On the stability of motion

ON THE STABILITY OF MOTION N. ROUCHE University of Louvain, Belgium

Ah&act--A theorem on uniform asymptotic stability of the origin for a differential system is proved, using two auxiliary functions: a Liapunov function and a “Liapunovlike” vector function. A theorem on instability is also proved. The results are applied to a system of Lagrange equattons with complete dtj?ipation and non energic forces: in this case, the Liapunov function is the total energy and the Liapunovlike vector function is the vector of the conjugate momenta.

I. INTRODUCTION BY way of introduction, let us recall the following elementary problem. The equation of a damped harmonic oscillator reads

mf f h2 + kx = 0,

(1)

where x is the elongation, m the mass, h the constant of viscous damping, k the spring constant and where the dot represents the time derivative. If we want to use Liapunov’s second method to prove that the origin of the x, ~2plane is asymptotically stable, we look for a positive definite function W(x, 2), whose time derivative computed along the solutions of (1) would be negative definite. To some extent, it is an odd fact that the total e6ergy H = fin%’ + +)kx’ is not a satisfactory function in such a simple situation, and this puzzles the physicist. In fact & = - hP is not negative definite, but only semi-de~nite. Of course, a satisfactory Liapunov function can be found, but unfortunately it has no particular physical significance. We observe essentially the same situation for the asymptotic stability of the origin in phase space, when the system is described by n Lagrange equations to which complete dissipation terms have heen added. The total energy is of no use in connection with Liapunov’s theorem on asymptotic stability and one has to resort to a more involved function (see, for instance, Salvadori [ 11). The theorem of Liapunov just mentioned is simple, but it is diflicult to construct an adequate Liapunov function. The motivation of the present research was to prove a more complicated theorem, but using Liapunov functions which are simpler to determine. As will be seen, the asymptotic stability of the origin for a Lagrangian system with complete dissipation will easily result from some obvious properties of the total energy and the conjugate momenta, all functions bearing physical meaning. In Section 2, we shall prove some abstract theorems on stability and instability, namely theorems concerning the equation 1 = X(x) (where x and X are real n-vectors) and without reference to mechanical systems. These theorems have been inspired by those of Matrosov [Z]. The fundamental idea of this paper may be outlined as follows. According to the classical 295

N. ROWHE

296

theorem of Liapunov, one has uniform asymptotic stability of the origin as soon as there exists a Liapunov function V(x) whose time derivative is sign definite. Roughly speaking, Matrosov considers a plain Liapunov function: its time derivative is therefore only sign semi-definite. But then a difficulty appears because of the existence of a set of points where v(x) = 0. We’ might call it the “dangerous set”. It is essential to prove that the solution x(t) cannot remain more than a given interval of time in this set. To prove this, Matrosov uses a scalar Liapunov-like function W(x) which has to be bounded and whose time derivative k(x) has to be non-vanishing definite on the dangerous set (for the definition of a non-vanishing definite function, cf. [2]). We propose here the following substantial change : to the auxiliary function W(x), substituting a vector function which should vanish and whose derivative should not vanish on the dangerous set. By doing this, we hope to obtain more flexibility in the applications. In Section 3, we use these results, as well as LaSalle’s theorem on asymptotic stability of invariant sets, in the study of Lagrangian systems with dissipative and non-energic forces. At last, in Section 4, we extend our previous results to non-autonomous systems. 2. THEOREMS

USING

VECTOR

LIAPUNOV

FUNCTIONS

2.1 Consider the differential equation i = X(x)

(2)

where x and X are real n-vectors and X is defined over the sphere B,, = {x : 11x 11< p’, p’ > O}. It will be convenient in the sequel to consider, along with B,., another sphere B, = {x : //x )I < p, p’ > p > 0). We use the norm

IIxII = l_&Icnlxi .. where the xi are the components of x. We assume that X satisfies sufficient regularity conditions in order that, through every point (to, x0) E I, x B,,, where I = [0, VJ], there passes one and only one maximal solution x(t; t,, x0) of (2). For this solution, we shall merely write x(t) whenever it is not necessary to remember its dependence with respect to the initial conditions t,, x0. At last we assume that X(0) = 0, in such a way that equation (2) admits the vanishing solution x E 0. In what follows, we use as the distance between two points x and y in R” d(xyY)=

llx-Yll

=

l<~~nlXi-yiJ~ . .

