Error Estimate for Linearization of a Quasilinear Periodic System of Finite-Difference Equations
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Journal of Mathematical Sciences, Vol. 249, No. 6, September, 2020
ERROR ESTIMATE FOR LINEARIZATION OF A QUASILINEAR PERIODIC SYSTEM OF FINITE-DIFFERENCE EQUATIONS E. V. Afinogentova Ogarev Mordovia State University 68, Bolshevistskaya St., Saransk 430005, Russia afin [email protected]
UDC 517.962.2
For a quasilinear system of difference-differential equations we obtain an estimate for the linearization error. The method is based on the difference analog of the second Lyapunov method and comparison theorem. Bibliography: 4 titles.
Discretization of continuous processes leads to the necessity to study properties of solutiosn to systems of finite-difference equations. The linearization method, widely used in the study of nonlinear nonlinear processes, can be applied to the following system of finite-difference equations: x(k + 1) = P (k)x(k) + q(k) + F (x(k)), k = 0, 1, 2, . . . , (1) x(0) = x0 , where P (k) is a t-periodic (n × n)-matrix, i.e., P (k + t) = P (k) for all k = 0, 1, 2, . . ., x ∈ Rn , F (x) is an n-dimensional vector-valued function, F (x) γxα , γ > 0, α > 1 for x < ρ, ρ is an arbitrary real constant, · denotes the Euclidean norm, {P (k)}∞ k=0 is a sequence of bounded nonsingular matrices bounded in the norm, and q(k) is a bounded vector-valued function, q(k) Q, where Q > 0. Linearizing (1), we obtain the problem z(k + 1) = P (k)z(k) + q(k),
k = 0, 1, 2 . . . ,
z(0) = x0 .
(2)
Our goal is to estimate the linearization error ε(k) = x(k)−z(k), k = 0, 1, 2, . . ., provided that the linear part is asymptotically stable. We consider the homogeneous linear system y(k + 1) = P (k)y(k),
k = 0, 1, 2, . . . ,
y(0) = x0 . Translated from Problemy Matematicheskogo Analiza 103, 2020, pp. 17-19. c 2020 Springer Science+Business Media, LLC 1072-3374/20/2496-0834
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(3)
As shown in [1], using the nondegenerate transformation y(k) = T (k)y(k), y(k) ∈ Rn , it is possible to reduce the system (3) to the linear system with constant coefficients y(k + 1) = By(k),
k = 0, 1, 2, . . . ,
(4)
y(0) = T −1 (0)x0 ,
where T (k) is a bounded (in the norm) matrix, together with its inverse (T (k) S, T −1 (k) S , k = 0, 1, 2, . . ., and B = T −1 (k + 1)P (k)T (k). We assume that the system (4) is asymptotically stable. By [2], there exists a Lyapunov function V (y) such that (a) y V (y) M y, M 1, (b) V (y ) − V (y ) M |y − y , (c) V (y(k + 1)) − V (y(k)) −χV (y(k)), 0 < χ < 1. Changing the variable x = T (k)x, where x ∈ Rn , we reduce the system (1) to the form x(k + 1) = Bx(k) + T −1 (k + 1)(q(k) + F (T (k)x(k))),
k = 0, 1, 2, . . . ,
x(0) = T −1 (0)x0 .
(5)
We consider the first difference of the function V on solutions to the system (5): V (x(k + 1)) − V (x(k)) −χV (x(k)) + M S (Q + γS α V α (x(k)).
(6)
We set ϕ(u) = −χu + M S (Q + γS α uα ). It is obvious that the equation ϕ(u) = 0
(7)
can have at most two solutions in the domain u 0. If 1/(α−1) χ χ(α − 1) , Q αM S αM S γS α
(8)
then Equation (7) has at least one solution in the domain u 0. T
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