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首页> 外文期刊>Memoirs on Differential Equations and Mathematical Physics >The Initial Problem for Linear Systems of Generalized Ordinary Differential Equations, Linear Impulsive and Ordinary Differential Systems. Numerical Solvability
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The Initial Problem for Linear Systems of Generalized Ordinary Differential Equations, Linear Impulsive and Ordinary Differential Systems. Numerical Solvability

机译：广义常微分方程线性系统，线性脉冲和常微分系统的初始问题。数值可解性

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For the system of generalized linear ordinary differential equations the initial problem dx = dA(t) · x + df(t) (t ∈ I), x(t0) = c0 is considered, where I ? R is an interval, A : I → R n×n and f : I → R n are, respectively, matrix- and vector-functions with components of local bounded variation, t0 ∈ I, c0 ∈ R n. Under a solution of the system is understood a vector-function x : I → R n with components of bounded local variation satisfying the corresponding integral equality, where the integral is understood in the Kurzweil sense. Along with a number of questions, we investigate the problems of the well-posedness and stability in Liapunov sense. Effective sufficient conditions, as well as effective necessary and sufficient conditions, are established for each of these problems. The obtained results are realized for the initial problem for linear impulsive system dx dt = P(t)x + q(t), x(τl+) ? x(τl?) = G(τl)x(τl) + u(τl) (l = 1, 2, . . .), where P and q are, respectively, the matrix- and the vector-functions with Lebesgue local integrable components, τl (l = 1, 2, . . .) are the points of impulse actions, and G(τl) (l = 1, 2, . . .) and u(τl) (l = 1, 2, . . .) are the matrix– and the vector-functions of discrete variables. Using the well-posedness results, the effective sufficient conditions, as well as the effective necessary and sufficient conditions, are established for the convergence of difference schemes to the solution of the initial problem for the linear systems of ordinary differential equations.
机译：对于广义线性常微分方程组，初始问题为dx = dA（t）·x + df（t）（t∈I），x（t0）= c0，其中I？ R是一个区间，A：I→R n×n和f：I→R n分别是矩阵和向量函数，具有局部有界变化的分量，t0∈I，c0∈R n。在该系统的解决方案下，可以理解向量函数x：I→R n，其局部有界变化的分量满足相应的积分等式，其中积分是从库兹韦尔意义上理解的。除了一些问题，我们还研究了利亚普诺夫意义上的适定性和稳定性问题。为这些问题中的每一个建立了有效的充分条件，以及有效的必要和充分条件。对于线性脉冲系统的初始问题dx dt = P（t）x + q（t），x（τl+）？ x（τl？）= G（τl）x（τl）+ u（τl）（l = 1，2，... ..），其中P和q分别是带有Lebesgue局部的矩阵函数和矢量函数可积分量τl（l = 1，2，.....）是冲动点，而G（τl）（l = 1，2，...）和u（τl）（l = 1，2，2， ....）是离散变量的矩阵和向量函数。利用适定性结果，建立了有效的充分条件，以及有效的必要条件和充分条件，以使差分方案收敛到常微分方程线性系统的初始问题的解。

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