The distance between a point x and a set E, and the distance between two sets E and F: are given respectively by d(x, E) = inf{d(x, y) :y E E}, d(E, F) = inf{d(x, y) :x E E, y E 8’).

2.2 LEMMA :lowerhound,for the transit time of a solution If t, and t, are in I, with t, < t2, if the solution x(t) is de$ned on [tl, t2] and if, for every .YE B,. : 1)X(u) /I Q A, where A is a constant, then d(x(t,), x(tJ) >, r > 0 implies that t, - t, 3 r/nA. By the mean value theorem lX~tl)-x,(t,)/=I~(ii)I(L2-tl) where

t,

d

si

d

t,.

(1
Because dxi dt (ri)

= (Xi(x(ri))I

< A,

we also have \xi(tZ) - xi(t,)l < A(t, - t ,). Hence, by summation,

the expected

result.

2.3 LEMMA: how to reduce un open covering of a compact space If G is a colizpact space and if the family of sets {pi : 1 < i < m} is an open covering of G, there eqists a 6 > 0 such that {pi\s, : 1 < i < m) where Si =

{.x

:

d(x,

is also an open covering of G. The symbol 9 means “the frontier the present lemma. we shall consider assuming that the set

F_Pi) < 6)

(1
(3)

of’. Let us insist on the fact that G is a space: in no point outside G. We do not lose generality by

rI = PI\ U Pj 2SjQm is not empty. The complement member. Therefore

of p, is contained

in the union

appearing

in the second

where % means “the complement of ‘. As complement of an open set, r, is closed ; as closed set of a compact space, it is compact. Observing that rI and 9p, are disjoint compact sets, we have d(r , , Fp,) > 0. If 6, is a positive number smaller than this distance and if s; = {x : d(.\-..Qp,)

d d,},

we still have an open covering of G if we replace (p,. p2,. . , p,,,} by {pl\s’,, p2. p3,. . . , y,;. Starting from this last covering, we successively determine in the same way qualltitich d2, ii,. . h,,, such that we get the open covering ipi\$: I < i 6 ITI) where s; = (.Y : d(.u,

Fpi)

< ~5;)

Finally, if 6 = min(b;, : I < i < ml, we obtain where si is given by (3).

(I didm). the

open

covering

{p,\s, : 1 < i < m j,

2.4 LEMMA : upper bound jbr the escape time of a solution,from a compact set Let there exist an A > 0 such that ji)r all x E B, : 11X(x) 11d A. Suppose that there exists a function W(x) : B,. + Rk (k a positive integer), with continuous partial derivatives, and a

298

N. ROWHE

.

co~~pact set F c B, such that for every .YE F : W(x) = 0 and W(x) # 0. Then there exist two quantities T > 0 and (T > 0 such that x(t) cannot remain in G = fx :d(u,F) < g}

(4)

for an interval of time equal to T. G is a subset of R” and even, as we shall see, a subset of BP,. Only later in the course of the proof, shall we identify it with the space G of lemma 2.3. The set F being compact, for every L > 0 there is a (I > 0, (a, < p’ - p), such that x~I~:df.~,F),
CL.

Let ~13,(1 < j < k) be the components of the vector W. Since ti is non-vanishing and continuous on F, every point of F has a neighbourhood where one of the wi is larger than a given positive number. Therefore, F being compact, there is a finite number of such neighbourhoods, constituting an open covering of F. Let this covering be designated as i ?fr, X2,. . . , 7~~). For every i, (1 d i < m) there exists a j, (1 < j d k) and a ti such that x E zi implies 1ti#) 1 > &. If we write 5 = min(c,, t2,. . . , <,j, we may also say that for every i, (1 d i d ~1 there exists aj. ( 1 G .j d k) such that X E pi j.

1

~j(Y,I

>

5.

(5)

Let us write

Since F and 9~ are disjoint compact sets, we have d(F, F-71) > 0. Let us choose CT< min (d(F, 9n), a,), where (T = a(L) --+0 when L --t 0. Then of course, G being defined by (4), x E G =+ 11W(x) /I < L.

