- Email: [email protected]
On Lyapunov stability of nonautonomous mechanical systems
10 July 2000
Physics Letters A 271 Ž2000. 413–418 www.elsevier.nlrlocaterpla
On Lyapunov stability of nonautonomous mechanical systems H. Tasso ) , G.N. Throumoulopoulos 1 Max-Planck-Institut fur Germany ¨ Plasmaphysik, Euratom Association, 85748 Garching bei Munchen, ¨ Received 22 February 2000; received in revised form 10 May 2000; accepted 5 June 2000 Communicated by M. Porkolab
Abstract A sufficient condition for the linear stability of nonautonomous dissipative mechanical systems with circulatory forces is derived. It is applied to autonomous systems transformed to nonautonomous ones by a time-dependent orthogonal transformation. This allows to obtain sufficient stability conditions in case the perturbed potential energy can have negative values, which was not possible up to now. This is, in particular, of importance for plasma equilibria with sheared flows. The nearness of the sufficient condition to necessity is discussed for a particular example. q 2000 Elsevier Science B.V. All rights reserved.
1. Introduction In a recent contribution by one of the authors w1x, a sufficient stability condition was obtained for dissipative and autonomous mechanical systems having circulatory forces. This was made possible by the use of a quadratic Lyapunov function containing the perturbed energy as well as the perturbed virial. We show here that a slightly modified Lyapunov function leads to a similar sufficient condition for nonautonomous systems. By applying this condition to autonomous systems transformed to nonautonomous ones by time-dependent orthogonal or unitary operators or matrices, one is able to obtain significant results for cases for
)
Corresponding author. Fax: q49 89 3299 1181. E-mail address: [email protected] ŽH. Tasso.. 1 Permanent address: University of Ioannina, Association Euratom – Hellenic Republic, Physics Department, Section of Theoretical Physics, GR 451 10 Ioannina, Greece.
which the perturbed potential energy can be negative. This leads to stability statements describing gyroscopic stabilization, combined with a dissipative one, in a quantitative way. If applied to discretized fluid systems, this result could be significant for answering questions about the stability of hydrodynamic and magnetohydrodynamic sheared flows. In Section 2 the generalized sufficient condition is derived. Its application to autonomous systems with negative potential energy is the topic of Sections 3, 4 and 5. In Section 6 the sufficient condition is compared to the necessary and sufficient one for a special case. Finally, the conclusions are summarized in Section 7.
2. The generalized sufficient condition Let us consider linearized mechanical systems with time-dependent coefficients generalizing those
0375-9601r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 0 0 . 0 0 3 8 6 - 8
H. Tasso, G.N. Throumoulopoulosr Physics Letters A 271 (2000) 413–418
414
given by Eq. Ž1. of w1x whose equation of motion can be written in the following compact form: N X x¨ q Ž DX q GX . x˙ q Ž K X q F X . x s 0,
Ž 1.
˙
˙
˙
˙
˙
First, if DX s K X s N X s 0 but F X / 0 and GX / 0 then the stability condition is equivalent to the one given by Eq. Ž7. of w1x i.e. F X y GX
where x is an n-dimensional displacement vector, N X is a positive symmetric matrix related to the inertia of the system, DX is a symmetric positive definite matrix describing damping effects, K X is the symmetric matrix due to potential forces, and GX and F X are skew symmetric matrices related to gyroscopic and circulatory forces respectively. Derivation of Eq. Ž1. in the autonomous case can be found in e.g. w2x for discrete systems and w3x for fluids and continuous systems. Note that for a real system all quantities appearing in Eq. Ž1. are real valued. The purpose of this contribution is to find stability conditions for the equilibrium position of Eq. Ž1. given by
The minimization of the Hermitian form associated with M X can be done using well known eigenvalues algorithms, which determine solely the sign of the lowest eigenvalue of M X . A positive sign implies also the positivity of the expression under the square brackets in Eq. Ž6., which means that this expression is then a Lyapunov function. Second, in the general case, the stability conditions are given by
x s x˙ s 0.
K X q DX y N X ) 0
Ž 2.
MX s
Let us take the scalar product of Eq. Ž1. with x˙ and x, respectively, to obtain 1 E Ž x˙ , N X x˙ .
Et
2
q
1 E Ž x,KXx.
Et
2 ˙X
y Ž x˙ , N x˙ .
X
X
y 12 Ž x , K x . q Ž x˙ , D x˙ . q Ž x˙ , F x . s 0, X
Ž 3.
X
E Ž x , N x˙ .
˙X
y Ž x , N x˙ . q
Et
˙X
1 E Ž x,D x.
Ž 4.
