Stability for a part of the variables

Lyapunov stability of the solution $ x = 0 $ relative not to all but only to certain variables $ x _ {1} \dots x _ {k} $, $ k < n $, of a system of ordinary differential equations

$$ \tag{1 } \dot{x} _ {s} = X _ {s} ( t, x _ {1} \dots x _ {n} ),\ \ s = 1 \dots n. $$

Here $ X _ {s} ( t, x) $ are given real-valued continuous functions, satisfying in the domain

$$ \tag{2 } t \geq 0,\ \ \sum _ {i = 1 } ^ { k } x _ {i} ^ {2} \leq \textrm{ const } ,\ \ \sum _ {j = k + 1 } ^ { n } x _ {j} ^ {2} < \infty $$

the conditions for the existence and uniqueness of the solution $ x ( t; t _ {0} , x _ {0} ) $; moreover,

$$ X _ {s} ( t, 0) \equiv 0,\ s = 1 \dots n, $$

and any solution is defined for all $ t \geq t _ {0} \geq 0 $ for which $ \sum _ {i = 1 } ^ {k} x _ {i} ^ {2} \leq H $.

Put $ x _ {i} = y _ {i} $ for $ i = 1 \dots k $; $ x _ {k + j } = z _ {j} $ for $ j = 1 \dots m $, $ n = k + m $ and $ m \geq 0 $; let

$$ \| y \| = \ \left ( \sum _ {i = 1 } ^ { k } y _ {i} ^ {2} \right ) ^ {1/2} ,\ \ \| z \| = \ \left ( \sum _ {j = 1 } ^ { m } z _ {j} ^ {2} \right ) ^ {1/2} , $$

$$ \| x \| = \left ( \sum _ {s = 1 } ^ { n } x _ {s} ^ {2} \right ) ^ {1/2} . $$

The solution $ x = 0 $ of the system (1) is called: a) stable relative to $ x _ {1} \dots x _ {k} $ or $ y $- stable if

$$ ( \forall \epsilon > 0) ( \forall t _ {0} \in I) ( \exists \delta > 0) ( \forall x _ {0} \in B _ \delta ) ( \forall t \in J ^ {+} ): $$

$$ \| y ( t; t _ {0} , x _ {0} ) \| < \epsilon , $$

i.e. for any given numbers $ \epsilon > 0 $( $ \epsilon < H $) and $ t _ {0} \geq 0 $ one can find a number $ \delta ( \epsilon , t _ {0} ) > 0 $ such that for every perturbation $ x _ {0} $ satisfying the condition $ \| x _ {0} \| \leq \delta $ and for every $ t > t _ {0} $ the solution $ y ( t; t _ {0} , x _ {0} ) $ satisfies the condition $ \| y \| < \epsilon $;

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

$$ ( \exists \epsilon > 0) ( \exists t _ {0} \in I) ( \forall \delta > 0) ( \exists x _ {0} \in B _ \delta ) ( \exists t \in J ^ {+} ): $$

$$ \| y ( t; t _ {0} , x _ {0} ) \| \geq \epsilon ; $$

c) $ y $- stable uniformly in $ t _ {0} $ if in definition a) for every $ \epsilon > 0 $ the number $ \delta ( \epsilon ) $ may be chosen independently of $ t _ {0} $;

d) asymptotically $ y $- stable if it is $ y $- stable and if for every $ t _ {0} \geq 0 $ there exists a $ \delta _ {1} ( t _ {0} ) > 0 $ such that

$$ \lim\limits _ {t \rightarrow \infty } \| y ( t; t _ {0} , x _ {0} ) \| = 0 \ \ \textrm{ for } \| x _ {0} \| \leq \delta _ {1} . $$

Here $ I = [ 0, \infty ) $, $ J ^ {+} $ is the maximal right interval on which $ x ( t; t _ {0} , x _ {0} ) $ is defined, $ B _ \delta = \{ {x \in \mathbf R ^ {n} } : {\| x \| < \delta } \} $; in case d), besides the conditions stated above it is assumed that all solutions of the system (1) exist on $ [ t _ {0} , \infty ) $.