(6)

Because, for fixed F, d(x, F) is a continuous function of x, G is closed. Since it is bounded, it is compact. Let us now consider G as a new space contained isometrically in R”. Compactness being an intrinsic property, G considered as a space is compact. For the sake of clarity. we hereafter designate the sets of the space G by Roman lower-case letters: for instance, we shall write pi for the set G n 7ci considered as a set of G. An immediate consequence is that, although G R zri is not necessarity an open set of R”, pi is an open set of G. Lemma 2.3 shows that the open covering by the pf = pj\si may be substituted to the open covering by the pi. If x(0) = x0 E G, then x0 belongs to one of the &, let us say pi. If x(t) remains in G, it cannot leave ph in a time less than &,%A, for it has to pass through p,\pk, an annular region of thickness S surrounding & (cf. lemma 2.2). Observe that 6 = 6(o) may be considered as a decreasing function of CT. On the other hand, we infer from (5) and (6) the existence of a Wj such that (7) From this we deduce that x(t) cannot remain in ph an interval of time longer than 2L/‘5. Indeed, suppose that x(t) stays in ph between the instants tl and t2, where t, < t,. AS w$t,)

-

W,(r,) = r +‘(r)dz, I

On the stability

we get, using the fact that

wj does not change 1y:.(t,)l

+ 1y(h)}

of motion

299

sign > ;! ) $b)I

dT.

Hence, using (7): 2L > 5(tz - ti). At last, in view of the properties of the functions a(L) and 6(o), it is possible to choose 2L/5 < G(a(L))/nA; thus, we can assert that, during the interval of time T = 2L/5, x(t) will leave p,, without crossing the annular region p,,\pA. Therefore, x(t) leaves G. 2.5 Comparison with “non-vanishing definite functions” It is interesting to compare the vectorial function W(x) of lemma 2.4 with the scalar definite on a given set”. The function qualified by Matrosov [2] as “non-vanishing properties of this last function enable one to prove that the solution x(t) escapes from the set after a certain lapse of time. Using the function W, we have proved that x(t) cannot remain more than a given lapse of time in a certain subset of the given set and that it cannot leave the subset while remaining in the set. Therefore, the conclusion is the same: x(t) escapes from the set after a finite period. The properties of the subset will not be used hereafter, but only those of the set. However, we shall show, by the example of mechanical systems, how the vectorial character of W can be taken advantage of in the applications. 2.6 THEOREM : sufficient conditions for asymptotic stability Let there exist two functions V(x) : Be. -+ R and W(x) : B,, + Rk (k a positive integer), continuous as well as their time derivatives V(x) and W(x) computed along th’e solutions of equation (2). Zf (a) (b) (c) (d)

$or all x E B, : 11 .x(u) 11< A, where A is a positive constant; V(u) is u Liapunovfunction and w’e write E = (_Y: .s E B,,. V(u) = 0) ; for every .Y E E : W(u) = 0; for every x EE\(O~, : W(x) # 0 (F is the closure Q’E , [Oi is the set containing the origin only) :

then, the vanishing solution x = 0 of equation (2) is asymptotically stable. Hypothesis (b) implies that the identically zero solution is stable: for every E, with p > E > 0, there exists an V(E) such that to E 1, /j x0 Ii -c q(c), t 3 to = Ii x(t; t,, x0)/j c E. Let us fix E and prove that to E I, /lx,, I/ < V(E) + /Ix(t; to, x0) 11-+ 0, t + co. We do not lose generality by assuming V(x) to be positive definite, in which case v(x) < 0. This implies that there exists a function a(r) : [0, p] + R, continuous, non-decreasing, equal to zero if and only if r = 0, and such that V(x) 2 a(/1 .x 11).

(8)

First, let us prove that for t, E I and 1)x0 II < q, th ere is no p > 0 such that for every t 2 to : V(x) > ,a If x,, = 0, this conclusion is obvious. Suppose that x0 # 0 and such a p exists. Due to the continuity of V, there is a v, with E > v > 0, such that II x 11< v *

V(x) < P-

Let us put F = {x : v d //x /I < E, d(x, E) = 0).