Adding Eqs. Ž3. and Ž4., we obtain X
X
X
1 E w Ž x˙ , N x˙ . q Ž x , K x . q Ž x , D x . q2 Ž x , N x˙ . x
Et
2 X
X X X X sy Ž x , K x . y Ž x˙ , Ž D y N . x˙ . q Ž x , Ž F yG . x˙ .
˙X
˙X
˙X
˙X
q Ž x , N x˙ . q 12 Ž x˙ , N x˙ . q 12 Ž x , D x . q 12 Ž x , K x . .
Ž 5.
X
By adding and subtracting Ž x, N x . to the expression under the square brackets, Eq. Ž5. can be written in the following form X X X X 1 E w Ž x , K x . q Ž x , Ž D y N . x . q ŽŽ x q x˙ . , N Ž x q x˙ .. x
Et
2 X
˙
X
X X X X sy Ž x , K x . y Ž x˙ , Ž D y N . x˙ . q Ž x , Ž F yG y N . x˙ .
˙X
2
0
) 0.
Ž 7.
Ž 8. ˙
F X y GX y N
˙
K X y 12 Ž DX q K X q N X .
2
˙
F X y GX q N X y 2
˙X
DX y N X y
) 0.
˙
NX 2
0 Ž 9.
X
q Ž x , K X x . s 0. X
DX y N X
y
Et
2
y Ž x , D x . y Ž x˙ , N x˙ . q Ž x ,GX x˙ . 1 2
2
F X y GX
˙
MX s
˙X
1 2
KX
˙X
˙X
q 12 Ž x , Ž D q K . x . q 12 Ž x˙ , N x˙ .
Ž 6.
From Eq. Ž6. we can extract two sufficient conditions for stability.
3. Autonomous systems with negative potential energy We would like to apply the results of the previous section to autonomous systems having negative values of the perturbed potential energy. This is a challenging problem in the presence of circulatory forces since the condition found in w1x does not apply or, more exactly is never verified and the condition of w4x needs very restrictive commutations conditions, which, in general, are not verified either. Situations of that kind occur whenever we look to the stability of moving fluids and plasmas. Eq. Ž1. with time-independent coefficients can be written as x¨ q Ž D q G . x˙ q Ž K q F . x s 0,
Ž 10 .
where the nonprimed operators are now time-independent and the inertial operator has been reduced to
H. Tasso, G.N. Throumoulopoulosr Physics Letters A 271 (2000) 413–418
415
the unit matrix as in w4x. Following w4x we introduce the orthogonal transformation
where Ž z, z . has been added under the square bracket X ˙ and Ž z, DX z˙. s 12 E Ž z , D z . y 12 Ž z, DX z . has been used.
x s LŽ t . z
Note that
Ž 11 . Ž 12 .
L Ž t . s exptT ,
where T is some skew symmetric matrix to be chosen in a suitable way later. Inserting Ž11. and Ž12. in Ž10. and multiplying the resulting equation by Ly1 we obtain z¨ q Ž DX q GX . z˙ q Ž K X q F X . z s 0, X
X
X
Ž 13 .
X
where D , G , F and K are, in general, time-dependent and are given by X
y1
X
y1
X
y1
D sL
DL,
Ž 14 . Ž 15 .
G sL
Ž G q 2T . L, F sL F q 12 Ž DT q TD . q 12 Ž GT y TG . L, Ž 16 . K X s Ly1 Ž K q T 2 q S . L,
Et
˙
DX s Ly1 w D,T x L, ˙X
y1
K sL
Ž 22 . Ž 23 .
w K q S,T x L,
where the square bracket denotes here the commutator. A sufficient condition for stability can be extracted from Eq. Ž21. using Eqs. Ž9., Ž14. and Ž17. i.e. K q S q T 2 q D y I ) 0,
Ž 24 . X
˙
MX s
˙
K X y 12 Ž DX q K X . X
X
F yG 2
X
F yG y 2
X
D yI
0
) 0.
Ž 25 .
Ž 17 .
with
4. A useful corollary
S s 12 Ž DT y TD . q 12 Ž GT q TG . .
Ž 18 .
Since we want to use the results of the previous section, we do not ask as in w4x to annihilate F X and consequently the square bracket in Eq. Ž16.. It was appropriate to do it in w4x because of the special kind of Lyapunov function used. It has also the disadvantage that T will be determined by a matrix equation i.e. by a system of linear equations with n2 unknown, which would be prohibitive for large values of n. Let us now take the scalar product of Eq. Ž13. by z and z˙ as in w1x to obtain
Ž z , z¨ . q Ž z , Ž DX q GX . z˙ . q Ž z , K X z . s 0, Ž 19 . X 1 E Ž z˙ , z˙ . q Ž z , K z . ˙ y 12 Ž z , K X z . q Ž z˙ , DX z˙ .
Et q Ž z˙ , F z . s 0. Adding Eqs. Ž19. and Ž20. we obtain 2
X
Ž 20 .
