The L p -Version of the Generalized Bohl Perron Principle for Vector Equations with Infinite Delay

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1 Advances in Dynamical Sysems and Applicaions ISSN , Volume 6, Number 2, pp (211) hp://campus.ms.edu/adsa The L p -Version of he Generalized Bohl Perron Principle for Vecor Equaions wih Infinie Delay Michael Gil Ben Gurion Universiy of he Negev P.. Box 653, Beer-Sheva 8415, Israel gilmi@bezeqin.ne We consider he vecor equaion ẏ() = Absrac d τ R(, τ)y( τ), where R(, τ) is an n n-marix-valued funcion. I is proved ha, if he nonhomogeneous equaion ẋ() = d τ R(, τ)x( τ) + f() ( ) wih f L p ([, ), C n ) (p 1) and he zero iniial condiion, has a soluion x L p ([, ), C n ), hen he considered homogeneous equaion is exponenially sable. AMS Subjec Classificaions: 34K2. Keywords: Funcional differenial equaion, linear equaion, exponenial sabiliy. 1 Inroducion Recall ha he Bohl Perron principle means ha he homogeneous ordinary differenial equaion (ODE) ẏ = A()y ( ) wih a variable n n-marix A(), bounded on [, ) is exponenially sable, provided he nonhomogeneous ODE ẋ = A()x + f() wih he zero iniial condiion has a bounded soluion for any bounded vecor valued funcion f, cf. [4]. In [7, Theorem 4.15] he Bohl Perron principle was generalized o a class of rearded sysems wih finie delays; besides he asympoic (no exponenial) sabiliy was proved. The resul from [7] was aferwards considerably developed, cf. Received February 1, 211; Acceped April 2, 211 Communicaed by Elena Braverman

2 178 Michael Gil he books [1, 1] and very ineresing papers [2, 3], in which he generalized Bohl Perron principle was effecively used for he sabiliy analysis of he firs and second order scalar equaions. In paricular, in [2] he scalar non-auonomous linear funcional differenial equaion ẋ() + a()x(h()) = is considered. The auhors give sharp condiions for exponenial sabiliy, which are suiable in he case ha he coefficien funcion a() is periodic, almos periodic or asympoically almos periodic, as ofen encounered in applicaions. In he paper [3], he auhors provide sufficien condiions for he sabiliy of raher general second-order delay differenial equaions. In he presen paper we derive a resul similar o he Bohl Perron principle in he erms of he norm of he space L p, which we will call he L p -version of he generalized Bohl Perron principle. In he case L our resul is deeply conneced wih he Bohl Perron principle. In Secion 3 below, we show ha he L p -version can be effecively used for he sabiliy analysis. In paricular, in he case p = 2 he sabiliy analysis can be reduced o he esimaes for he characerisic marix-valued funcions. Such esimaes can be found, for insance, in [5, 6]. As i is well-known, he basic mehod for he sabiliy analysis of vecor ime-varian equaions is he direc Lyapunov mehod. By ha mehod many very srong resuls are obained. Bu finding Lyapunov s ype funcionals for equaions wih infinie delay is usually difficul. A he same ime, in Secion 3 we sugges explici sabiliy condiions. Besides, we improve and generalize [5, Theorem 9.2.2] and [8, Theorem 5.2.2] in he case of linear equaions wih coninuous delay. Le C n be a complex Euclidean space wih he scalar produc (.,.), he Euclidean norm. n = (.,.) and he uni marix I; for an n n-marix A, A n = sup v C n Av n v n is he specral norm. Denoe by L p (ω) L p (ω, C n ) (p 1) he space of funcions u defined on a se ω R wih values in C n and he finie norm u L p = u L p (ω) := [ u() p nd] 1/p ω (1 p < ), and u L (ω) = vrai sup u(x) n. In addiion, C(ω) C(ω, C n ) is he space of coninuous funcions defined on ω wih values in C n and he finie sup-norm. C. For a x ω linear operaor M acing from L p (ω) ino L p (ω 1 ), pu M L p = sup u L p (ω) Mu L p (ω 1 ). Consider in C n he equaion ẏ() = u L p (ω) d τ R(, τ)y( τ) ( ; ẏ() = dy/d), (1.1) where R(, τ) is an n n-marix-valued funcion defined on [, ) 2, which is coninuous in for each τ and righ-coninuous in τ for each. The inegral in (1.1) is undersood as he improper vecor Riemann Sieljes inegral. For he deails see for insance [8, p ].