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 $ ( k = n) $. 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 $ y $- stability. At the basis of this method there are a number of theorems generalizing the classical theorem of Lyapunov.

Consider a real-valued function $ V ( t, x) \in C ^ {1} $, $ V ( t, 0) = 0 $, and at the same time its total derivative with respect to time, using (1):

$$ \dot{V} = \ \frac{\partial V }{\partial t } + \sum _ {s = 1 } ^ { n } \frac{\partial V }{\partial x _ {s} } X _ {s} . $$

A function $ V ( t, x) $ of constant sign is called $ y $- sign-definite if there exists a positive-definite function $ W ( y) $ such that in the region (2),

$$ V ( t, x) \geq W ( y) \ \ \textrm{ or } \ \ - V ( t, x) \geq W ( y). $$

A bounded function $ V ( t, x) $ is said to admit an infinitesimal upper bound for $ x _ {1} \dots x _ {p} $ if for every $ l > 0 $ there exists a $ \lambda ( l) $ such that

$$ | V ( t, x) | < l $$

for $ t \geq 0 $, $ \sum _ {i = 1 } ^ {p} x _ {i} ^ {2} < \lambda $, $ - \infty < x _ {p + 1 } \dots x _ {n} < \infty $.

Theorem 1.

If the system (1) is such that there exists a $ y $- positive-definite function $ V ( t, x) $ with derivative $ \dot{V} \leq 0 $, then the solution $ x = 0 $ is $ y $- stable.

Theorem 2.

If the conditions of theorem 1 are fulfilled and if, moreover, $ V $ admits an infinitesimal upper bound for $ x $, then the solution $ x = 0 $ of the system (1) is $ y $- stable uniformly in $ t _ {0} $.

Theorem 3.

If the conditions of theorem 1 are fulfilled and if, moreover, $ V $ admits an infinitesimal upper bound for $ y $, then for any $ \epsilon > 0 $ one can find a $ \delta _ {2} ( \epsilon ) > 0 $ such that $ t _ {0} \geq 0 $, $ \| y _ {0} \| \leq \delta _ {2} $, $ 0 \leq \| z _ {0} \| < \infty $ implies the inequality

$$ \| y ( t; t _ {0} , x _ {0} ) \| < \ \epsilon \ \textrm{ for } \textrm{ all } t \geq t _ {0} . $$

Theorem 4.

If the system (1) is such that there exists a $ y $- positive-definite function $ V $ admitting an infinitesimal upper bound for $ x _ {1} \dots x _ {p} $( $ k \leq p \leq n $) and with negative-definite derivative $ \dot{V} $ for $ x _ {1} \dots x _ {p} $, then the solution $ x = 0 $ of the system (1) is asymptotically $ y $- stable.

For the study of $ y $- 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 $ y $- stability have been established; for example, the converses of theorems 1, 2 as well as of theorem 4 for $ p = k $. Methods of differential inequalities and Lyapunov vector functions have been applied to establish theorems on asymptotic $ y $- 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 $ C $ be a class of trajectories, $ x ( t ; t _ {0} , x _ {0} ) $ a trajectory in $ C $. This trajectory is stable relative to $ C $ if for a given $ \epsilon > 0 $ there exists a $ \delta > 0 $ such that for each trajectory $ \widetilde{x} ( t ; t _ {0} , \widetilde{x} {} _ {0} ) $ one has that $ \| x _ {0} - \widetilde{x} {} _ {0} \| \leq \delta $ implies $ \| x( t ; t _ {0} , x _ {0} ) - \widetilde{x} ( t ; t _ {0} , \widetilde{x} {} _ {0} ) \| \leq \epsilon $. If $ C $ is not the class of all trajectories, such a $ x ( t; t _ {0} , x _ {0} ) $ 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