Nonautonomous differential systems: several functions in the stability problem

Nonautonomous Differential Systems: Several Functions in the Stability Problem* N. Ianiro Istituto di Meccanica e Macchine Fat. Ingegnmiu Universiti d&Aquilu L’Aquila, Italy and C. Maffei Istituto di Matematica Universiti di Cam&no Came&w, Italy

Transmitted by V. -than

ABSTRACT Results of asymptotic stability and instability for nonautonomous differential systems with unbounded right hand side are obtained by using several auxibary functions and topological properties of the SUmit set of solutions. The theorem established is applied to prove the asymptotic stability of the stationary solution of the Lienard equation with unbounded dampings.

1.

INTRODUCTION

Several authors, as is well known, have studied the asymptotic stability problem for the stationary solution of a differential equation

i=f(Q),

XER”,

0.1)

when one isn’t able to apply the classical Liapunov theorem, i.e. when the derivative of the auxiliary function V(t,x) is equal to or less than a function V(r) which is only negative semidefinite along the solutions of (1.1) *Work perfomed

under the auspices of the Italian Council of Research (C.N.R.).

APPLIED MATHEMiiTlCS AND COMPUTATION

Q Elsevier

7:71-80 (1980)

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North Holland, Inc., 1980

52 Vanderbilt Ave., New York, NY 10017

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N. IANIRO AND C. htAFFE1

V. M. Matrosov [6] in the case of nonautonomous differential equations with bounded right hand side, proved asymptotic stability results by introducing several auxiliary functions on the neighborhood of those sets in which the function V*(x) is equal to zero. In the same framework [8], the boundness condition on the right hand side has been removed and Matrosov’s theorem generalized. J. P. La Salle, in a result known as the “invariance principle” [3], if the right hand side of (1.1) is autonomous or periodic, obtained information about the asymptotic behavior of the solutions by combining the auxiliary function’s properties with geometrical and invariance properties of the G-limit set of solutions. This result has been generalized to nonautonomous right hand sides ([I], [2], [9], [lo], for example) using the techniques of limiting equations, i.e. of those equations which describe the limiting behavior of the original nonautonomous law of motion, In this paper we consider the general problem of asymptotic stability, following Matrosov’s ideas, but using at the same time topological properties of the Q-limit set of solutions. To be more precise, comparing the two different approaches to the problem due to Matrosov and La Salle, we give in Sets. 3 and 4 asymptotic stability and instability theorems. The introduction of several auxiliary functions in the neighborhood of those sets in which V*(X) = 0 enables us to say that the solution cannot stay, for all t, near those sets; the problem arises of excluding infinite oscillations of the solution between the different components of the set in which V*(X) is zero. This problem is solved by taking into account the fact that the Q-limit set of solutions is a connected set. Considering a generalized Lienard scalar equation, we give in Sec. 5 an application of our result. Notice that our approach to the problem doesn’t need to verify invariance properties and enables us to consider also the case in which the function V* depends explicitly on the time.

2.

BASIC FACTS Consider a system of ordinary differential equations: i=f(t,x),

(24

where 3c denotes a real n-vector with the norm j]x]] =$rf + . . . + X, ; the vector-valued function f is defined for every x in an open set I c R” and for every t E I = [0, co). We suppose that f( t, X) is continuous and f( t, 0) = 0. Let us denote by x(t) = x(t, t,,,~,,) a noncontinuable solution of the equation (2.1) passing through (to, x0).

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We shall make use of the following notations: is the distance between the two points ~(x~Y)~Ilx~YIl~~~-1(4~Yi)2 x and y; p(x, E) = inf{ p(x, y), y E E } is the distance between a point x and a set E; for any yE(O,p(O,aI)), B, is the set {xEI: /lx]]
aV(t,x)

finally, for every continuous function v* : B,-+R, we set E, =O}. Let us assume the following

= {x E B, : V*(X)

DEFINITION2.1. A function W E 6?‘([Z X BY], R ) is definitely divergent on E,. if for every v E (0, y), for every to > 0 and for every continuous function +: Z+B,, there exist two positive constants e= e(y, Y), A = A(Y, 7) such that if v<]]+(r)]]t,,, one has

In particular, if in the definition we consider only those functions +(t) which are solutions of (2.1), then the function W(t,x) is said to be definitely divergent on Ep along the motion. In this case if a function W( t,~) has a derivative @t, x) definitely not equal to zero (in the sense of Matrosov) on the set E,, then W( t, x) is definitely divergent on E, along the motion.

