POSITIVE AND MONOTONE SYSTEMS IN A PARTIALLY ORDERED SPACE

Transcription

1 Urainian Mahemaical Journal, Vol. 55, No. 2, 2003 POSITIVE AND MONOTONE SYSTEMS IN A PARTIALLY ORDERED SPACE A. G. Mazo UDC We invesigae properies of posiive and monoone differenial sysems wih respec o a given cone in he phase space. We formulae crieria for he sabiliy of linear posiive sysems in erms of monoonically inverible operaors and develop mehods for he comparison of sysems in a parially ordered space. 1. Inroducion The posiiviy (monooniciy) of a dynamical sysem is equivalen o he posiiviy (monooniciy) of a cerain operaor (describing he moion of his sysem) wih respec o a given cone in he phase space [1 3]. The Lyapunov and Riccai differenial equaions are examples of posiive sysems wih respec o he cone of symmeric nonnegaive-definie marices. The properies of posiive and monoone sysems are used in various problems of analysis and synhesis. The invesigaion of he sabiliy of a class of linear auonomous posiive sysems reduces o he soluion of algebraic equaions deermined by he operaor coefficiens of he given sysems [4 6]. In he presen paper, we invesigae condiions for he posiiviy and monooniciy of differenial sysems in a parially ordered Banach space. We propose mehods for he analysis of he sabiliy of linear posiive sysems based on he soluion of algebraic equaions wih monoonically inverible operaors. We also presen analogs of comparison sysems in a parially ordered space. 2. Operaors in a Space wih a Cone A convex closed se K of a real normed space is called a cone if K I K = { 0 } and α X + β Y K for all X, Y K and α, β 0. A space wih a cone is parially ordered: X Y ( X < Y ) Y X K ( Y X K 0 ), where K 0 is he se of inerior poins of K. A cone K is called normal if i follows from he inequaliy 0 X Y ha X c Y, where c is a universal consan. If = K K, hen he cone K is reproducing. Le a cone K 1 1 ( K 2 2 ) be specified in a Banach space 1 ( 2 ). An operaor M : 1 2 is called monoone (monoone on he cone K ) if M X M Y for X Y ( X Y 0 ). The monooniciy of a linear operaor is equivalen o is posiiviy: X 0 M X 0. The inequaliy beween operaors M L means ha he operaor L M is posiive. If M 1 K 2, hen he operaor M is everywhere posiive. A linear operaor M is called monoonically inverible if, for every Y K 2, he equaion M X = Y has a soluion X K 1. If K 2 is a normal reproducing cone and M 1 M M 2, hen he monoone inveribiliy of he operaors M 1 and M 2 implies he monoone inveribiliy of he operaor M ; in his case, M 1 2 M 1 1 M 1 [1]. We specify he class of linear operaors [5] Insiue of Mahemaics, Urainian Academy of Sciences, Kiev. Translaed from Urains yi Maemaychnyi Zhurnal, Vol. 55, No. 2, pp , February, Original aricle submied April 27, /03/ $ Plenum Publishing Corporaion 199