1 E Ž z , K X z . q Ž z , Ž DX y I . z . q Ž z˙ q z , z˙ q z .
Et
2 X
s y Ž z , K z . y Ž z˙ , Ž DX y I . z˙ . ˙
˙
q 12 Ž z , Ž K X q DX . z . q Ž z˙ , Ž GX y F X . z . ,
Ž 21 .
To satisfy inequality Ž25. we need at least that ˙
˙
K X y 12 Ž DX q K X . ) 0, X
D y I ) 0.
Ž 26 . Ž 27 .
Using Eqs. Ž22. and Ž23. and the orthogonality property of L, conditions Ž26. and Ž27. are equivalent to K q S q T 2 y 12 w D q K q S,T x ) 0, D y I ) 0.
Ž 28 . Ž 29 .
Since a commutator cannot have a definite sign, it is safe in view of Ž24., Ž28. and Ž29. to take K q S q T 2 ) 0, D y I ) 0.
Ž 30 . Ž 31 .
Conditions Ž30. and Ž31. are significant in two respects. First, they imply condition Ž24. and second, they have a numerical advantage i.e. the matrices appearing in them are time independent and have the same character than those involved in Eq. Ž10., in particular an eventual band structure inherited from the discretization of a continuous fluid system. This is not the case of L defined in Eq. Ž12. and consequently of M X defined by relation Ž25.. These remarks are important for numerical applications. Indeed, the search for the lowest eigenvalue
H. Tasso, G.N. Throumoulopoulosr Physics Letters A 271 (2000) 413–418
416
of M X may be prohibitive because it is not sparse, while the matrices appearing in Ž30. and Ž31. are as sparse as those appearing in Eq. Ž10.. One should, however, keep in mind that Ž30. and Ž31. imply Ž24. ˙ ˙ and Ž25. only if DX q K X and F X y GX are small enough. For practical reasons it makes sense to test Ž30. and Ž31. at the first place. This implies already that Ž24. is verified and gives to Ž25. the potentiality to be verified. Note that T is still undetermined in Eq. Ž30., which calls for a judicious and efficient choice of it like the one displayed in the next section.
5. Gyroscopic and dissipative stabilization combined In order to exploit conditions Ž30. and Ž31. we have to make some choice of T so that these conditions become particularly useful. If we take T s y G2 we can first calculate some quantities entering Ž24. and Ž25. i.e. S s y 14 Ž DG y GD . y
G2
,
2
Ž 32 .
GX s 0,
Ž 33 .
X
y1
X
y1
D sL
K sL X
DL,
ž ž
˙
G2
˙
DG y GD y
4
4
DG q GD
DX s yLy1 D, K X s Ly1 K y
Ky
4 G 2
/
/
L,
DG y GD 4
G 2
L,
Ž 39 . Ž 40 .
The following example aims at roughly testing the necessity of the conditions Ž24. and Ž25. for the stability of autonomous systems in connection with Ž10.. The example consists of a particular choice of a 2 = 2 matrix system represented by a specification of x, N, D, G, K, and F as follows x1 , x2
ž /
Ž 41 .
D s dI, G s gi, K s kI and F s fi, where
Ž 37 . ,y
) 0,
6. Nearness to necessity of the sufficient condition
Ž 35 . Ž 36 .
L,
4
This means that even if the potential energy represented here by K has negative values the modified ‘potential energy’ given by Ž39. can be positive. To be sure that the original system is stable we need, however, that there is enough dissipation as demanded by condition Ž40.. Our choice of T illustrates best the gyroscopic stabilization combined with the dissipative one. An optimization of the choice of T for specific applications is left for future work.
xs L,
G2
D y I ) 0.
Ž 34 .
Ky
F s Ly1 F y
then conditions Ž30. and Ž31. can be verified provided
Ž 38 .
where the square bracket w A, B x denotes again the commutator of A and B. Note that yG 2 is a positive operator since G is a skew symmetric matrix describing the gyroscopic force. This allows us to say that K y G 2 4 can be positive even if K itself has negative values. We can then conclude that if the commutators in Eqs. Ž37. and Ž38. are small enough and if F X is small enough,
Is
ž
1 0
0 , 1
/
is
ž
0 1
y1 . 0
/
Ž 42 .
In analogy with the physical problem d is positive. Also, as in Section 5 we take for the transformation matrix T s yGr2 for which GX s 0. It then follows
w D,G x s 0,
Ž 43 .
L s e tT s I q tT q s cos
gt
Ž tT . 2! gt
2
q...
ž / ž / ž / ž / 2
Ly1 s cos
I y sin
gt 2
2
I q sin
i,
gt 2
Ž 44 . i,
Ž 45 .
