ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

9. Ordinary Differential Equations and Dynamical Systems

1991 MS Classification: 34, 45, 65 Munembe, Joao Sebastiao Paulo, Perm State University, Russia Computationally verifiable conditions of the existence of stabilizable solutions for linear differential systems with periodic parameters We consider the system ẋ(t) − P (t)x[t − Δ(t)] + f(t), t∈[0, ∞), (1) where x(ξ) = 0, if ξ < 0, n×n-matrix P, functions Δ and f are T-periodic, Δ(t)≥0. A solution x_{α}(t), α∈R^{n} of (1) defined by the condition x(0) = α, is said to be stabilizable to a T-periodic function y : [0, ∞)→R^{n} if lim_{N→∞} max_{t∈[NT,(N+1)T]} ∥ x_{α}(t) − y(t)∥_{R^{n}} = 0. The conditions of the existence of stabilizable solutions x_{α} as well as an equation defining the T-periodic function y are obtained. The sufficiently high precision of the constructed approximation C̃(t, s) to the Cauchy matrix C(t, s) of (1) on [0, T] allows to establish the existence of a stabilizable solution x_{α} and to give efficiently computable upper estimates for the values θ(α, N) = max_{t∈[NT,(N+1)T]} ∥ x_{α}(t)y(t)∥. Reference: V. P. Maksimov and J. S. P. Munembe, On the question of enclosing solutions of linear functional differential systems, Memoirs of Differential Equations and mathematical Physics 12(1997), pp149-156, Tbilisi.