(9)

According to lemma 2.4, there exist two positive quantities T and 0 such that x(t) cannot remain in G = {x : d(.u, F) < c} for a lapse of time equal to T. Consider now the two overlapping sets U

= {x : v < .Y < rs,d(.u, E) < CT;,

U* = {x : v d .Y d 6, d(x, E) 3 a/2). Since U c G, x(t) cannot remain in U for a lapse of time equal to T. Let us then consider interval J = [r1,t2] c I, of length T and suppose that for every t E J : v d /I.x(t) I/ d E.

an

(10)

We have only two possibilities: (i) either, for every t E J : x(t) E U* ; since u7,: I - v(7,) = 7 li(r)dr,

T1 if we write 1 = inf( 1V(.Y) a quantity

certainly

> 0, we get v(7,) - v(T,) < - TE.

(12)

(ii) or there is a t E J such that x(t) # U* ; for this value of t, x(t) E U. But there is another t E J for which x(t) $ U. At the first of these instants d(.u, E) < a/2, while at the second d(x, E) > 0. Because x(t) is continuous, there are in J two instants TV and rb for which d(x(z,), E) = a/2 and d(x(zb), E) = o. Of course one also has d(.u(r,), x(rb))> a/2. Suppose for definiteness that z0 > T;. From lemma 2.2 we deduce that 7b - 7. > a/2nA. Thus V(T,) - v(7,) < v(7;) - V(7,) = i li(T)dT< - lo,‘2nA.

rl!

(13)

From (12) and (13) we conclude that hypothesis (10) cannot be maintained ; for otherwise. after a finite number of intervals like I, V would become negative. Since, because of stability, /i .Y/j cannot reach the value c, there is a t for which /j x(t) Ij < v and therefore V(t) < p (cf. (9)). Because V is non increasing V(t) -+ 0 as t -P -l. . The same is true of u( j,.x/,1(cf. (Xl) and also of ,.Y!,.Q.E.D. 2.7 THEOREM: sufficient conditionsfir instability Let there exist two functions V(.u) :B,, +, R and y(x) : B,, + Rk (k a posirive intrger), continuous as well us their time derivatives V(.Y) and W(.x) computed along the solutions of’ equation (2). Zf (a) jbv all .YE B, : Ij X(x) I/ < A, where A is a positive constant; (b) V(0) = 0 and, in every neighbourhood of the origin, there are points where V(x) > 0; V(x) = 01; (c) 3(x) 2 0; we write E = {x : xEB,,

On thr stahilitv

301

c~‘motion

(d) ,fbr every .Y E E : W(.u) = 0; (e) .for every .YE fl{O) : t+(x) f 0; then, the uunishing solution x z 0 of equation (2) is unstable. Let us assume that the vanishing solution is stable: i.e. for every E > 0, there is an V(E)> 0 such that to E 1, /I -yojj < )?w, t a to * 11.$t ; to, x0) I/ < E. We choose an .Y: such that 0 < 11.Y: 11cc V(E) and that V(x$) > 0 (hypothesis (b)). There exists a v > 0 with the property that for every t > t o : v < Ij .u(t ; to, .u$) /j < E.

(14)

Indeed, select v in order that /I .YI/ < v a V(x) < V(.u$. Proposition (14) follows since, by hypothesis (c), I/ is non-decreasing. Using lemma 2.4 and defining the sets U and [i* as in theorem 2.6, we can prove, as in this theorem, the existence of two positive quantities T and 0 such that u(t) cannot remain in U for a lapse of time equal to T. Considering a time interval J of length T, one proves again as in theorem 2.6, that either .x(t) remains in U* during J and the increase of k’ is at least TI, or there is a t E J such that u(t) 6 U*, and the increase of I/ is at least lo/ZnA. But V, which is continuous on the closure of I?,, is bounded and this bound would not be respected after a finite number of intervals like .I. Therefore, the initial assumption of stability has to be discarded, Q.E.D. 3. ASYMPTOTIC

STABILITY

AND

INSTABILITY

FOR

MECHANICAL

SYSTEMS

3.1 Type of system We shall now apply LaSalle’s theorem [3] as well as theorems 2.6 and 2.7, to the study of Lagrangian systems. Let us consider a mechanical system subject to holonomous, scleronomous constraints, described by n generalized coordinates ql, q,, . . , q,,. Let 52, and Sz, be neighbourhoods of the origin in the respective spaces of the coordinates qi and the generalized velocities di. We put s2 = s2, x Sz, and assume furthermore that (a) some of the forces derive from a force function U(q,, q2,. . . , 4.) possessing continuous derivatives in ag; this force function is negative definite in s2, and has in s2, no other stationary point than the origin; (b) other forces, designated by Qe (1 < i G n), are functions of the qi and de with continuous first derivatives, and are such that, in a

the expression in the first member of this inequality vanishes if and only if fj, = q2 = . . . = lj, = 0;

(c) all the other forces, designated by Si, are non-energic, i.e.