H. Tasso, G.N. Throumoulopoulosr Physics Letters A 271 (2000) 413–418
G2
K X s Ly1 K y
F X s Ly1 F y
ž
Ls kq
4
DG q GD
g2 4
/
I,
ž
Ls fy
4
Ž 46 . dg 2
/
i
Ž 47 .
and
X
M s
ž
kq
2
g
/
4
y 12 f y
ž
1 2
I
dg
/
2
i
ž
fy
dg 2
/
Ž d y 1. I
i
0
.
Ž 48 .
4
)0
Ž 49 .
and d y 1 ) 0. The matrix M X has two double eigenvalues l1 , l2 which satisfy the relations
ž
l1 q l2 s k q
g2 4
/
q Ž d y 1.
Ž 50 .
and
l1 l2 s y
1 16
2
2
2
d g y 4 d Ž g q fg q 4 k .
q4 Ž f 2 q g 2 q 4 k .
Ž 51 .
In order that l1 ) 0 and l2 ) 0, which guarantees M X ) 0, the expression under the square brackets in Ž51. must be negative. Introducing the parameter e s f y dgr2 this requirement yields d)1q
1
e2
4 k qg 2 4
.
Ž 52 .
Relations Ž49. and Ž52. are the sufficient conditions for stability. We now make the ansatz x j s c j exp v t Ž j s 1,2. in Ž10. to obtain the characteristic polynomial
v 4 q 2 d v 3 q Ž 2 k q d 2 q g 2 . v 2 q 2 Ž dk q gf . v q k 2 q f 2 s 0.
d)
ž
e2 k qg 2 4
1r2
/
.
Ž 54 .
The sufficient conditions are ‘nearer’ to the necessary ones the lower than unity the difference DŽ e . between the right-hand sides of Ž52. and Ž54.. The relation DŽ e . F 1 is satisfied for Žy2 4 k q g 2 F e F 2 4 k q g 2 .. In particular, for the class of values of d, g, k and f such that e s " 4 k q g 2 , at which DŽ e . s 0, conditions Ž49. and Ž52. are necessary and sufficient. For plasmas of low density, e.g. magnetic-confinement plasmas, a more appropriate possible analogy for the matrix of inertia is N s d I, where d - 1, and therefore the sufficient conditions may be even nearer to the necessary ones. Note that for this case in the context of the above example the nearness requirement DŽ e . - 1 weakens to DŽ e . - d .
(
(
Note that M becomes time independent. The prerequisite conditions Ž30. and Ž31. are satisfied provided g2
Applying the Routh–Hurwitz criterion w5,6x to Ž53. we find that the necessary and sufficient conditions are Ž49. and
(
X
kq
417
Ž 53 .
7. Conclusions It has been shown that a sufficient condition for stability of dissipative mechanical systems with circulatory forces, previously derived for autonomous systems in w1x, can be extended to nonautonomous systems with slight modifications. Applied to autonomous systems transformed by time-dependent orthogonal matrices into apparently nonautonomous systems, the condition allows to make stability statements for cases with negative perturbed potential energy. This was always a challenging problem and occurs, in particular, for fluids and plasmas in motion. The commutation assumed in w4x to obtain necessary and sufficient stability conditions is not needed for the derivation of the sufficient condition in this Letter. Small enough commutators Žsee sentence before Eq. Ž39.. are, however, assumed in order to obtain the sufficient conditions Ž39. and Ž40., the full conditions being given by Ž24. and Ž25.. The conditions obtained by Lyapunov functions as Hermitian forms are of great advantage for appli-
418
H. Tasso, G.N. Throumoulopoulosr Physics Letters A 271 (2000) 413–418
cations. They permit to avoid the investigation of a non-Hermitian eigenvalue problem, which is very hard to do analytically as well as numerically. The conditions obtained by the Lyapunov method are, however, usually only sufficient, so that part of the answer is missing. A comparison of the results of this Letter with the full eigenvalue problem, which can be carried through analytically for the case of 2 = 2 matrices shows that the sufficient condition obtained in this Letter can be, for some parameter values, very near to necessity. Whether this efficiency will deteriorate for larger systems, e.g. for discretized plasma and fluid flows, remains to be seen. This problem, however, cannot
be cleared before applications to real situations are undertaken. This is now a possible task.
References w1x w2x w3x w4x w5x
H. Tasso, Phys. Lett. A 257 Ž1999. 309. P.C. Muller, Stabilitat ¨ ¨ und Matrizen, Springer, Berlin, 1977. H. Tasso, Z. Naturforsch. 33a, 1978, p. 257. H. Tasso, ZAMP 31 Ž1980. 536. F.R. Gantmacher, Matrizenrechnung II, VEB Deutscher Verlag der Wissenschaften, Berlin, 1959. w6x C.T. Chen, Introduction to Linear System Theory, HRW Series in Electrical Engineering, Electronics, and Systems, New York, 1970.