2 200 A. G. MAZKO M = L P, P K 1 K 2 L K 1, where K 2 is a normal reproducing cone. A crierion for he monoone inveribiliy of such operaors is he inequaliy ρ ( T ) < 1, where ρ ( T ) is he specral radius of he pencil of operaors T ( λ ) = P λ L. If he cone K 2 is solid, hen his inequaliy is equivalen o he solvabiliy of he equaion M X = Y in he form X 0 for some Y > Posiive and Monoone Sysems Le be a Banach space parially ordered by a cone K and le X ( ) be a sae of a dynamical sysem wih coninuous or discree ime 0. A sysem is called (, 0 )-posiive if he relaion X ( 0 ) = X 0 K implies ha X ( ) K. A sysem is posiive if i is (, 0 )-posiive for all > 0 0. This propery of a sysem is equivalen o he posiiviy of he operaor of moion of he sysem V (, 0 ) : deermining he ransiion from he sae X 0 o he sae X ( ) = V (, 0 ) X 0 for > 0 0. A sysem is called monoone (monoone on he cone K ) if is operaor of moion V (, 0 ) is monoone (monoone on he cone K ) for > 0 0. Consider he linear differenial sysem X () + M() X () = G ( ), 0, (1) where M ( ) is a bounded operaor acing in a parially ordered space wih cone K. Assume ha every iniial condiion X ( 0 ) = X 0 of sysem (1) deermines he unique soluion X ( ) = W (, ) X + WsGsds (, ) ( ), (2) where W (, s ) = W(, 0) [ Ws (, 0) ] represenable as he series 1 is he evoluion operaor and 0 0. The linear operaor W (, 0 ) is W (, 0 ) = I M( ) d + M( ) M( ) d d uniformly convergen wih respec o he operaor norm. According o (2), X ( ) K for any iniial condiion X 0 K if W (, 0 ) 0, WsGsds (, ) ( ) 0. (3) 0 Here, he firs inequaliy means he monooniciy of he operaor wih respec o he cone K, and he second inequaliy means ha he value of he funcion belongs o he given cone. The converse saemen can easily be proved wih regard for he fac ha he cone K is closed. Consequenly, sysem (1) is posiive if and only if relaions (3) hold for > 0 0.

3 POSITIVE AND MONOTONE SYSTEMS IN A PARTIALLY ORDERED SPACE 201 Using formulas (2) and (3), one can esablish he equivalence of he following saemens: (a) for any funcion G ( ) 0, sysem (1) is (, 0 )-posiive for 0; (b) sysem (1) is monoone; (c) he operaor W (, s ) is monoone for > s 0; (d) i follows from he relaions Z ( ) + M() Z() 0 and Z ( 0 ) = Z 0 0 ha Z ( ) 0 for > 0. If G ( ) 0, hen each of saemens (a) (d) is equivalen o he posiiviy of sysem (1). Lemma 1. The evoluion operaor W (, s ) is monoone for s 0 if and only if he exponenial operaor e Mh () is monoone for 0 and h 0. Proof. We use he procedure of represenaion of he operaor W (, s ) in he form of a muliplicaive inegral [7]. Pariioning he segmen [ s, ] by he poins n = s + h n, where h n = ( s)/ n, = 0,, n, for large values of n we ge W (, s ) = W (, ) W (, ) W (, ) nn n1n n1n n 2n 1n 0 n, M( θ ) h W ( n, 1 n) = e n n + o( h ), = 1,, n, where θ n [ 1 n, n] are cerain inermediae poins. Therefore, W (, s ) = lim M( ) h M( ) h e nn n e n n [ θ θ 1 ]. n If e Mh () 0 for all 0 and h 0, hen he operaor W (, s ) is he limi of a cerain sequence of monoone operaors and is monoone by virue of he closedness of he cone of linear monoone operaors. The converse saemen is proved by analogy on he basis of he relaions n M( θ ) h/ n W (, h / n ) = e n + o( 1 / n), where θ n [ h/ n, ], n = 1, 2,. The lemma is proved. () n = lim [ W (, h/ n) ], e Mh n Lemma 2. If M ( ) = M1() + M2() and he operaors WM 1 (, s) and WM 2 (, s) are monoone for s 0, hen he operaor W (, s) is also monoone for s 0. M