3 The Generalized Bohl Perron Principle 179 Everywhere below i is assumed ha he variaion of R(, τ) in τ is uniformly bounded on [, ): where V (R) := sup d τ R(, τ) n <, η d τ R(, τ) n = lim d τ R(, τ) n, η and he laer inegral on a finie segmen [, η] means he limi, if i exiss, of he sums m 1 k=1 R(, s (m) k+1 ) R(, s(m) k ) n ( = s (m) < s (m) 1 <... < s (m) m = η) as max s (m) k+1 k s(m) k. Take he iniial condiion y() = φ() ( < ) (1.2) for a given φ C(, ) L 1 (, ). A soluion of problem (1.1), (1.2) is an absoluely coninuous funcion, which saisfies ha equaion on (, ) almos everywhere and condiion (1.2). The exisence resuls and sabiliy definiions can be found for insance in [8, p. 138]. Consider also he nonhomogeneous equaion ẋ() = d τ R(, τ)x( τ) + f() ( ) (1.3) wih a given vecor funcion f() and he zero iniial condiion So problem (1.3), (1.4) can be wrien as ẋ() = 2 Main Resul x() = ( ). (1.4) d τ R(, τ)x( τ) + f() ( ). (1.5) If f L p (, ), p 1, hen a soluion of problem (1.3), (1.4) is defined as a locally absoluely coninuous funcion x(), wih ẋ() L p (, 1 ) for any posiive finie 1, which saisfies condiion (1.4) and equaion (1.3) on (, ) almos everywhere.

4 18 Michael Gil We will say ha (1.1) has he ɛ-propery, if sup (e ɛτ 1) d τ R(, τ) n as ɛ (ɛ > ). Equaion (1.1) is said o be exponenially sable, if here are posiive consans ν and m, such ha y() n m φ C(,) e ν ( ) for any soluion y() of problem (1.1), (1.2). Now we are in a posiion o formulae our main resul. Theorem 2.1. For a p 1 and any f L p (, ), le he nonhomogeneous problem (1.3), (1.4) has a soluion x L p (, ). If, in addiion, equaion (1.1) has he ɛ- propery, hen i is exponenially sable. The proof of his heorem is divided ino a series of lemmas presened in his secion. Inroduce he operaor E : L p (, ) L p (, ) by Eu() = d τ R(, τ)u( τ) ( ; u L p (, )). Lemma 2.2. For any p 1, he inequaliy E L p V (R) is rue. Proof. For a u L 1 (, ) we have Eu L 1 (, ) = d s R(, s)u( s) n d Hence, d s R(, s) n u( s) n d. Eu L 1 (, ) sup τ d s R(τ, s) n sup s [, ) V (R) u L 1 (, ). u( s) n d = This proves he lemma in he case p = 1. Furhermore, for a v L (, ), we have sup τ Ev L (, ) = vrai sup d s R(τ, s) n sup s [, ) d s R(, s)v( s) n v( s) L (, ) = V (R) v L (, ). Hence, by he Riesz Torino inerpolaion heorem [9] we ge he required resul.

