Stability in the presence of persistently acting perturbations

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A property of the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086980/s0869801.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086980/s0869802.png" />, of an initial value problem

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086980/s0869803.png" /> (*)

which consists of the following. For every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086980/s0869804.png" /> there is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086980/s0869805.png" /> such that for every point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086980/s0869806.png" /> satisfying the inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086980/s0869807.png" />, and every mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086980/s0869808.png" /> satisfying the conditions

a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086980/s0869809.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086980/s08698010.png" /> are continuous on the set

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086980/s08698011.png" />

b)

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086980/s08698012.png" />

the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086980/s08698013.png" /> of the initial value problem

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086980/s08698014.png" />

is defined for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086980/s08698015.png" /> and satisfies the inequality

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086980/s08698016.png" />

Bohl's theorem [1]. Suppose that the initial value problem (*) has a solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086980/s08698017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086980/s08698018.png" />, satisfying the conditions:

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086980/s08698019.png" />) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086980/s08698020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086980/s08698021.png" /> are continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086980/s08698022.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086980/s08698023.png" />;

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086980/s08698024.png" />) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086980/s08698025.png" />;

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086980/s08698026.png" />) the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086980/s08698027.png" /> is differentiable with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086980/s08698028.png" /> at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086980/s08698029.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086980/s08698030.png" />, uniformly with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086980/s08698031.png" />, i.e.

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086980/s08698032.png" />
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086980/s08698033.png" />

Then in order that this solution of the initial value problem be stable in the presence of persistently acting perturbations, it is necessary and sufficient that the upper singular exponent (cf. Singular exponents) of the system of variational equations of the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086980/s08698034.png" /> along the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086980/s08698035.png" /> be less than zero.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086980/s08698036.png" /> does not depend on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086980/s08698037.png" /> (an autonomous system) and the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086980/s08698038.png" /> is periodic or constant, and also if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086980/s08698039.png" /> is periodic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086980/s08698040.png" /> and the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086980/s08698041.png" /> is periodic with the same (or a commensurable) period or is a constant, then: 1) the condition stated in Bohl's theorem on uniform differentiability is superfluous (it follows from the remaining conditions of the theorem); and 2) the upper singular exponent of the system of variational equations of the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086980/s08698042.png" /> along the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086980/s08698043.png" /> can be calculated efficiently.

References

[1] P. Bohl, "Ueber Differentialgleichungen" J. Reine Angew. Math. (1914) pp. 284–318
[2] I.G. Malkin, "Theorie der Stabilität einer Bewegung" , R. Oldenbourg , München (1959) (Translated from Russian)
[3] Yu.L. Daletskii, M.G. Krein, "Stability of solutions of differential equations in Banach space" , Amer. Math. Soc. (1974) (Translated from Russian)


Comments

Stability under persistently acting perturbations is also called stability under persistent perturbations and total stability.

References

[a1] W. Hahn, "Stability of motion" , Springer (1965) pp. §56 (Translated from German)


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