4 202 A. G. MAZKO The proof is based on Lemma 1 and he relaions [5] 1 2 τ = lim [ E( τ / ) ], E ( h ) = ( e M h e M h + e M h e M h ). 2 e ( M + M ) The posiiviy of a sysem can be used for he esimaion of is soluions. If funcions X 1 ( ) and X 2 ( ) saisfy he inequaliies X () + M() X () G 1 ( ), X () + M() X () G 2 ( ), X 1 ( 0 ) X 2 ( 0 ), hen, under condiions (3), he following relaions are rue: X2() X1() W (, 0) [ X2( 0) X1( 0) ] + WsGsds (, ) ( ) 0, where G ( ) = G ( ) G ( ). This yields he following saemen: 2 1 Lemma 3. Le X ( ) be a soluion of he posiive sysem (1) and le funcions X 1 ( ) and X 2 ( ) saisfy he inequaliies X () + M() X () α G ( ) and X () + M() X () β G ( ), where α 1 and β 1. Then he relaion X 1 ( 0 ) X ( 0 ) X 2 ( 0 ) implies ha X 1 ( ) X ( ) X 2 ( ) for 0. If α = 0, hen he lower bound X 1 ( ) of a soluion of sysem (1) in Lemma 3 does no depend on he righ-hand side G ( ) of sysem (1). In he case α = β = 1, he saemen of Lemma 3 is rue if W (, s ) 0, s 0. We generalize he differenial sysem (1) as follows: X + M() X = G ( X, ), 0, (4) where G is a nonlinear operaor. The posiiviy (monooniciy) of sysem (4) is equivalen o he posiiviy (monooniciy) of he operaor of shif along he rajecories X ( ) = V(, 0) X0. The soluions of sysem (4) saisfy he inegral equaion X ( ) = W (, ) X + WsGXs (, ) ( ( ), sds ), (5) where W (, s ) is he evoluion operaor of he linear sysem (1). I follows from (5) ha sysem (4) is posiive if he operaor G ( X, ) is everywhere posiive and he operaor W (, s ) is monoone for s 0. Assume ha he operaor W (, s ) is monoone wih respec o a solid cone K and le F 0 and F be he families of operaors G defined by he condiions

5 POSITIVE AND MONOTONE SYSTEMS IN A PARTIALLY ORDERED SPACE 203 X 0, ϕ K *, ϕ ( X ) = 0 ϕ ( G ( X, ) ) 0 and X Y, ϕ K *, ϕ ( X Y ) = 0 ϕ ( G ( X, ) G ( Y, ) ) 0, respecively; here, K * is he conjugae cone of linear funcionals. Then he posiiviy (monooniciy) of sysem (4) can be esablished for G F 0 ( G F ). Lemma 4. Le X ( ) be a soluion of sysem (4) wih a monoone operaor W (, s ), and le funcions X 1 ( ) and X 2 ( ) saisfy he relaions X + M() X = G 1 ( X 1, ), X + M() X = G 2 ( X 2, ), G 1 ( X, ) G ( X, ) G 2 ( X, ), G 1, G 2 F, 0. Then he relaion X 1 ( 0 ) X ( 0 ) X 2 ( 0 ) implies ha X 1 ( ) X ( ) X 2 ( ) for 0. Examples. The nonlinear differenial sysem x + A() x = g ( x, ), x R n, 0, where A ( ) is a marix wih nonposiive diagonal elemens, is posiive wih respec o he cone of nonnegaive vecors K if he vecor funcion g ( x, ) saisfies he condiions [2] x 0, x i = 0 g i ( x, ) 0, i = 1,, n, and i is monoone wih respec o he same cone if g ( x, ) is quasimonoone and nondecreasing wih respec o x (he Ważewsi condiion), i.e., x y, x i = y i g i ( x, ) g i ( y, ), i = 1,, n. If boh resricions on g ( x, ) are saisfied for 0 x y, hen he given sysem is monoone on he cone K. Examples of differenial equaions posiive wih respec o he cone of symmeric nonnegaive-definie marices are he Lyapunov and Riccai differenial equaions and he more general equaion T T X A() X XA() B () XB () = XC() X + D (), where A ( ), B ( ), C ( ) = C ( ) T 0, and D ( ) = D ( ) T 0 are given marices. In he case C ( ) 0, he given sysem also possesses he propery of monooniciy. In he example considered, he operaor M ( ) has he following srucure:

