On the asymptotic stability in functional differential equations

Ignatyev, A.

doi:10.1090/S0002-9939-99-05094-7

On the asymptotic stability in functional differential equations
HTML articles powered by AMS MathViewer

by A. O. Ignatyev PDF

Proc. Amer. Math. Soc. 127 (1999), 1753-1760 Request permission

Abstract:

Consider a system of functional differential equations $dx/dt=f(t,x_{t})$ where $f$ is the vector-valued functional. The classical asymptotic stability result for such a system calls for a positive definite functional $V(t,\varphi )$ and negative definite functional ${dV}/{dt}$. In applications one can construct a positive definite functional $V$, whose derivative is not negative definite but is less than or equal to zero. Exactly for such cases J. Hale created the effective asymptotic stability criterion if the functional $f$ in functional differential equations is autonomous ($f$ does not depend on $t$), and N. N. Krasovskii created such criterion for the case where the functional $f$ is periodic in $t$. For the general case of the non-autonomous functional $f$ V. M. Matrosov proved that this criterion is not right even for ordinary differential equations. The goal of this paper is to prove this criterion for the case when $f$ is almost periodic in $t$. This case is a particular case of the class of non-autonomous functionals.

References

M. S. Berger and Y. Y. Chen, Forced quasiperiodic and almost periodic solution for nonlinear systems, Nonlinear Anal. 21 (1993), no. 12, 949–965. MR 1249212, DOI 10.1016/0362-546X(93)90118-C
Tadasi Nakayama, On Frobeniusean algebras. I, Ann. of Math. (2) 40 (1939), 611–633. MR 16, DOI 10.2307/1968946
A. R. Collar, On the reciprocation of certain matrices, Proc. Roy. Soc. Edinburgh 59 (1939), 195–206. MR 8, DOI 10.1017/S0370164600012281
Sonoko Mori, On $\textrm {S}5$ type modal logics based on intermediate logics with finitely many valued linear models, J. Tsuda College 10 (1978), 35–43. MR 479948
C. Corduneanu, Almost Periodic Functions, 2nd edition, Chelsea Publ. Co., New York, 1989.
A. M. Fink, Almost periodic differential equations, Lecture Notes in Mathematics, Vol. 377, Springer-Verlag, Berlin-New York, 1974. MR 0460799, DOI 10.1007/BFb0070324
Wolfgang Hahn, Stability of motion, Die Grundlehren der mathematischen Wissenschaften, Band 138, Springer-Verlag New York, Inc., New York, 1967. Translated from the German manuscript by Arne P. Baartz. MR 0223668, DOI 10.1007/978-3-642-50085-5
Jack Hale, Theory of functional differential equations, 2nd ed., Applied Mathematical Sciences, Vol. 3, Springer-Verlag, New York-Heidelberg, 1977. MR 0508721, DOI 10.1007/978-1-4612-9892-2
V. B. Kolmanovskiĭ and V. R. Nosov, Stability of functional-differential equations, Mathematics in Science and Engineering, vol. 180, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London, 1986. MR 860947
N. N. Krasovskiĭ, Stability of motion. Applications of Lyapunov’s second method to differential systems and equations with delay, Stanford University Press, Stanford, Calif., 1963. Translated by J. L. Brenner. MR 0147744
B. M. Levitan and V. V. Zhikov, Almost periodic functions and differential equations, Cambridge University Press, Cambridge-New York, 1982. Translated from the Russian by L. W. Longdon. MR 690064
V. M. Matrosov, On the theory of stability of motion, J. Appl. Math. Mech. 26 (1962), 1506–1522. MR 0150404, DOI 10.1016/0021-8928(62)90189-2
Nicolas Rouche, P. Habets, and M. Laloy, Stability theory by Liapunov’s direct method, Applied Mathematical Sciences, Vol. 22, Springer-Verlag, New York-Heidelberg, 1977. MR 0450715, DOI 10.1007/978-1-4684-9362-7
George Seifert, On uniformly almost periodic sets of functions for almost periodic differential equations, Tohoku Math. J. (2) 34 (1982), no. 2, 301–309. MR 664736, DOI 10.2748/tmj/1178229256
T. Yoshizawa, Stability theory and the existence of periodic solutions and almost periodic solutions, Applied Mathematical Sciences, Vol. 14, Springer-Verlag, New York-Heidelberg, 1975. MR 0466797, DOI 10.1007/978-1-4612-6376-0