302

(d) the kinetic continuous

N. ROWHE

energy T is a function of the qi and the gi with derivatives in C?; one has in Sz : T > 0, and T = 0 if and only if 4, = Lj2 =

of second order

. . = 4, = 0.

We may summarize hypothesis (b) by merely stating that we consider “complete dissipation”. This is the case, for instance, if the’ forces Qi derive from a Rayleigh function, i.e. if Qi = aF/&j,, where F is a function of the qi and gi, analytic in the gi, and if the expansion of F in powers.of the 4, begins with a quadratic form which is 2 0 and vanishes if and only ifd, =gz=... = 4, = 0 (cf. N. Rouche et ul. [4]). A particular case of non-energetic forces is obtained when the Si are linear in the &: the gyroscopic forces. Notice that we do not require the kinetic energy to be a quadratic form with respect to the dP This implies that the Lagrangian is L = T, + U where aT G4i-T L

T,= c

I
is the Legendre transform equations of motion are

of T with respect d aL

3L

dt &ji

aqi

(15)

to the gi, called

the kinetic

coenergy.

The

(16)

- Qi + Si.

Hypothesis (a) and (d) imply that the total energy H = T - U is a positive definite function at the origin of the phase space (i.e. the space of the qi and di). Furthermore, because

A =

C Qi~i

1SiQn

d 0,

where fi is computed along the solutions of equation (16), H is a Liapunov function the origin is stable. Let us now show, in two ways, that it is asymptotically stable.

and

3.2. First proof of asymptotic stability There are several variants of LaSalle’s theorem which we intend to use now. Let us recall the following, concerning equation (2) : 11‘there exists a real function V(x), possessing continuous partiul derivatives und n lower bound, and such thut Y = {x : V(x) < I,1 > 0; be u bounded region;

ifdV/dt,

computed

along the solutions oj(2), is < 0; if

E = {x : xe Y,dV/dt

= 0)

and if M is the lurgest invuriant subset of E, then x,, E Y *

x(t ; t,, x0) -+ M

when

t + K.

More precisely, we shall use the corollary stating that, if M is the origin, and if it is stable, then it is asymptotically stable. To deal with the mechanical system, we first notice that, H being positive definite, there exists a region Y around the origin of the phase space and a number 1 such that H < 1 for every point in Y and H = 1 on the frontier of Y. The set E

On the .vtabilir_v of’motion

where fi = 0 is the hyperplane ID defined by the equations A vector of m has, in the phase space, the components

whereas the components in phase space) are

At the moment

of crossing

303

d1 = GZ =

q1,92,. . 3qn, 090, . . -0, of the phase velocity (velocity of the point representing

ID, this phase velocity 0, 0,

. = 4, = 0.

(17) the system

has the components

. , 0, ij,, ijl,

. . ) ij”.

(18)

The scalar product of the vectors (17) and (18) is zero. In order to prove that IID,with the exception of the origin, contains no invariant subset, it is sufficient to prove that, outside the origin, the vectors (17) and (18) are different from zero. This is obvious for (17). Consider now the equaiions of motion at the moment of crossing w. Written explicitely, these equations are (19) By a well-known

argument

(see for instance

Lanczos

[5]), (15) implies that

?T,Jc?q, = - dT/aq,. But from hypothesis(d), we know that, in ~TI: (7T/aqi = 0. Therefore ?TC/?qi = 0. Let us now prove that all the Qj also vanish in a. Hypothesis (b) implies that, for any fixed choice of the qi, the function R = is minimum

at the origin

of the di. Therefore

CYR

G=

c

C

1
Qicii

at this point

aQi. ,qqi+ Qj=o

(1<
J

14iQn

and in ZD: Qj = 0. By a similar argument, it can be proved that the Si vanish in a. If we assume that, at a given point of w, differing from the origin of the phase space, one has iii ZX0

(1 didn)