Physics Letters A 271 Ž2000. 413–418 www.elsevier.nlrlocaterpla
On Lyapunov stability of nonautonomous mechanical systems H. Tasso ) , G.N. Throumoulopoulos 1 Max-Planck-Institut fur Germany ¨ Plasmaphysik, Euratom Association, 85748 Garching bei Munchen, ¨ Received 22 February 2000; received in revised form 10 May 2000; accepted 5 June 2000 Communicated by M. Porkolab
Abstract A sufficient condition for the linear stability of nonautonomous dissipative mechanical systems with circulatory forces is derived. It is applied to autonomous systems transformed to nonautonomous ones by a time-dependent orthogonal transformation. This allows to obtain sufficient stability conditions in case the perturbed potential energy can have negative values, which was not possible up to now. This is, in particular, of importance for plasma equilibria with sheared flows. The nearness of the sufficient condition to necessity is discussed for a particular example. q 2000 Elsevier Science B.V. All rights reserved.
1. Introduction In a recent contribution by one of the authors w1x, a sufficient stability condition was obtained for dissipative and autonomous mechanical systems having circulatory forces. This was made possible by the use of a quadratic Lyapunov function containing the perturbed energy as well as the perturbed virial. We show here that a slightly modified Lyapunov function leads to a similar sufficient condition for nonautonomous systems. By applying this condition to autonomous systems transformed to nonautonomous ones by time-dependent orthogonal or unitary operators or matrices, one is able to obtain significant results for cases for
)
Corresponding author. Fax: q49 89 3299 1181. E-mail address: [email protected] ŽH. Tasso.. 1 Permanent address: University of Ioannina, Association Euratom – Hellenic Republic, Physics Department, Section of Theoretical Physics, GR 451 10 Ioannina, Greece.
which the perturbed potential energy can be negative. This leads to stability statements describing gyroscopic stabilization, combined with a dissipative one, in a quantitative way. If applied to discretized fluid systems, this result could be significant for answering questions about the stability of hydrodynamic and magnetohydrodynamic sheared flows. In Section 2 the generalized sufficient condition is derived. Its application to autonomous systems with negative potential energy is the topic of Sections 3, 4 and 5. In Section 6 the sufficient condition is compared to the necessary and sufficient one for a special case. Finally, the conclusions are summarized in Section 7.
2. The generalized sufficient condition Let us consider linearized mechanical systems with time-dependent coefficients generalizing those
0375-9601r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 0 0 . 0 0 3 8 6 - 8
H. Tasso, G.N. Throumoulopoulosr Physics Letters A 271 (2000) 413–418
414
given by Eq. Ž1. of w1x whose equation of motion can be written in the following compact form: N X x¨ q Ž DX q GX . x˙ q Ž K X q F X . x s 0,
Ž 1.
˙
˙
˙
˙
˙
First, if DX s K X s N X s 0 but F X / 0 and GX / 0 then the stability condition is equivalent to the one given by Eq. Ž7. of w1x i.e. F X y GX
where x is an n-dimensional displacement vector, N X is a positive symmetric matrix related to the inertia of the system, DX is a symmetric positive definite matrix describing damping effects, K X is the symmetric matrix due to potential forces, and GX and F X are skew symmetric matrices related to gyroscopic and circulatory forces respectively. Derivation of Eq. Ž1. in the autonomous case can be found in e.g. w2x for discrete systems and w3x for fluids and continuous systems. Note that for a real system all quantities appearing in Eq. Ž1. are real valued. The purpose of this contribution is to find stability conditions for the equilibrium position of Eq. Ž1. given by
The minimization of the Hermitian form associated with M X can be done using well known eigenvalues algorithms, which determine solely the sign of the lowest eigenvalue of M X . A positive sign implies also the positivity of the expression under the square brackets in Eq. Ž6., which means that this expression is then a Lyapunov function. Second, in the general case, the stability conditions are given by
x s x˙ s 0.
K X q DX y N X ) 0
Ž 2.
MX s
Let us take the scalar product of Eq. Ž1. with x˙ and x, respectively, to obtain 1 E Ž x˙ , N X x˙ .
Et
2
q
1 E Ž x,KXx.
Et
2 ˙X
y Ž x˙ , N x˙ .
X
X
y 12 Ž x , K x . q Ž x˙ , D x˙ . q Ž x˙ , F x . s 0, X
Ž 3.
X
E Ž x , N x˙ .
˙X
y Ž x , N x˙ . q
Et
˙X
1 E Ž x,D x.
Ž 4.
Adding Eqs. Ž3. and Ž4., we obtain X
X
X
1 E w Ž x˙ , N x˙ . q Ž x , K x . q Ž x , D x . q2 Ž x , N x˙ . x
Et
2 X
X X X X sy Ž x , K x . y Ž x˙ , Ž D y N . x˙ . q Ž x , Ž F yG . x˙ .