3.

THE MAIN RESULT: AN EQUIASYMPTOTIC THEOREM

STABILITY

Let us consider a system of ordinary differential equations f=F(t,x),

(3-I)

where F(t,x) is assumed to be continuous on [0, 00) X r, r an open set of R”. Assume on F the following condition:

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CONDITION(A). For every compact set K cl? and for every continuous function r:I+K, for every a>0 there exist T= T(a,K,x)>O, {=l((~,K,r) >0 such that if lll~‘tF(~,~(~))d~ll>a for s>T, then t>{ (see [5]).

LEMMA3.1. Assume that Condition (A) holds and that with respect to the system (3.1) there exists a (?’ functionV: I X B,-+R having the following properties : (a) the function V is positive definite, (b) for every v E (0,~)~ xE CV,vVt > to, we have +(t,x) < - &(t)v*(x) < 0, where V*(x) is a nonnegative continuous function, and &Cl,(t)i-s nonnegative, continuous, and such that forany infinite system S of closed nonintersecting intervals of the semiaxis [0, 00) of identical fixed length, we have jsJ/,( t) dt = co. Then if x(t) E C&

(i.e., the positive

fort > to, une has

limit set !G!(x(t)) is contained

in EP).

PROOF. From the continuity of the function V* and from the hypotheses (a) and (b) it easily follows that there must exist a divergent sequence {t,} such that for every E > 0, p( x( t,,), EV) 0 such that p(x(r,,), E,)>E for n-cc. Assume that TV>&, k-l,2 ,..., n. Consider now the two divergent sequences {S,} and { 8;}, tk
P(QW%=)= ;

and p(x(fIi),E,)

= E

k-l

, 2 ,...,n

and E/2 < p(x(s),E,) =G.F for s E [8,, &!I. One has

where 4.=inf{V*(x),x~C,,,,p(x,E,)>E/2}>0. But p(x(Q, x(Q))?E/2, and from Condition (A) it follows that there exist a k(E, B,,x) = k and &E, B,, x) = 2 such that 0: - 0, > g for k > k. From hypothesis (b) one has V-+ - 00 for t-+co, and this contradicts hypothesis (a). The result is then proved.

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THEOREM 3.1. Assume that hypotheses (a) and (b) of Lemma and that V(x) = V,(x) V,(x). Moreouer assume that:

3.1 hold

(c) the functionV( t, x) admits an infinitely small upper limit; (d) the set Evl={xEB,:O< Ilxll 0, every to > 0, and euey continuuus there exists a positive constant c = c(k) such that p@(t), E,,) > k for all t 2 to, one has >c(t-

function + : Z-B,, if G(t) E Cv,y and

t,,).

Then the null solution of (3.1) is equiasymptotically

stable.

PROOF. From hypotheses (a), (b), (c) it foBows that the origin is uniformly stable, i.e., for any number p ~(0,y) there exists a 6( /3) >0 such that t,EZand ~~x,J~<~(/I)implythatx(t)existsforaBt>t,,and Ilx(t)ll to. Let u = 6(p), t,, E I, and x0 be such that ((xJ tb: the origin will be attractive. The proof is obtained by contradiction and can be reached in three steps. Step 1. We shah prove that for every k >0 there exists an unbounded sequence {t,}, t,ER+, such that x(t,)EN,(EvJ={x~BB,:6(v)< llrll O and a T(k) such that p(r(t), Evl) > k for all t > T(k); from property (iii) it follows that

But taking into account hypothesis (i), this is a contradiction. So we have proved by means of an auxiliary function that the limit set fl(x(t)) has a nonempty intersection with Evl n Cs(v,,s Step 2. One can easily see, now, from hypotheses (i) and (ii) that the solution x(t) cannot remain from a certain t > to onward in the set N,(h), where E= E(~(Y),P) is the same as in Definition 2.1.