(20)

then. because d(dT, &j,) dt is linear with respect to the 4,. equation (191 would read at that point aI//~qi = 0. But this is impossible by hypothesis (a): I/ is stationary at no point outside q, = q2 = . = qn = 0. Therefore (20) is absurd: the hyperplane is crossed at right-angle and with non-vanishing velocity by the point representing the system in phase space. Thus, apart from the origin, 1~ contains no invariant set. 3.3 Second proof of asymptotic stability; the case of instability In paragraph 3.2, we had to prove that a given set was not invariant, and there was, in the statement of LaSalle’s theorem, no indication of a procedure to achieve this. On the other hand, theorem 2.6 contains information on this point : the only use of the function W is to expel x(t) from a given set, thus proving that it is not invariant. The role of the vector

N ROUCHE

304

function W will be played by the set of conjugate momenta aTJ&j, (1 d i < n). According to hypothesis (d) and formula (15) these quantities tend to zero when the point in phase space approaches E. Furthermore, their time derivatives along the solutions of the problem are given by (19). But we know that, on E dCIT, dt

au

aqi= a4,

(1,
and the second members are different from zero, except at the origin, which is therefore asymptotically stable. If we replace hypothesis (a) of paragraph 3.1 by the statement that, in every neighbourhood of the origin, there are points where U(q) > 0, then theorem 2.7 shows that the origin is unstable.

4. THE CASE

OF NON-AUTONOMOUS

EQUATIONS

4.1 We consider

now the differential

equation d.u ~ = X(t,x) dr

where x and X are real n-vectors and where X is defined over I x B,,, with I and B,, as in 2.1. We suppose here also that X satisfies sufficient regularity conditions in order that, through every point of I x B,., there passes one and only one maximal solution .u(t : t,, .x0) of (21). In this context, lemma 2.2 remains true with X(x) replaced by X(t, s) and the proof is the same; lemma 2.3 of course is not affected. In order to proceed any farther, a new definition is necessary. 4.2 Definition : non vanishing definite vector jimction The vector function f+ft, x) : I x B, --+Rk, where k is a positive integer, is non cunishinq d&ite on a set E if, for every pair of positive numbers v and E with v c E < I’, there is a positive number t(v, E) and an open covering {rrr, 7c2,. . , q,,J of the set F = {x : v d 11x I/ < E,d(.u, E) = 0) such that for every i, (1 < i < m), there is a component Wj of W with the property that t E I, x E ?ri =a 1Wj(L x) ) > 5. This is a generalization of the concept of non vanishing scalar function appearing in Matrosov [2]. Lemma 2.4 can now be replaced by the following. 4.3 LEMMA Let there exist an A > 0 such that for all t E I and x E B, : // X(t, x) 1; < A. Supposr thut there exist a function W(t, x) : I x B, + Rk und a set E c B, with the followiny properties (a) for every L -+ 0, there is a x(L) > 0 with I/ WL x) // < L; (b) k@(t, x) is non vanishing definite on E ;

x < p’ - p, such

that

d(x, E) < 1 -

On thr stability 01 motion

305

then,,for uny choice of v and E, there are positive quantities T and o such that x(t) cannot remain in G = (x : d(x, F) < o}, where F is defined by (22), for a lapse of time equal to T. The proof is substantially the same as the one of lemma 2.4. 4.4 THEOREM :suflcient conditions for uniform asymptotic stability Let there exist two functions V(t, x) : I x B,. + R and W(t, x) : I x BP8 + Rk (k u positive integer), conrinuous as well as their time derivatives p(t, x) and l@(t. x) computed aloig the solutions of(21). [f (a) jbr ull Y E B, : (1X(t, x) 11d A, where A is a positive constant; (b) V(t, x) is positive definite ; V(t, x) -+ 0 uniformly in t when x -+ 0; (c) there exists u continuous function V*(x) : B,. -+ R such that $‘(t,x) li(f,O) = 0: we write E = {x : V*(x) = 0); (d) for every L > 0, there is u x > 0 such that 4x3 E) < x * (e)

d V*(x) ,< 0;

11Wt. x) Ij < L.’

