View source for Stability for a part of the variables

[[Lyapunov stability|Lyapunov stability]] of the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086960/s0869601.png" /> relative not to all but only to certain variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086960/s0869602.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086960/s0869603.png" />, of a system of ordinary differential equations

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Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086960/s0869605.png" /> are given real-valued continuous functions, satisfying in the domain

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086960/s0869606.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

the conditions for the existence and uniqueness of the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086960/s0869607.png" />; moreover,

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and any solution is defined for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086960/s0869609.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086960/s08696010.png" />.

Put <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086960/s08696011.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086960/s08696012.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086960/s08696013.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086960/s08696014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086960/s08696015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086960/s08696016.png" />; let

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The solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086960/s08696019.png" /> of the system (1) is called: a) stable relative to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086960/s08696020.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086960/s08696022.png" />-stable if

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i.e. for any given numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086960/s08696025.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086960/s08696026.png" />) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086960/s08696027.png" /> one can find a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086960/s08696028.png" /> such that for every perturbation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086960/s08696029.png" /> satisfying the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086960/s08696030.png" /> and for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086960/s08696031.png" /> the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086960/s08696032.png" /> satisfies the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086960/s08696033.png" />;

b) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086960/s08696035.png" />-unstable in the opposite case, i.e. if

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c) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086960/s08696040.png" />-stable uniformly in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086960/s08696041.png" /> if in definition a) for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086960/s08696042.png" /> the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086960/s08696043.png" /> may be chosen independently of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086960/s08696044.png" />;

d) asymptotically <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086960/s08696046.png" />-stable if it is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086960/s08696047.png" />-stable and if for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086960/s08696048.png" /> there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086960/s08696049.png" /> such that

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Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086960/s08696051.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086960/s08696052.png" /> is the maximal right interval on which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086960/s08696053.png" /> is defined, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086960/s08696054.png" />; in case d), besides the conditions stated above it is assumed that all solutions of the system (1) exist on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086960/s08696055.png" />.

The statement of the problem of stability for a part of the variables was given by A.M. Lyapunov [[#References|[1]]] as a generalization of the stability problem with respect to all variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086960/s08696056.png" />. For a solution of this problem it is particularly effective to apply the method of Lyapunov functions, suitably modified (cf. [[#References|[2]]], and [[Lyapunov function|Lyapunov function]]) for the problem of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086960/s08696057.png" />-stability. At the basis of this method there are a number of theorems generalizing the classical theorem of Lyapunov.

Consider a real-valued function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086960/s08696058.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086960/s08696059.png" />, and at the same time its total derivative with respect to time, using (1):

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A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086960/s08696061.png" /> of constant sign is called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086960/s08696063.png" />-sign-definite if there exists a positive-definite function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086960/s08696064.png" /> such that in the region (2),

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A bounded function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086960/s08696066.png" /> is said to admit an infinitesimal upper bound for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086960/s08696067.png" /> if for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086960/s08696068.png" /> there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086960/s08696069.png" /> such that

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for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086960/s08696071.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086960/s08696072.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086960/s08696073.png" />.

===Theorem 1.===
If the system (1) is such that there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086960/s08696074.png" />-positive-definite function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086960/s08696075.png" /> with derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086960/s08696076.png" />, then the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086960/s08696077.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086960/s08696078.png" />-stable.

===Theorem 2.===
If the conditions of theorem 1 are fulfilled and if, moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086960/s08696079.png" /> admits an infinitesimal upper bound for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086960/s08696080.png" />, then the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086960/s08696081.png" /> of the system (1) is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086960/s08696082.png" />-stable uniformly in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086960/s08696083.png" />.

===Theorem 3.===
If the conditions of theorem 1 are fulfilled and if, moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086960/s08696084.png" /> admits an infinitesimal upper bound for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086960/s08696085.png" />, then for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086960/s08696086.png" /> one can find a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086960/s08696087.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086960/s08696088.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086960/s08696089.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086960/s08696090.png" /> implies the inequality

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===Theorem 4.===
If the system (1) is such that there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086960/s08696092.png" />-positive-definite function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086960/s08696093.png" /> admitting an infinitesimal upper bound for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086960/s08696094.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086960/s08696095.png" />) and with negative-definite derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086960/s08696096.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086960/s08696097.png" />, then the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086960/s08696098.png" /> of the system (1) is asymptotically <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086960/s08696099.png" />-stable.

For the study of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086960/s086960100.png" />-instability, Chetaev's instability theorem (cf. [[Chetaev function|Chetaev function]]) has been successfully applied, as well as certain other theorems. Conditions for the converse of a number of theorems on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086960/s086960101.png" />-stability have been established; for example, the converses of theorems 1, 2 as well as of theorem 4 for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086960/s086960102.png" />. Methods of differential inequalities and Lyapunov vector functions have been applied to establish theorems on asymptotic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086960/s086960103.png" />-stability in the large, on first-order approximations, etc. (cf. [[#References|[3]]], ).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.M. Lyapunov,   ''Mat. Sb.'' , '''17''' :  2  (1893)  pp. 253–333</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  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)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  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</TD></TR></table>



====Comments====
Stability for a part of the variables is also called partial stability and occasionally conditional stability, [[#References|[a1]]]. However, the latter phrase is also used in a different meaning: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086960/s086960104.png" /> be a class of trajectories, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086960/s086960105.png" /> a trajectory in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086960/s086960106.png" />. This trajectory is stable relative to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086960/s086960107.png" /> if for a given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086960/s086960108.png" /> there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086960/s086960109.png" /> such that for each trajectory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086960/s086960110.png" /> one has that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086960/s086960111.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086960/s086960112.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086960/s086960113.png" /> is not the class of all trajectories, such a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086960/s086960114.png" /> is called conditionally stable, [[#References|[a2]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Hahn,   "Stability of motion" , Springer  (1965)  pp. §55</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  S. Lefshetz,   "Differential equations: geometric theory" , Dover, reprint  (1977)  pp. 78, 83</TD></TR></table>