˙X
˙X
˙X
˙X
q Ž x , N x˙ . q 12 Ž x˙ , N x˙ . q 12 Ž x , D x . q 12 Ž x , K x . .
Ž 5.
X
By adding and subtracting Ž x, N x . to the expression under the square brackets, Eq. Ž5. can be written in the following form X X X X 1 E w Ž x , K x . q Ž x , Ž D y N . x . q ŽŽ x q x˙ . , N Ž x q x˙ .. x
Et
2 X
˙
X
X X X X sy Ž x , K x . y Ž x˙ , Ž D y N . x˙ . q Ž x , Ž F yG y N . x˙ .
˙X
2
0
) 0.
Ž 7.
Ž 8. ˙
F X y GX y N
˙
K X y 12 Ž DX q K X q N X .
2
˙
F X y GX q N X y 2
˙X
DX y N X y
) 0.
˙
NX 2
0 Ž 9.
X
q Ž x , K X x . s 0. X
DX y N X
y
Et
2
y Ž x , D x . y Ž x˙ , N x˙ . q Ž x ,GX x˙ . 1 2
2
F X y GX
˙
MX s
˙X
1 2
KX
˙X
˙X
q 12 Ž x , Ž D q K . x . q 12 Ž x˙ , N x˙ .
Ž 6.
From Eq. Ž6. we can extract two sufficient conditions for stability.
3. Autonomous systems with negative potential energy We would like to apply the results of the previous section to autonomous systems having negative values of the perturbed potential energy. This is a challenging problem in the presence of circulatory forces since the condition found in w1x does not apply or, more exactly is never verified and the condition of w4x needs very restrictive commutations conditions, which, in general, are not verified either. Situations of that kind occur whenever we look to the stability of moving fluids and plasmas. Eq. Ž1. with time-independent coefficients can be written as x¨ q Ž D q G . x˙ q Ž K q F . x s 0,
Ž 10 .
where the nonprimed operators are now time-independent and the inertial operator has been reduced to
H. Tasso, G.N. Throumoulopoulosr Physics Letters A 271 (2000) 413–418
415
the unit matrix as in w4x. Following w4x we introduce the orthogonal transformation
where Ž z, z . has been added under the square bracket X ˙ and Ž z, DX z˙. s 12 E Ž z , D z . y 12 Ž z, DX z . has been used.
x s LŽ t . z
Note that
Ž 11 . Ž 12 .
L Ž t . s exptT ,
where T is some skew symmetric matrix to be chosen in a suitable way later. Inserting Ž11. and Ž12. in Ž10. and multiplying the resulting equation by Ly1 we obtain z¨ q Ž DX q GX . z˙ q Ž K X q F X . z s 0, X
X
X
Ž 13 .
X
where D , G , F and K are, in general, time-dependent and are given by X
y1
X
y1
X
y1
D sL
DL,
Ž 14 . Ž 15 .
G sL
Ž G q 2T . L, F sL F q 12 Ž DT q TD . q 12 Ž GT y TG . L, Ž 16 . K X s Ly1 Ž K q T 2 q S . L,
Et
˙
DX s Ly1 w D,T x L, ˙X
y1
K sL
Ž 22 . Ž 23 .
w K q S,T x L,
where the square bracket denotes here the commutator. A sufficient condition for stability can be extracted from Eq. Ž21. using Eqs. Ž9., Ž14. and Ž17. i.e. K q S q T 2 q D y I ) 0,
Ž 24 . X
˙
MX s
˙
K X y 12 Ž DX q K X . X
X
F yG 2
X
F yG y 2
X
D yI
0
) 0.
Ž 25 .
Ž 17 .
with
4. A useful corollary
S s 12 Ž DT y TD . q 12 Ž GT q TG . .
Ž 18 .
Since we want to use the results of the previous section, we do not ask as in w4x to annihilate F X and consequently the square bracket in Eq. Ž16.. It was appropriate to do it in w4x because of the special kind of Lyapunov function used. It has also the disadvantage that T will be determined by a matrix equation i.e. by a system of linear equations with n2 unknown, which would be prohibitive for large values of n. Let us now take the scalar product of Eq. Ž13. by z and z˙ as in w1x to obtain
Ž z , z¨ . q Ž z , Ž DX q GX . z˙ . q Ž z , K X z . s 0, Ž 19 . X 1 E Ž z˙ , z˙ . q Ž z , K z . ˙ y 12 Ž z , K X z . q Ž z˙ , DX z˙ .
Et q Ž z˙ , F z . s 0. Adding Eqs. Ž19. and Ž20. we obtain 2
X
Ž 20 .
