DSpace at VNU: Note on the Asymptotic Stability of Solutions of Differential Systems

V N U J O U R N A L O F S C IE N C E , M a t h e m a tic s - Physics T.xx, N q - 2004 N O TE ON TH E A S Y M P T O T IC ST A B IL IT Y OF SOLUTIONS OF D IF F E R E N T IA L S Y S T E M S D a o T h i L ie n Thai Nguyen Teacher Training College A b s t r a c t We shall discuss the asymptotic behavior of solutions of differential systems Some new notions of stability and examples will be given and some stability conditions will be proved Consider the differential system Ỉ =x(t ’x) (1) X ( t , ) = , t e I = [a, + oo), a > , where X G R n , D = {(£, x) I t / , ||x|| < H , } H > and suppose th a t function X : D— » R n (t, x) I— >X ( t , x) is continuous and satisfies condit ion of uniqueness of solution in D T h e re is a vast literature on the theory and applications of L iapu no v’s second m eth od (see, for example, [1], [2], [3], [4], [5], [GJ, [7], [8 ], [9]) Here we shall discuss on the ” degree” of the asym ptotic behavior of solution of differential system ( ) As well known, if there exist the num bers TV > , > such t h a t ||x (í,ío ,xo )|| ^ Ar.||xo||.e_7(í_ío)) Ví ^ Í , (2) the zero solution of the system ( ) is exponential asym ptotically stable However, there exist some m otions which is not exponentially stable b u t it tends to zero more fast th an P( t ) = (j-f-yx —> 0, as t —> 4-0 , (A > 0) First, we give some definitions D e f in itio n s a n d e x a m p l e s D e f in itio n T he trivial solution of (1) is said to be quasi-asymptotically stable of order A(A € M+) if given any > an d any to € I th ere exist a ỏ = ỗ(to,e) and a T = T ( t , ), such th a t ||x i( í;ío ,z o ) || < e(i - t 0) ~ x (3) for all t > to + T ( t , e) if ||X()|| < Ố, or there exist th e num bers N > an d T > such th a t ||x(i, to, £o) II ^ iV||xo||(i - io)_A T ypeset by 16 17 N o t e on the a s y m p t o t i c s t a b i l i t y o f s o l u t i o n s o f d i f f e r e n tia l s y s t e m s tor rill t y* t (1 + T D e f in itio n 1.2 T h e trivial solution of (1) is said to 1)0 (]uasi-unif()iin-as!im.-ptoticalh) stable, of order A if the num bers Ò and T in Definition are independent of toD e f in itio n 1.3 Thu trivial solution of (1) is said to he equi-asymptotically stable of order A if it is stable in the sense of Liapunov and quasi-asympfotioally stable OÍ order A D e f in itio n 1.4 T h e trivial solution of (1) is said to h r quasi-exponential asymptotically stable if t here exists a > and given any f > and any t,[) £ 1 hero exist a i) — f) > 1111(1 ;i T = T(t{), f ) > such that if ||.r()|ị < Ổ, tin'll lk(M (),J-u)|| < (4 ) for all t ^ t() 4* T E x a m p le Considering the ('([nation dx \V(' S('(' t hat r = 7.1 (It 21 is a solution of which T he general solution of (5) is x( t ) = “ Ti t2 for all / > 1, thus the trivial solution X = is quasi-asyinptotically stable of oi(l(T E x a m p le Consular th(* equation ^ = _ r v , (It t ^ ! (0 ) \Y(' luivr ('asilv t licit for t() > ro (ỈT x f t 2dt ** x(t ) = to - *o) + ^ This implies 3|x*ol Ị I an d a T = T(io) slK-h th a t W ') l < ĨỊ— Tli(‘i('f()i(' tilt' Z0+ ị { V ( t + h , x { t + h)) - V { t , x ( t ) ) } ll We see easily ([7]) that By the same calculation , we obtain the relation lịm h-+ + r { V { t + h , x ( t + h)) — V ( t , x ) } = Hm h-+ 0+ 'l - { V ( t + h , x + h X ( t , x ) ) — V( t , x ) } h In the case V ( t , x ) has continuous partial derivates of the first order, it is evident th a t If+ Function V ( t , x ) is called Liapunov one T h e o r e m 2.1 Suppose that there exists a L iapunov function V ( t , x ) defined on D , saủisfying the following conditions ( i) v ụ , ) = f ii) ||x|| A $: Ị/(£,x), VÀ G R+ f Hi) V ^ ( t , x ) ^ —ĩỉhỵihEl? where m G N, m > Then, th e solution X = o f (1) is equi-asyinptotically stable o f order \ ( m — ) Proof