Stability for a part of the variables

Lyapunov stability of the solution relative not to all but only to certain variables , , of a system of ordinary differential equations

(1)

Here are given real-valued continuous functions, satisfying in the domain

(2)

the conditions for the existence and uniqueness of the solution ; moreover,

and any solution is defined for all for which .

Put for ; for , and ; let

The solution of the system (1) is called: a) stable relative to or -stable if

i.e. for any given numbers () and one can find a number such that for every perturbation satisfying the condition and for every the solution satisfies the condition ;

b) -unstable in the opposite case, i.e. if

c) -stable uniformly in if in definition a) for every the number may be chosen independently of ;

d) asymptotically -stable if it is -stable and if for every there exists a such that

Here , is the maximal right interval on which is defined, ; in case d), besides the conditions stated above it is assumed that all solutions of the system (1) exist on .

The statement of the problem of stability for a part of the variables was given by A.M. Lyapunov [1] as a generalization of the stability problem with respect to all variables . For a solution of this problem it is particularly effective to apply the method of Lyapunov functions, suitably modified (cf. [2], and Lyapunov function) for the problem of -stability. At the basis of this method there are a number of theorems generalizing the classical theorem of Lyapunov.

Consider a real-valued function , , and at the same time its total derivative with respect to time, using (1):

A function of constant sign is called -sign-definite if there exists a positive-definite function such that in the region (2),

A bounded function is said to admit an infinitesimal upper bound for if for every there exists a such that

for , , .

Theorem 1.

If the system (1) is such that there exists a -positive-definite function with derivative , then the solution is -stable.

Theorem 2.

If the conditions of theorem 1 are fulfilled and if, moreover, admits an infinitesimal upper bound for , then the solution of the system (1) is -stable uniformly in .

Theorem 3.

If the conditions of theorem 1 are fulfilled and if, moreover, admits an infinitesimal upper bound for , then for any one can find a such that , , implies the inequality

Theorem 4.

If the system (1) is such that there exists a -positive-definite function admitting an infinitesimal upper bound for () and with negative-definite derivative for , then the solution of the system (1) is asymptotically -stable.

For the study of -instability, Chetaev's instability theorem (cf. Chetaev function) has been successfully applied, as well as certain other theorems. Conditions for the converse of a number of theorems on -stability have been established; for example, the converses of theorems 1, 2 as well as of theorem 4 for . Methods of differential inequalities and Lyapunov vector functions have been applied to establish theorems on asymptotic -stability in the large, on first-order approximations, etc. (cf. [3], ).

References

[1]	A.M. Lyapunov, Mat. Sb. , 17 : 2 (1893) pp. 253–333
[2]	V.V. Rumyantsev, "On stability of motion for a part of the variables" Vestn. Moskov. Univ. Ser. Mat. Mekh. Astron. Fiz. Khim. : 4 (1957) pp. 9–16 (In Russian)
[3]	A.S. Oziraner, V.V. Rumyantsev, "The method of Lyapunov functions in the stability problem for motion with respect to a part of the variables" J. Appl. Math. Mech. , 36 (1972) pp. 341–362 Prikl. Mat. i Mekh. , 36 : 2 (1972) pp. 364–384

Comments

Stability for a part of the variables is also called partial stability and occasionally conditional stability, [a1]. However, the latter phrase is also used in a different meaning: Let be a class of trajectories, a trajectory in . This trajectory is stable relative to if for a given there exists a such that for each trajectory one has that implies . If is not the class of all trajectories, such a is called conditionally stable, [a2].

References

[a1]	W. Hahn, "Stability of motion" , Springer (1965) pp. §55
[a2]	S. Lefshetz, "Differential equations: geometric theory" , Dover, reprint (1977) pp. 78, 83