1 E Ž z , K X z . q Ž z , Ž DX y I . z . q Ž z˙ q z , z˙ q z .
Et
2 X
s y Ž z , K z . y Ž z˙ , Ž DX y I . z˙ . ˙
˙
q 12 Ž z , Ž K X q DX . z . q Ž z˙ , Ž GX y F X . z . ,
Ž 21 .
To satisfy inequality Ž25. we need at least that ˙
˙
K X y 12 Ž DX q K X . ) 0, X
D y I ) 0.
Ž 26 . Ž 27 .
Using Eqs. Ž22. and Ž23. and the orthogonality property of L, conditions Ž26. and Ž27. are equivalent to K q S q T 2 y 12 w D q K q S,T x ) 0, D y I ) 0.
Ž 28 . Ž 29 .
Since a commutator cannot have a definite sign, it is safe in view of Ž24., Ž28. and Ž29. to take K q S q T 2 ) 0, D y I ) 0.
Ž 30 . Ž 31 .
Conditions Ž30. and Ž31. are significant in two respects. First, they imply condition Ž24. and second, they have a numerical advantage i.e. the matrices appearing in them are time independent and have the same character than those involved in Eq. Ž10., in particular an eventual band structure inherited from the discretization of a continuous fluid system. This is not the case of L defined in Eq. Ž12. and consequently of M X defined by relation Ž25.. These remarks are important for numerical applications. Indeed, the search for the lowest eigenvalue
H. Tasso, G.N. Throumoulopoulosr Physics Letters A 271 (2000) 413–418
416
of M X may be prohibitive because it is not sparse, while the matrices appearing in Ž30. and Ž31. are as sparse as those appearing in Eq. Ž10.. One should, however, keep in mind that Ž30. and Ž31. imply Ž24. ˙ ˙ and Ž25. only if DX q K X and F X y GX are small enough. For practical reasons it makes sense to test Ž30. and Ž31. at the first place. This implies already that Ž24. is verified and gives to Ž25. the potentiality to be verified. Note that T is still undetermined in Eq. Ž30., which calls for a judicious and efficient choice of it like the one displayed in the next section.
5. Gyroscopic and dissipative stabilization combined In order to exploit conditions Ž30. and Ž31. we have to make some choice of T so that these conditions become particularly useful. If we take T s y G2 we can first calculate some quantities entering Ž24. and Ž25. i.e. S s y 14 Ž DG y GD . y
G2
,
2
Ž 32 .
GX s 0,
Ž 33 .
X
y1
X
y1
D sL
K sL X
DL,
ž ž
˙
G2
˙
DG y GD y
4
4
DG q GD
DX s yLy1 D, K X s Ly1 K y
Ky
4 G 2
/
/
L,
DG y GD 4
G 2
L,
Ž 39 . Ž 40 .
The following example aims at roughly testing the necessity of the conditions Ž24. and Ž25. for the stability of autonomous systems in connection with Ž10.. The example consists of a particular choice of a 2 = 2 matrix system represented by a specification of x, N, D, G, K, and F as follows x1 , x2
ž /
Ž 41 .
D s dI, G s gi, K s kI and F s fi, where
Ž 37 . ,y
) 0,
6. Nearness to necessity of the sufficient condition
Ž 35 . Ž 36 .
L,
4
This means that even if the potential energy represented here by K has negative values the modified ‘potential energy’ given by Ž39. can be positive. To be sure that the original system is stable we need, however, that there is enough dissipation as demanded by condition Ž40.. Our choice of T illustrates best the gyroscopic stabilization combined with the dissipative one. An optimization of the choice of T for specific applications is left for future work.
xs L,
G2
D y I ) 0.
Ž 34 .
Ky
F s Ly1 F y
then conditions Ž30. and Ž31. can be verified provided
Ž 38 .
where the square bracket w A, B x denotes again the commutator of A and B. Note that yG 2 is a positive operator since G is a skew symmetric matrix describing the gyroscopic force. This allows us to say that K y G 2 4 can be positive even if K itself has negative values. We can then conclude that if the commutators in Eqs. Ž37. and Ž38. are small enough and if F X is small enough,
Is
ž
1 0
0 , 1
/
is
ž
0 1
y1 . 0
/
Ž 42 .
In analogy with the physical problem d is positive. Also, as in Section 5 we take for the transformation matrix T s yGr2 for which GX s 0. It then follows
w D,G x s 0,
Ž 43 .
L s e tT s I q tT q s cos
gt
Ž tT . 2! gt
2
q...
ž / ž / ž / ž / 2
Ly1 s cos
I y sin
gt 2
2
I q sin
i,
gt 2
Ž 44 . i,
Ž 45 .