For any < < H we have V(£,x) ^ e* for t G / = [a, 4-0 ) and X such th a t 11a: 11 = e due to the condition (ii) For the fixesd to G / , we can choose a Ổ = S(to,e) > such t h a t ||xo|| < Õimplies V ( t 0i x 0) < e* because of the continuity of V ( t , x ) and K (io ,0 ) = Suppose th a t a solution X = x(t,^o,xo) of (1) such th a t II.XoII < s satisfies ||x (ii, to, xq) € at some tị From (ill), it follows th a t V ( t u x ( t x , t 0, x 0)) < V { t 0, x 0) and hence e* ^ V ( t u x ( t i , t 0, x 0)) ^ V ( t , x 0) < e* This is a contradiction and hence, if ||x0 || < s then ||x(í, to, xo)|| < e, for all t ^ to th a t is X = is stable in sense of Liapunov Now given Q > 0, we assume t h a t x(£,io»£o) is a solution of (1) satisfying condition ||xo|| ^ a Applying Theorem 4.1 in [7], by (iii) we have t m V ( t,x ( M o ,* o ) ) < V ( i o , x o ) ( f ) ^ t 0™V( t 0i x 0)(t - to)-™ to the (7) N o t e on the a s y m p t o t i c s t a b i l i t y o f s o l u t i o n s o f d if f e r e n tia l s y s t e m s 19 for all t sufficiently large Let M ( t , a ) = max||Xo||^a V(to, To) and let T ( t ,€ ,a ) be such th at 0< M^ a ) < Ẻ t — to c for all t ^ to + T ( t 0, e , a) T h e n from (7), it follows th a t for t ^ i + T ( t 0, e, a) , ||x (M o ,£ o )|| * ^ V { t , x ( t , t 0, x 0)) ^ t 0m V ( t o , x 0) ( t - t 0) < tom M ( t o ' a \ t - t o ) - m+1 < e* (t — to )~ m+1 t — to = > ||x(í, Í , z 0)|| < e(í - ío)_A(m_1)’ which proves equi-asym ptotical stability of order A(m — 1) of the solution £ = 0, and the theorem is proved Ill the case m = th e zero solution is equi-asymptotically stable of order A T h e o r e m 2.2 Suppose th a t there exists a Liapunov function V ( t , x ) defined on D, sat­ isfying condition (i) and (ii) o f Theorem 2.1 and besides the following (Hi)’ V!xA t , x ) ^ - C V ( t , x ) , where c > is a constant Then the solutions = o f (1) is quasi-exponential asym ptotically stable Proof It is sufficient to prove th e inequality ||x (i,io ,x o )|| < ee_Q(t_to), for all t sufficiently large, w ith some positive num ber a For this, we give p > and assume that x{t,t , x 0) is a solution of ( ) satisfying th e condition ||io|| = p Due to the theorem 4.1 in [7], by (iii)’ the following inequality is valid V { t , x ( t , t 0, x 0)) ^ V { t 0, x o ) e - c{t- to\ (8) for all t sufficiently large Let = m a x { v (to, Xo), Ị|xo|| = /3},0 < Cl < c and let T( t o , e , ) such th a t „ „ M(to, )e - c(t- ‘o) ^ ^ - < e for all t ^ t + T{ t o, e , P) T h e n from ( ) it follows th a t \\x(t, to, £o)|| * ^ V ( t , x { t , t o , x o ) ) < e * e ~ c ( t ~to) = > ||x(i, to, £o)II < ee_ACl(i-to), for all t ^ t + T { t 0,e/3), (here a = ACi) T h e theorem is proved By the same arg um ents used in th e proof of the above theorem s we can prove the two following theorem s for th e quasi-uniform asymptotical stability of the zero solution 20 D a o Th i L i e n T h e o r e m 2.3 Suppose that there exists a Liapunov function V ( t , x ) defined on D which satisfies the following conditions ( i) ||x ||A ^ V ( t , x ) ^ Ò( 11X11), where b(r) is a continuous increasing and positive definite function, X G R+ ( i i ) V ^ { t , x ) ^ —mV(tfX) ĩ where m e N , m > Then, the solution X = o f the equation (1) is quasi-uniform-asymptotically stable o f order A(rn — ) T h e o r e m 2.4 Suppose that there existss a Liapunov function V (t , X) defined on D which satisfies the following conditions ( ỉ) ||x ||A ^ ^ ò(||x||), where b(r) is a continuous increasing and positive definite function, A R + ( ii) V ^ ( t , x ) ^ —c V ( t , x ) , where c > is a constant Then the solution X = o f (1) is quasi-exponential asymptotically stable 2.2 Let consider now the linear system dx x = = A( t ) x, (9) where A(t) is acontinuous n X n m atrix on I and to E / Note s!; = { x e Rn : < ||x|| sc /I}, where < h < H , > We