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N. IANIRO AND C. MAFFEI

Step 3. Finally we can show that the solution x(t) cannot oscillate endlessly around the set Ev,. In fact, let us consider the two divergent sequences {r,}, {kJ,>, rk
and for all s, E [TV,0,. one has

Consider now

we can observe that V,(x(s,)), s, E [rk, ok], is definitely different from zero from a certain k onward. Otherwise there would exist a subsequence, say {So}, such that Vs(x(sk))+O as sk-+oc, so that V,(limk,,,x(sk))= V,(?)=O, and this implies that there exists a limit point of x(t) whose distance from EvI is greater than s/2. But this is impossible because the Wimit set has a nonempty intersection with E,, (see step l), it is a connected set (see for example [5]) contained in Evlv, by Lemma 3.1, and hypothesis (d) holds. So we have

from a certain k onward, where b, = inf{ V,(r(s))V,(x(s)), X(S) E C8(P,,p,p(x(s), E, n Cscy,,s) > e/2} > 0; then V-, - cc for t+ co, and this contradicts (a). So a t& > to can be found such that ]Ix( tA)ll<6(v), and this proves the attractivity. The theorem follows now from the fact that uniform stability and attractivity imply equiasymptotic stability. REMARK1. If { and T in Condition (A) are independent of the solution r(t) taking into account the hypothesis of infinitely small upper limit for V( t, x), then in the previous theorem the asymptotic stability is uniform.

REMARK 2. Let us compare Condition (A) with the two conditions placed on the right hand side of (3.1) in [2] and in [I.

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CONDITION (B) (see [2]). For every compact set K cr and for every continuous function x : I+ K, for every LI> 0 and s > 0 there exists 6 = S(o) > 0 such that

(i.e., the function m(s) = /iF( 7, x( 7)) dr is uniformly continuous).

CONDITION (C) (see [7]). For every compact set K cI’ continuous function x : I+ K one has

(ll~+‘~(7,~(7))d~ll~O S

and for every

uniformly for tE[O,l].

for s+co

Notice first that Condition (C) strictly implies Condition (B); consider in fact the function

f(t) =

2A,tcos(t2) if

tER+/S,

where

Z=[O,~],Zk=[~

v2(2k+l)+l

, m

v2(2k+2)+1

1,

and Ak is the length of Zk. This function verifies Condition (B) but not Condition (C). Compare now Condition (B) with Condition (A): the first implies the second; in fact this last condition can be considered as uniform continuity requirement from a given t on. Moreover, notice that, with respect to the asymptotic stability problem, there is no relevant difference between the two conditions: in fact one needs oscillations of a fixed length for the solutions of (3.1) [see hypothesis (b) of Lemma 3.11 only for t-co.

N. IANIRO AND C. MAFFJZI

78 4.

AN INSTABILITY THEOREM

THEOREM4.1. Assume with respect to the system (3.1) that there exist a t,, E I, y > 0, and a C?’function V: Z X B,+R satisfying the following pmperties:

(a) for every 11E (0, y) there exists x E B, such that V(to, x) > 0; (b) V admits an infinitely small upper limit; (c) V(t,x)>~Jt)Vr(x)Vs(x)>O, where V,(r)Vs(x)= P(x) and &(t) satisfies hypothesis (b) of Lemma 3.1. Suppose further that (d), (e) of The orem 3.1 hold. Then the null solution of (3.1) is unstable.

PROOF. Assume by contradiction that the hypotheses hold and that the origin is stable. Then for every E E (0, y) and to E I, a S(E) > 0 can be found such that llx,,ll<6(~) implies Ilx(t)ll 0. From hypothesis (c) one has

thus A=b-l(V(t,,x,))< I/x(t)//<& for all t>t,, where b is the k-class function of hypothesis (b). The proof follows then as in Theorem 3.1.

5.