H. Tasso, G.N. Throumoulopoulosr Physics Letters A 271 (2000) 413–418
G2
K X s Ly1 K y
F X s Ly1 F y
ž
Ls kq
4
DG q GD
g2 4
/
I,
ž
Ls fy
4
Ž 46 . dg 2
/
i
Ž 47 .
and
X
M s
ž
kq
2
g
/
4
y 12 f y
ž
1 2
I
dg
/
2
i
ž
fy
dg 2
/
Ž d y 1. I
i
0
.
Ž 48 .
4
)0
Ž 49 .
and d y 1 ) 0. The matrix M X has two double eigenvalues l1 , l2 which satisfy the relations
ž
l1 q l2 s k q
g2 4
/
q Ž d y 1.
Ž 50 .
and
l1 l2 s y
1 16
2
2
2
d g y 4 d Ž g q fg q 4 k .
q4 Ž f 2 q g 2 q 4 k .
Ž 51 .
In order that l1 ) 0 and l2 ) 0, which guarantees M X ) 0, the expression under the square brackets in Ž51. must be negative. Introducing the parameter e s f y dgr2 this requirement yields d)1q
1
e2
4 k qg 2 4
.
Ž 52 .
Relations Ž49. and Ž52. are the sufficient conditions for stability. We now make the ansatz x j s c j exp v t Ž j s 1,2. in Ž10. to obtain the characteristic polynomial
v 4 q 2 d v 3 q Ž 2 k q d 2 q g 2 . v 2 q 2 Ž dk q gf . v q k 2 q f 2 s 0.
d)
ž
e2 k qg 2 4
1r2
/
.
Ž 54 .
The sufficient conditions are ‘nearer’ to the necessary ones the lower than unity the difference DŽ e . between the right-hand sides of Ž52. and Ž54.. The relation DŽ e . F 1 is satisfied for Žy2 4 k q g 2 F e F 2 4 k q g 2 .. In particular, for the class of values of d, g, k and f such that e s " 4 k q g 2 , at which DŽ e . s 0, conditions Ž49. and Ž52. are necessary and sufficient. For plasmas of low density, e.g. magnetic-confinement plasmas, a more appropriate possible analogy for the matrix of inertia is N s d I, where d - 1, and therefore the sufficient conditions may be even nearer to the necessary ones. Note that for this case in the context of the above example the nearness requirement DŽ e . - 1 weakens to DŽ e . - d .
(
(
Note that M becomes time independent. The prerequisite conditions Ž30. and Ž31. are satisfied provided g2
Applying the Routh–Hurwitz criterion w5,6x to Ž53. we find that the necessary and sufficient conditions are Ž49. and
(
X
kq
417
Ž 53 .
7. Conclusions It has been shown that a sufficient condition for stability of dissipative mechanical systems with circulatory forces, previously derived for autonomous systems in w1x, can be extended to nonautonomous systems with slight modifications. Applied to autonomous systems transformed by time-dependent orthogonal matrices into apparently nonautonomous systems, the condition allows to make stability statements for cases with negative perturbed potential energy. This was always a challenging problem and occurs, in particular, for fluids and plasmas in motion. The commutation assumed in w4x to obtain necessary and sufficient stability conditions is not needed for the derivation of the sufficient condition in this Letter. Small enough commutators Žsee sentence before Eq. Ž39.. are, however, assumed in order to obtain the sufficient conditions Ž39. and Ž40., the full conditions being given by Ž24. and Ž25.. The conditions obtained by Lyapunov functions as Hermitian forms are of great advantage for appli-
418
H. Tasso, G.N. Throumoulopoulosr Physics Letters A 271 (2000) 413–418
cations. They permit to avoid the investigation of a non-Hermitian eigenvalue problem, which is very hard to do analytically as well as numerically. The conditions obtained by the Lyapunov method are, however, usually only sufficient, so that part of the answer is missing. A comparison of the results of this Letter with the full eigenvalue problem, which can be carried through analytically for the case of 2 = 2 matrices shows that the sufficient condition obtained in this Letter can be, for some parameter values, very near to necessity. Whether this efficiency will deteriorate for larger systems, e.g. for discretized plasma and fluid flows, remains to be seen. This problem, however, cannot
be cleared before applications to real situations are undertaken. This is now a possible task.
References w1x w2x w3x w4x w5x
H. Tasso, Phys. Lett. A 257 Ž1999. 309. P.C. Muller, Stabilitat ¨ ¨ und Matrizen, Springer, Berlin, 1977. H. Tasso, Z. Naturforsch. 33a, 1978, p. 257. H. Tasso, ZAMP 31 Ž1980. 536. F.R. Gantmacher, Matrizenrechnung II, VEB Deutscher Verlag der Wissenschaften, Berlin, 1959. w6x C.T. Chen, Introduction to Linear System Theory, HRW Series in Electrical Engineering, Electronics, and Systems, New York, 1970.