k@(t, x) is non vunishing dejinite on E;

then, the vanishing solution x = 0 of equation (21) is uniformly usymptotically stable. The proof is a straightforward modification of the one of theorem 2.6. In particular, one has to replace the definition (11) of 1 by the following: 1 = inf { 1V*(x)/ : Y E U*}. 4.5 THEOREM :sufficient conditions for instability Let there exist two functions V(t, x) : 1 x B,. -+ R and W(t, x) : I x B,, --+Rk (k a positive integer), continuous as well us their time derivatives p(t, x) and w (t, x) computed along the solutions ofequution (21). Jf (a) jbr ull x E B, : /j X(t, x) 11< A, where A is a positive constunt; (b) V(t, x) -+ 0 when x -+ 0, uniformly for t E I; in every neighbourhood of the origin and .for t larger thun (I given t, there are points where V(t, x) > 0; (c) there is u continuous function V*(x) : B, -+ R such that i/ (I, x) > V*(x) 2 0; we write E = (.Y : V*(x) = 0) ; p(t, 0) = 0; (d) .for every L > 0, there is a x > 0 such that

d(.x, E) Q x * (e)

*(t,

/IWt, x) 11< L;

x) is non vanishing definite on E;

then. the vanishing solution x = 0 of (21) is unstable. In the proof, one has to pay attention to the fact that the hypothesis “ub absurdo” should be stability, not uniform stability (of course, the stability considered in connection with theorems 2.6 and 2.7 was always uniform, for the corresponding differential equation was autonomous). Thus one should say: let us assume that the vanishing solution is stable, i.e. for every E > 0 and every to E I, there is an ~(5 to) such that /I xl) j/ < v(a to), t b to -

IIx(t ; to, x0) /I < E.

Corresponding to the to mentioned in hypothesis (b), we choose an xg such that Ii .Y: /j < Y](E,toI and that V(to, xg*) > 0. The rest of the proof comes as a trivial modification of that of theorem 2.7. B

306

N.ROUCHE

4.6 Let us notice at last that these two theorems enable us to extend our results on asymptotic stability and instability of Lagrangian systems to the case where the forces of both types Qi and Si depend on E,provided that

A~knowi~,~~Pmcnt-The author wishes to express his gratitude to K. Peiffer for many useful discussions REFERENCES SALVADORI. SullS stabilitii asintotica delle posizioni d’equilibrio di un &sterna in presenza di forze dissipative anche non lineari. Rc. Accud. SeLfis. mat., Nupo/i 30, 145-152 (1963). [2] V. M. MATROSOV, Ob ustoichivosti dvizheniia [on the stability of motion]. Prikl. Mar. Mekh. 26, 885-895

(I] L.

( l462). [3] J. LASALLE and S.LEFSCHETZ, S&i&y 141 N.

ROUCHE et E. VAN LAMSWEERDE, La

b,vLiapunov’s Direcr Method, with Applications. Academic Press (1961). fonction de Rayleigh et la mecanique des systbmes dissipatifs. Jnl M&.

J. 323 338 (1966). LANCZOS, T/W Variational Principles of Mechanics, 2nd edition. University of Toronto Press (1962).

[!I] C.

(Received 6 July 1967)

R&urn&-On demontre un theorbme de stabilite asymptotique uniforme de I’origine pour un systbme differentiel, en utilisant deux fonctions auxiliaires: une fonction de Liapounoff et une fonction du kenre des fonctions de Liapounoff, mais vectorieile. On dimontre aussi un thCorZme ~ins~biiit~. Ces resultats sont appliques a des equations de Lagrange comportant des forces de dissipation complete et des forces non Cnergetiques: dans ce cas, c’est I’energie totale qui est la fonction de Liapounoff et le vecteur des moments conjugues qui est la fonction vectorielle auxiliaire. ~nsammenfassnng-Fin Theorem fiber die gleichformige asymptotische Stabilitat des Ursprungs fiir ein Differentialsystem wird mit zwei Hjlfsfunktionen bewiesen: Einer Liapunov-Funktion und einer dieser lhnlichen Vektorfunktion. Ebenfalls bewiesen wird ein Theorem tiber die Instabilitkt. Die Ergebnisse werden auf ein System Lagrange’scher Gleichungen mit vollkommener Dissipation und nichtenergetischen Kraften angewendet: In diesem Falf entspricht die Liapunov-Funktion der totalen Energie und die vektorielle Hilfsfunktion dem Vektor der kanonisch konjugierten Impulse.

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