5 The Generalized Bohl Perron Principle 181 Lemma 2.3. Under he hypohesis of Theorem 2.1, any soluion of he homogeneous problem (1.1), (1.2) is in L p (, ). Proof. Wih a µ >, pu φ µ () = { e µ φ() if, φ() if <. Clearly, φ p L p (,) φ L 1 (,) φ p 1 C(,). So φ µ L p (, ). By he previous lemma E φ µ L p (, ). Subsiue y = φ µ + x uno (1.1). Then we have problem (1.3), (1.4) wih f = µe µ φ() + E φ µ. By he condiion of he lemma x L p (, ). We hus ge he required resul. Lemma 2.4. If a soluion y() of problem (1.1), (1.2) is in L p (, ) (p 1), hen i is bounded on [, ). Moreover, if p <, hen y p C(, ) pv (R) y p 1 L p (, ) y L p (, ) pv (R) y p L p (, ). Proof. By (1.1) and Lemma 2.2, ẏ L p (, ) V (R) y L p (, ). For simpliciy, in his proof we pu y() n = y(). The case p = is obvious. In he case p = 1 we have d y( 1 ) y() = d 1 ẏ( 1 ) d 1 < ( ), d 1 since ẏ L 1. Assume now ha 1 < p <. Then by he Gólder inequaliy p y() p = y( 1 ) p 1 ẏ( 1 ) d 1 p[ d y( 1 ) p d 1 = p d 1 y( 1 ) q(p 1) d 1 ] 1/q [ y( 1 ) p 1 d y( 1) d 1 d 1 where q = p/(p 1). Since q(p 1) = p, we ge he inequaliies As claimed. ẏ( 1 ) p d 1 ] 1/p, y() p p y p 1 L p (, ) ẏ L p (, ) p y p 1 L p (, ) V (R) y L p (, ) ( ). Proof of Theorem 2.1. By he variaion of consans formula, x = W f, where W f() = G(, s)f(s)ds. Here G(, s) is he fundamenal soluion o (1.1). For all finie T > and w L p (, ), le P T be he projecions defined by { w() if T, (P T w)() = if > T

6 182 Michael Gil and P = I. Operaor W is bounded in L p (, T ), since [, T ] is finie. Thus, he operaors W T = P T W are bounded on L p (, ). Bu W T srongly converge o W as T. So by he Banach Seinhaus heorem, W is bounded in L p (, ). Furhermore, subsiuing ino (1.1) he equaliy y ɛ () = e ɛ y() (ɛ > ; > ), y ɛ () = y() ( ) (2.1) we obain he equaion ẏ ɛ () = E ɛ y ɛ (), (2.2) where E ɛ w() = ɛw() + Besides, y ɛ () = e ɛ φ() ( ). We hus ge Hence, for a finie T. Here W L p (,T ) = e ɛτ d τ R(, dτ)w( τ) ( > ). y ɛ y = E ɛ y ɛ Ey = E(y ɛ y) + (E ɛ E)y ɛ. y ɛ y L p (,T ) W L p (,T ) (E ɛ E)y ɛ L p (,T ) (E ɛ E)y ɛ L p (,T ) ɛ y ɛ L p (,T ) + sup W w L p (,T )/ w L p (,T ). Bu w L p (,T ) v(ɛ) y ɛ L p (,T ), (1 e ɛτ )d τ R(, τ)y ɛ ( τ) L p (,T ) where So v(ɛ) := ɛ + sup (e ɛτ 1) d τ R(, τ) n. y ɛ L p (,T ) y L p (, ) + v(ɛ)( y ɛ L p (,T ) + φ L p (,)) W L p (,T ). Take ɛ, such ha v(ɛ) W L p (,T ) < 1, for all sufficienly large T. This is possible due o he ɛ-propery. We have y ɛ L p (,T ) ( y L p (, ) + v(ɛ) W L p (,T ) φ L p (,))(1 v(ɛ) W L p (,T )) 1. Leing T, we ge y ɛ L p (, ). By Lemma 2.4 hence i follows ha a soluion y ɛ of (2.2) is bounded on [, ), if ɛ is small enough. Now (2.1) proves he heorem.