6 204 A. G. MAZKO M ( ) = L ( ) P ( ), L ( ) X = A() X XA () T T, P ( ) X = B() X B(). In his case, he monooniciy of he evoluion operaor W M (, s) follows from Lemma 2 and he relaions WL(, s) X = Ω(, s) XΩ(, s) T 0, W P (, s ) X 0 X = X T 0, where Ω (, s ) is he marizan of he sysem ẋ = A ( ) x, s 0. A linear marix differenial equaion of he form T T X A() X XA() B () XB () = 0 is nown as he equaion of second momens for he Iô sochasic sysem dx() A() x() d = B() x() dw(), where w are componens of he sandard Wiener process. This equaion possesses he properies of posiiviy and monooniciy and is used in he heory of sabiliy of sochasic sysems. 4. Sabiliy of Linear Posiive Sysems Le a normal reproducing cone K be specified in he phase space of he linear auonomous sysem Z ( ) + MZ () = 0, 0, (6) where M is a bounded operaor. The posiiviy of his sysem is equivalen o he monooniciy of he operaor e M wih respec o he cone K for 0. Therefore, one can use he properies of one-parameer posiive semigroups [3] for he invesigaion of condiions of sabiliy of he posiive sysem (6). We define he limi of growh of he operaor exponenial and he specral bound as follows: γ M = lim 1 ln e M <, α M = inf { Re λ : λ σ ( M ) }. I follows from he heorem on he mapping of he specrum of a bounded operaor ha γ M = α M. The specral radius of a monoone linear operaor is a poin of is specrum (see he Krein Bonsall Karlin heorems [1]). Therefore, for he posiive sysem (6), we have α M σ ( M ). Lemma 5. If sysem (6) is posiive, hen he relaions ( M + γ I) 1 0 and γ > γ M ( M + γ I) 1 0 for every γ γ 0, hen sysem (6) is posiive and γ 0 > γ M. are equivalen. If Proof. If sysem (6) is posiive, hen, for every γ > γ M, he following relaion is rue:

7 POSITIVE AND MONOTONE SYSTEMS IN A PARTIALLY ORDERED SPACE 205 γ ( M + γ I) 1 M = e e d 0 0. Conversely, if he operaor M + γ I is monoonically inverible for every γ γ 0, where γ 0 hen = lim ( M + I) e M 1 [ ] 0, =, 0. is a real number, Le us show ha, for he posiive sysem (6), he operaor M + γ I is no monoonically inverible for γ γ M. Assume ha he operaors M 1 = M + γ 1 I and M 2 = M + γ 2I are monoonically inverible for some numbers γ 1 and γ 2 such ha γ 1 < γ M < γ 2. Then i follows from he relaion M 1 M + γ M I M 2 and he heorem on a wo-sided esimae of monoonically inverible operaors ha he operaor M + γ MI is also monoonically inverible [1]. However, his conradics he condiion α M = γ M σ ( M ). Consequenly, under he condiion of he posiiviy of sysem (6), he operaor M + γ I is monoonically inverible only for γ > γ M. The lemma is proved. Lemma 6. If ( M αi) 1 0 for every α α 0, hen he specrum of he operaor M lies in he half-plane Re λ > α 0. Proof. I follows from he inveribiliy of he operaor M α I ha he operaor M does no have real poins of is specrum on he inerval (, α 0 ]. The specral radius of he monoone operaor ( M αi) 1 is equal o 1/ ( α α), where α * is a real poin of he specrum σ ( M ) such ha λ α α α > 0 λ σ ( M ). In his case, α * > α 0 α and α * does no depend on α. If Re λ α 0, hen one can choose α so ha he opposie inequaliy is saisfied: λ α < α α. Consequenly, Re λ > α 0 for λ σ ( M ). In his case, α * coincides wih α M. The lemma is proved. If α M > 0, hen sysem (6) is exponenially sable, i.e., Z () βe γ ( 0 ) Z0, 0, where γ and β are posiive consans independen of he choice of a soluion and γ < α M. For he posiive sysem (6) having he parial soluion Z ( ) = e α M ( 0 ) V, where V 0, he converse saemen is also rue. Taing ino accoun Lemmas 5 and 6 for γ 0 = α 0 = 0, we formulae he following saemen: Theorem 1. If he operaor M + γ I is monoonically inverible for every γ 0, hen sysem (6) is posiive and exponenially sable. If sysem (6) is posiive, hen i is exponenially sable if and only if he operaor M is monoonically inverible. Noe ha he exponenial sabiliy of sysem (6) follows from he monoone inveribiliy of he operaors M and M + γ 0 I, where γ 0 > 0 is a sufficienly large number. Indeed, for γ [ 0, γ 0 ], every operaor M + γ I

8 206 A. G. MAZKO mus be monoonically inverible; in his case, λ+ γ 0 α γ 0 > 0 λ σ ( M ) (see he proofs of Lemmas 5 and 6). For sufficienly large γ 0, his yields he inequaliy α M > 0, under which sysem (6) is exponenially sable. The fac ha he monoone inveribiliy of he operaor M is a necessary condiion for he exponenial sabiliy of he posiive sysem (6) can also be esablished by using he resuls of [6]. If he operaor M is monoonically inverible wih respec o a solid cone K, hen here exis elemens X > 0 and Y > 0 saisfying he equaion MX = Y, and he posiive sysem (6) mus be exponenially sable [6, p. 38]. The nown crieria for he asympoic mean-square sabiliy of he Iô sochasic sysems (see Sec. 3) are corollaries of Theorem 1. Now consider nonauonomous sysems of he form M + X ( ) + M () X () = 0, 0. (7) Sysem (7) is called posiive reducible if here exiss a Lyapunov ransformaion X ( ) = Q ( ) Z ( ) ha reduces i o he posiive auonomous sysem (6). In his definiion, Q ( ) is a uniformly bounded differenial operaor having he uniformly bounded inverse Q 1 ( ) and saisfying he operaor differenial equaion Q ( ) + M () Q () Q () M = 0, 0. In his case, condiions for he sabiliy of sysems (6) and (7) are equivalen [7]. Consequenly, he posiive-reducible sysem (7) is exponenially sable if and only if he operaor M is monoonically inverible. Examples of reducible sysems are he ω-periodic sysems (7) for which M ( + ω ) = M ( ), W ( + ω ) = W ( ) W ( ω ), Q ( ) = W ( ) e M, M = ω 1 ln W ( ω), and he specrum of he monodromy operaor W ( ω ) = W ( ω, 0 ) does no surround zero. Consider he subclass of sysems (7) described by a funcionally commuaive operaor M ( ), i.e., M ( ) M ( τ ) M ( τ) M ( ), 0, τ 0. (8) In his case, he evoluion operaor is defined by he relaions [8] Assume ha here exiss he limiing bounded operaor W (, s ) = e Ns (,), N (, s ) = M( τ ) dτ, s. (9) where ϕ ( ) > 0 is a funcion such ha ϕ ( ) as. s 1 M = lim M( ) d ϕ () τ τ, (10) 0

