# 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

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].