7 The Generalized Bohl Perron Principle Equaions wih Coninuous Delay In his secion we illusrae Theorem 2.1 in he case p = 2 and show ha i enables us o apply he Laplace ransform. Consider in C n he equaion ẏ() = A(τ)y( τ)dτ + K(, τ)y( τ)dτ ( ), (3.1) where A(τ) is a piece-wise coninuous marix-valued funcion defined on [, ) and K(, τ) is a piece-wise coninuous marix-valued funcion defined on [, ) 2. Besides, A(s) n Ce µs and K(, s) n Ce µs (C, µ = cons > ;, s ). (3.2) Then, clearly, (3.1) has he ɛ-propery. I is also supposed ha he operaor K defined by Kw() = K(, τ)w( τ)dτ is bounded in L 2 = L 2 (, ). To apply Theorem 2.1, consider he equaion ẋ() = A(τ)x( τ)dτ + K(, τ)x( τ)dτ + f() ( ) (3.3) wih f L 2. To esimae soluions of he laer equaion, we need he equaion u() = A(τ)u( τ)dτ + h() ( ) (3.4) wih h L 2 (, ). Applying o (3.4) he Laplace ransform we have zû(z) = Â(z)û(z) + ĥ(z) where are Â(z), û(z) and ĥ(z) are he Laplace ransforms of A(), u() and h(), respecively, and z is he dual variable. Then û(z) = (zi Â(z)) 1 ĥ(z). I is assumed ha de (zi Â(z)) is a sable funcion, ha is all is zeros are in he open lef half plane, and θ := sup (iωi Â(iω)) 1 n < 1 ω R K. (3.5) L 2 As i was above menioned, various esimaes for θ can be found in [5, 6]. By he Parseval equaliy we have u L 2 θ h L 2. By his inequaliy, from (3.3) we ge x L 2 θ f + Kx L 2 θ f L 2 + θ K L 2 x L 2. Hence (3.5) implies ha x L 2. Now by Theorem 2.1 we ge he following resul.

8 184 Michael Gil Corollary 3.1. Le he condiions (3.2) and (3.5) hold. Then (3.1) is exponenially sable. For example, K saisfies condiions (3.2) and (3.5), if K(, s) n e νs a() (, s ) wih a bounded funcion a L 2 (, ) saisfying he inequaliy θ a L 2 (, ) < 2ν. References [1] Azbelev, N.V. and P.M. Simonov, Sabiliy of Differenial Equaions wih Afereffecs, Sabiliy Conrol Theory Mehods Appl. v. 2, Taylor & Francis, London, 23. [2] Berezansky, L. and Braverman, E. On exponenial sabiliy of a linear delay differenial equaion wih an oscillaing coefficien. Appl. Mah. Le. 22 (29), no. 12, [3] Berezansky, L., Braverman, E. and Domoshnisky, A. Sabiliy of he second order delay differenial equaions wih a damping erm. Differ. Equ. Dyn. Sys. 16 (28), no. 3, [4] Daleckii, Yu L. and Krein, M. G.. Sabiliy of Soluions of Differenial Equaions in Banach Space, Amer. Mah. Soc., Providence, R. I [5] Gil, M.I. Sabiliy of Finie and Infinie Dimensional Sysems, Kluwer, NewYork, [6] Gil, M.I. L 2 -absolue and inpu-o-sae sabiliies of equaions wih nonlinear causal mappings, J. Robus and Nonlinear sysems, 19, (29), [7] Halanay, A. Differenial Equaions: Sabiliy, Oscillaion, Time Lags, Academic Press, New York, 1966 [8] Kolmanovskii, V. and A. Myshkis, Applied Theory of Funcional Differenial Equaions, Kluwer, Boson, [9] Krein, S.G. Funcional Analysis, Nauka, Moscow, In Russian. [1] Tyshkevich, V. A. Some Quesions of he Theory of Sabiliy of Funcional Differenial Equaions, Naukova Dumka, Kiev, In Russian.