9 POSITIVE AND MONOTONE SYSTEMS IN A PARTIALLY ORDERED SPACE 207 Theorem 2. Suppose ha condiions (8) are saisfied and sysem (6) wih operaor (10) is posiive. Then he monoone inveribiliy of operaor (10) yields he asympoic sabiliy of sysem (7). Proof. Relaions (8) (10) yield M ( ) N (, τ ) = N (, τ ) M ( ), M N (, 0 ) = N (, 0 ) M, M (, 0 ) = (, 0 ) M, (, 0 ) = 1 (, 0) ϕ( ) N M, where (, 0 ) 0 as. Therefore, in view of (9), an arbirary soluion of sysem (7) can be represened in he form X ( ) = e ϕ( )[ M+ (, 0 )] ϕ() M ϕ() (, 0 ) X0 = e e X0. I follows from he monoone inveribiliy of he operaor M of he posiive sysem (6) ha α M > 0 (see he proof of Lemma 5). For every ε > 0, here exiss 1 0 such ha (, 0 ) < ε for > 1. In his case, he following esimae is rue: ()( ) () ()( 2 ) X () β ϕ e α M ε e ϕ ε X0 = β ϕ e α M ε X 0, where β > 0 is a cerain consan. Seing ε < α M/ 2 and aing ino accoun ha ϕ ( ), we esablish ha X () 0 as. Consequenly, sysem (7) is asympoically sable. The heorem is proved. Example. Consider he marix sysem (7) wih M ( ) = a () b (), b () a () where a ( ) and b ( ) are given funcions. The marix M ( ) obviously saisfies he condiion of funcional commuaiviy (8). Assume ha ϕ ( ) = bsds (), 0 1 asds () ϕ( ) c,. 0 Then he limiing marix (10) has he form M = c 1 1. c The auonomous sysem (6) wih marix M is posiive wih respec o he cone of nonnegaive vecors (see Sec. 2). The condiion of he monoone inveribiliy of he marix M reduces o he inequaliy c > 1. In his case, according o Theorem 2, he original sysem (7) is asympoically sable.

10 208 A. G. MAZKO 5. Robus Sabiliy One of imporan applied problems is he problem of he sabiliy of a given family of sysems wih indefinie parameers. Consider a family of differenial sysems of he form X + M() X = G( X, ), M () M ( ) M ( ), 0, (11) G () M () X G( X, ) G () M () X. (12) 1 1 In his family, we selec he following wo sysems: X + M() + M () X [ ] = G 1 ( ), 0, (13) [ ] = G 2 ( ), 0. (14) X + M() + M () X Lemma 7. Suppose ha he evoluion operaor of sysem (13) is monoone and inequaliies (12) are saisfied for X K. Then soluions X ( ) 0 of every sysem (11), (12) are bounded by soluions of sysems (13) and (14), i.e., X 1 ( 0 ) X ( 0 ) X 2 ( 0 ) X 1 ( ) X ( ) X 2 ( ), 0. If inequaliies (12) are saisfied for X, hen he posiiviy of sysem (13) yields he posiiviy of every sysem (11), (12) and, furhermore, 0 X 1 ( 0 ) X ( 0 ) X 2 ( 0 ) 0 X 1 ( ) X ( ) X 2 ( ), 0. Proof. Subracing relaions (13) and (11) from (11) and (14), respecively, and using (12), we obain he differenial inequaliies [ ] [ M () M ()] X (), H () + M() + M () H () [ ] M M X H () + M() + M () H () H () + M() + M () H () [ () ()] 1 (), [ ] [ M () M ()] X2 (), [ ] [ M () M ()] X (), H () + M() + M () H () where H 1 ( ) = X ( ) X 1 ( ) and H 2 ( ) = X 2 ( ) X ( ). In his case, he following relaions are rue: M () + M 1 () M () + M 1 () M () + M 2 () M () + M 2 (). If sysem (13) is posiive, hen, according o (3), is evoluion operaor WM+ M (, s) mus be monoone. 1 The monooniciy of he operaor WM+ M (, s) yields he monooniciy of he operaors W s 1 M+ M 1 (, ), WM+ M 2 (, s), and WM+ M 2 (, s) (see Sec. 2).