have the following converse theorem for the system (9) T h e o r e m 2.5 Suppose th at there exist a M > and A e M+ sucii that lk ( M o ,z o ) || ^ A /||x0||(í - Í + ) " \ (10) for all t ^ to, where x(t, to, Xo) is a solution o f (9) Then there exists a Liapunov function V( t , x ) which satisfies the following conditions ( i) ||a?p ^ V(t yX) ^ M* \ \ x \ \ J ( ii) \\ V(t , x) - V ( t , x ')II ^ L\\x - x '\\,V x ,x ' £ S ( Hi) V(9)(t , x) ^ Proof Let V ( t , x ) be defined bv V ( t , x ) = sup IIx( t + r , i , x ) | | ^ ( r + 1)_A T^o It is clear th a t V ( t , x ) ^ ||x ( t ,t ,z ) ||* = | | x p Oil the other hand, because of (10) we have ||x (i + T , t , x ) II ^ M\ \ x \\(t + ) ~ A N o t e on the a s y m p t o t i c s t a b i l i t y o f s o lu t i o n s o f d i f f e r e n t ia l s y s t e m s 21 for all r ^ Hcnce V( t , x) = s u p ||x ( t + t , í , x ) P ( t + ) ~ A < s u p M * | | x | | * ( r + )_À_1 = M * | | x p r^O r^O The condition (i) is proved Since the system is linear, we have the relation x (r, t, x) - x (r, t, x') = x (r, í, X - x') (11) Hcncc for all x , x ' € S7h the inequalities following will be hold |V ( i,x ) - V ( i , x ;)| = = | s u p | | x ( i + T , t , x ) \ \ i { T + l ) ~ x - s u p ||x ( i + T ,i,x ' ) ll* (r + 1) A| ^ s u p { |||x ( f + T,i, x)li* - ||*(í + r l t >x ' ) | | ỉ | } ( r + i r A ^ s u p L i { | ||x ( £ + T, t , x) \ \ - \\x{t + t , í , x ' ) | | | } ( t + )_A T^o ^ s u p L i{ ||a ; ( i + T , t , x ) - x { t + T ,t ,x ') l l} ( r + ) ~ A T^O where L\ is a positive number This implies by (11) IV ( t , x ) - V ( t , x l)I ^ s u p L i { ||x ( i + r , i , x - x , ) | | } ( r + ) " A T^o ^ s u p L \ M \ \ ( x - x')H(-r + 1)~2A = Lỵ M\ \ ( x - *')||, T^o for all x , x ' € SỈỊ By p uttin g L = L \ M wc have (ii) Now we shall prove the continuity of v ( t , x ) The conditions (i), (ii) imply V ( t X) IS continue at We shall prove this in X# Take a num ber Ỗ^ that 0, we have \ V( t + ỏ , x ' ) - V ( t , x ) \ ^ sufficiently small the right hand part of (13) will be arbitrary small Hence we have th e continuity of V ( t , x ) Finally, we shall establish condition (iii) It is clear t h a t for h > V( t + h, x( t + h, t , x) ) — sup IIx( t + h + T^o ^ sup ||x(í + T^o T, T ,t + h , x ( t + h, t , x ) ) II * ( r + 1) -A t, x)|| * ( r + 1)_A = V ( t , x ) T h a t is V j t + h, x ( t + h, ỵ x)) - y (£, x) < h Thus V ^ Ạ t, x) ^ This completes the proof R eferences Hatvani L., On the stability of solutions for ordinary differential equations with mechanical application Alkalm Mat Lap 1990/1991, V.15, N ° 1/2, p 1-90 La-Salle J p., Lefschetz s., Stability by Liapunov's Direct M ethod with Application Academic Press, New York, 1961 Lakshm ikantham V., Leela s., M artynyuk A A Stability Analysis o f Nonlinear Sys­ tems N.Y Dekker, 1989 Peiffer K., Rouche N., Liapunov’s second m etho d applied to partial stability, J Mec, 1969, V , N ° 2, p 323-334 Rouche N., Ha bets p., Laloy M., Stability Theory by Liapunov's Direct Method Springer-Verlag New York - Heidelberg Berlin, 1997 Vorotnikov V I., Partial Stability and Control Boston : Birkhauser, 1998, 442p Yoshizawa T., Stability theory by L ia p u n o v ’s second m ethod, T h e m athem atical society of Japan, 1966  ... ^ Ạ t, x) ^ This completes the proof R eferences Hatvani L., On the stability of solutions for ordinary differential equations with mechanical application Alkalm Mat Lap 1990/1991, V.15, N °... e theorem is proved By the same arg um ents used in th e proof of the above theorem s we can prove the two following theorem s for th e quasi-uniform asymptotical stability of the zero solution... ptotical stability of order A(m — 1) of the solution £ = 0, and the theorem is proved Ill the case m = th e zero solution is equi-asymptotically stable of order A T h e o r e m 2.2 Suppose th a t there