AN APPLICATION:

UNBOUNDED

DAMPINGS

As an application to the Theorem 3.1, we consider the generalized Lienard scalar equation: i+h(t,x,i)f+f(x)=O or the equivalent system 3i.=y,

j=

-h(t,q)y--f(r),

I

t>O,

(~>y)-,o

(5.1)

Assume that: (i) h( t, x, y) is continuous for (x, y) E B,, t E Z and verifies Condition (A) of sec. 3;

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(ii) h(t,x,y) W(Wx, y), w here k( x, y) is a nonnegative continuous function for (x, y) E BY and &cl,(t)verifies hypothesis (b) of Lemma 3.1; (iii) f(x) is continuous and g(x) >0 if x#O; (iv) the set E, = {(x, y) E Z?,: 1x1#O, y =0} is a connected component with respect to the topological space E = {(x, y) E B, : 1XI#O, yk( x, y) = 0} ; the null solution of (5.1) is equiasymptoticahy stable. Set in fact, V(x, y) = y2/2 + /gf(&‘)d.$; along the solutions of (5.1) one has V(x,y)= - h(t,x,y)y2 < - rC;(t)k(x,y)y2 GO. Therefore the function V(x, y) verifies hypotheses (a), (b), and (c) of Theorem 3.1 and this implies the uniform stability of the origin. Consider the two functions Wr(x, y) = xy and W2(x, y) = x; for every Y E (0, y) WI and W, are bounded in CG(vj,y. We shah show that the function W, is also definitely divergent on E,={(x,y)~B,:Ixl#o, y=O}:

From hypothesis (i) one can see that for every a > 0 and for every continuous function +:Z+B, such that S(~)<(+(t)(O such that

and then

where m = m(6(v),y) = min{ $(x),x fl=(tt())/t. Thus

E C8(~,,y}, c is a positive constant, and

it is sufficient to choose c > l/(1 - LY) to be sure that W, is definitely divergent on E,, where A(6( Y), y) of Definition 2.1 is equal to m/c + m m. Moreover, from the fact that W2 = y it is also easily seen that W, verifies property (iii) of Theorem 3.1. From Theorem 3.1 it follows that the origin is equiasymptotically stable.

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REMARK. This result can be compared with the one obtained in [7J by N. Onuchic and with the result in [l] due to Z. Artstein. In [7], as we have already observed, stronger conditions are given on the right hand side of the system; the equiasymptotic stability for the origin follows from invariance properties for the Mirnit set. The asymptotic stability result in [l] is obtained under assumptions on h(t,x, y) which imply Condition (A) restricted to the motion; the hypotheses on the right hand side in [l] ensure also that W( t, x, y) is definitely divergent along the motion, without needing to know the solutions; and this is all we need in order to obtain the result.

It is a pleasure to thank L. Salvadori and P. Negrini for many helpful discussions.

REFERENCES 1. 2. 3. 4. 5.

6. 7.

8. 9. 10.

Z. Artstein, Topological dynamics of ordinary differential equations and Kurzweilequations,J. LXfferential Equations 23:224-243, (1977). Z. Artstein, Uniform asymptotic stabilityvia the limitingequations,J. Dij@rential Equations 27:172-189 (1978). J. P. La SaBe and S. Lefschetz, Stability by Liapwwu’s Direct Method with Applications, Academic Press, New York, 1961. J. P. La Salle, Stability of nonautonomous systems _l. Nonlinear Analysis Theory, Methods and Applications 1:83-90 (1976). J. P. La Salle, New stability results for nonautonomous systems, in Dynumica2 Systems-Proceedings of the University of Florida lntemational Symposium (A. Ft. Bednareck and L. Cesari, Eds.), Academic Press, New York, 1977, pp. 175-183. V. M. Matrosov, On the stability of motion, 1. Appl. Math. Mech. 26: 1337-1353 (1962). N. Onuchic, Invariance properties for ordinary differential equations: stability and instability, J. Nonlinear Analysis Theory, Methods and Applicutiuns 269-76 (1977). L. Salvadori, Sulla stabihm de1 movimento, Matematiche (Catania) 24, No. 1 (1969). G. Ft. Sell, Nonautonomous differential equations and topological dynamics I, II, Trans. Amer. Math. Sot. 127:241-263 (1967). T. Yoshizawa, Stability Theory by Liupum’s Second Method, Math. Sot. Japan, Tokyo, 1966.