11 POSITIVE AND MONOTONE SYSTEMS IN A PARTIALLY ORDERED SPACE 209 If X ( ) 0 or X 1 ( ) 0, hen he relaion H 1 ( 0 ) 0 implies ha H 1 ( ) 0, i.e., X 1 ( ) X ( ) for 0. Similarly, if X ( ) 0 or X 2 ( ) 0, hen he relaion H 2 ( 0 ) 0 yields H 2 ( ) 0, i.e., X ( ) X 2 ( ) for 0. Therefore, he posiiviy of sysem (13) yields he posiiviy of every sysem of family (11), (12). In he case X ( ) 0, inequaliies (12) are used only for X K. The lemma is proved. Theorem 3. If sysem (14) is asympoically sable and sysem (13) is posiive, hen every linear sysem from family (11), (12) is asympoically sable and posiive. The proof is carried ou using Lemma 7 and he properies of a normal reproducing cone. Noe ha, for saionary sysems, he monoone inveribiliy of he operaors M and M yields he monoone inveribiliy of every operaor M from he segmen M M M. In paricular, i follows from Theorems 1 and 3 ha he operaor M is monoonically inverible if he relaions e M 0 and Re λ > 0 hold for every 0 and λ σ( M ). 6. Comparison Sysems In sabiliy heory, one uses mehods of comparison based on he mapping of he space of saes of he original sysem o he spaces of saes of well-sudied auxiliary sysems. As comparison sysems, one can use he classes of posiive and monoone sysems, in paricular, sysems saisfying he condiions of heorems of he Chaplygin and Ważewsi ype [9 11]. In his case, Theorems 1 3 and Lemmas 3, 4, and 7 may urn ou o be helpful. Consider he differenial sysem ẋ = f ( x, ), x X, 0, (15) where f is an operaor ha guaranees he exisence and uniqueness of soluions x ( ). In he space parially ordered by a normal reproducing cone K, we consruc sysems of he form Ẋ = F ( X, ), X, 0, (16) which are used as comparison sysems for sysem (15). Le Σ + denoe he class of sysems (16) for which one can esablish a correspondence beween heir soluions and soluions of he differenial inequaliies Ż F ( Z, ), Z, 0, such ha he relaion X ( 0 ) Z ( 0 ) implies ha X ( ) Z ( ) for > 0 0. Le E ( x, ) be an operaor ha coninuously maps a cerain neighborhood of he poin x = 0 X for 0 o he space. If he expression E ( x, ) and is generalized derivaive along he soluions of sysem (15) saisfy he relaion DE ( x, ) ( 15 ) F ( E ( x, ), ), (17) hen sysem (16) from he class Σ + is an upper comparison sysem for sysem (15), i.e.,

12 210 A. G. MAZKO E ( x ( 0 ), 0 ) X ( 0 ) E ( x ( ), ) X ( ), 0. (18) In (17), he derivaive along he soluions of sysem (15) can be defined as follows: 1 DE ( x, ) ( 15 ) = lim sup ( (, ), ) (, ) h h E x + hf x + h E x 0+ [ ] By analogy, one can define he class of sysems Σ and lower comparison sysems; in his case, all inequaliy signs used in he space are replaced by he opposie ones. I is obvious ha, under he condiion F ( 0, ) 0, every sysem of he class Σ + mus be posiive, and every sysem of Σ + U Σ mus be monoone. If he equaliy DE ( x, ) ( 15 ) = F ( E ( x, ), ) (19) holds in (17), hen, using he definiion of he propery of monooniciy of sysem (16), we obain X 1 ( 0 ) E ( x ( 0 ), 0 ) X 2 ( 0 ) X 1 ( ) E ( x ( ), ) X 2 ( ), 0, (20) where X 1 ( ) and X 2 ( ) are cerain soluions of he given sysem. This means ha, under condiion (19), he monoone sysem (16) is simulaneously a lower comparison sysem and an upper comparison sysem for sysem (15). Esimaes (18) and (20) can be used for he comparison of he dynamical properies of sysems (15) and (16). For example, if he operaor E is chosen so ha E ( x, ) 0 only for x = 0, hen relaion (18) and he