space conrollabiliy of linear neural sysems. SIAM Journal on Conrol and Opimizaion 14, (16)-[9] Kanorovich, L. M., Akilov, G. P., Funcional Analysis, Pergamon Press, Oxford. Kharovskii, V., Pavlovskaya, A., 213. Complee conrollabiliy and conrollabiliy for linear auonomous sysems of neural ype. Auomaion and Remoe Conrol 74, Onwuau, J. U., On he null-conrollabiliy in funcion space of nonlinear sysems of neural funcional differenial equaions wih limied conrols. Journal of Opimizaion Theory and Applicaions 42 (3), Onwuau, J. U., Null conrollabiliy of nonlinear infinie neural sysem. Kyberneika 29, Rivera Rodas, H., Langenhop, C., A sufficien condiion for funcion space conrollabiliy of a linear neural sysem. SIAM Journal on Conrol and Opimizaion 16, Sinha, A.S. C., Null conrollabiliy of nonlinear infinie delays sysems wih resrained conrols. Inernaional Journal of Conrol 42, Umana, R., 28. Null conrollabiliy of nonlinear infinie neural sysems wih muliple delays in conrol. Journal of Compuaional Analysis & Applicaions 1, Umana, R., 211. Null conrollabiliy of nonlinear neural Volerra inegrodifferenial sysems wih infinie delay. Differenial Equaions & Conrol Processes (ISSN ) 2, Underwood, R. G., Chukwu, E. N., Null Conrollabiliy of Nonlinear Neural Differenial Equaions. Journal of Mahemaical Analysis and Applicaions 129(2),