Stability in the presence of persistently acting perturbations
A property of the solution ,
, of an initial value problem
![]() | (*) |
which consists of the following. For every there is a
such that for every point
satisfying the inequality
, and every mapping
satisfying the conditions
a) and
are continuous on the set
![]() |
b)
![]() |
the solution of the initial value problem
![]() |
is defined for all and satisfies the inequality
![]() |
Bohl's theorem [1]. Suppose that the initial value problem (*) has a solution ,
, satisfying the conditions:
)
and
are continuous on
for some
;
)
;
) the mapping
is differentiable with respect to
at the point
for
, uniformly with respect to
, i.e.
![]() |
![]() |
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 along the solution
be less than zero.
If does not depend on
(an autonomous system) and the solution
is periodic or constant, and also if
is periodic in
and the solution
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
along the solution
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) |
Stability in the presence of persistently acting perturbations. V.M. Millionshchikov (originator), Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Stability_in_the_presence_of_persistently_acting_perturbations&oldid=17291