fac ha X ( ) 0 imply ha x ( ) 0 as. In consrucing comparison sysems posiive or monoone on he cone, he operaor E can be chosen o be everywhere posiive. For example, for a linear sysem ẋ = A ( ) x, seing E ( x, ) = xx T 0 and using (17), we obain he upper marix comparison sysem T X A() X XA () = G ( X, ) 0, X R n n. (21) This sysem is posiive wih respec o he cone of nonnegaive-definie marices. The asympoic sabiliy of he marix equaion (21) yields he asympoic sabiliy of he original sysem. Lower and upper comparison sysems of he form (16) can be consruced in differen parially ordered spaces 1 and 2 simulaneously. In his case, he properies of he corresponding operaors E 1 ( x, ) and E 2 ( x, ) and he order relaions defined by he cones K 1 1 and K 2 2 in he relaions E 1 ( x ( ), ) X 1 ( ) and E 2 ( x ( ), ) X 2 ( ) mus be coordinaed wih he purpose of sudying cerain characerisics of he original sysem (15). For example, one can require ha he sysem of inequaliies E 1 ( x, ) 0 and E 2 ( x, ) 0 be saisfied only for x = 0. In his case, one should expec ha x ( ) 0 as if X 1 ( ) 0 and X 2 ( ) 0. If he operaors E 1 and E 2 coincide, hen his propery follows from he lemma on wo policemen in a parially ordered space [1]. The problem of comparison of sysems (15) and (16) becomes more complicaed wihou he requiremen of he uniqueness of soluions. In [9, 10], comparison sysems of he form (16) were considered in a parially ordered space R n using he noion of maximum and minimum soluions ogeher wih he generalized Dini derivaives of he vecor funcion E ( x, ) on he soluions of he original sysem (15). In his case, he funcion.

13 POSITIVE AND MONOTONE SYSTEMS IN A PARTIALLY ORDERED SPACE 211 E ( x, ) mus be locally Lipschizian wih respec o x, and he vecor funcion F ( X, ) mus be quasimonoonically nondecreasing in X wih respec o he cone K R n, i.e., F F. This propery guaranees ha sysem (16) belongs o he classes Σ + and Σ ; in he case of he cone of nonnegaive vecors, i reduces o he Ważewsi condiions (see Sec. 3). REFERENCES 1. M. A. Krasnosel sii, E. A. Lifshis, and A. V. Sobolev, Posiive Linear Sysems [in Russian], Naua, Moscow (1985). 2. M. A. Krasnosel sii, Operaor of Shif along Trajecories of Differenial Equaions [in Russian], Naua, Moscow (1966). 3. P. Clémen, H. J. A. M. Heijmans, S. Angenen, C. J. van Duijn, and B. de Pager, One-Parameer Semigroups, Norh-Holland, Amserdam (1987). 4. A. G. Mazo, Sabiliy of linear posiive sysems, Ur. Ma. Zh., 53, No. 3, (2001). 5. A. G. Mazo, Localizaion of Specrum and Sabiliy of Dynamical Sysems [in Russian], Preprin No. 28, Insiue of Mahemaics, Urainian Academy of Sciences, Kiev (1999). 6. G. N. Mil shein, Exponenial sabiliy of posiive semigroups in a linear opological space, Izv. Vyssh. Uchebn. Zaved., Ser. Ma., No. 9, (1975). 7. Yu. L. Dalesii and M. G. Krein, Sabiliy of Soluions of Differenial Equaions in a Banach Space [in Russian], Naua, Moscow (1970). 8. B. P. Demidovich, Lecures on he Mahemaical Theory of Sabiliy [in Russian], Naua, Moscow (1967). 9. V. M. Marosov, L. Yu. Anapol sii, and S. N. Vasil ev, Mehod of Comparison in he Mahemaical Theory of Sysems [in Russian], Naua, Novosibirs (1980). 10. V. Lashmianham, S. Leela, and A. A. Marynyu, Sabiliy of Moion: Mehod of Comparison [in Russian], Nauova Duma, Kiev (1991). 11. N. S. Posniov and E. F. Sabaev, Marix comparison sysems and heir applicaions o problems of auomaic conrol, Avomaia Telemehania, No. 4, (1980).