Malkhaz Ashordia

The Initial Problem for Linear Systems of Generalized Ordinary Differential Equations, Linear Impulsive and Ordinary Differential Systems. Numerical Solvability

abstract:
For the system of generalized linear ordinary differential equations the initial problem
\begin{gather*} d x=d A(t)\cdot x+df(t)\;\;\;(t\in I),\\ x(t_0)=c_0 \end{gather*}
is considered, where $I\subset \mathbb{R}$ is an interval, $A:I\to \mathbb{R}^{n\times n}$ and $f:I\to \mathbb{R}^n$ are, respectively, matrix- and vector-functions with components of local bounded variation, $t_0\in I$, $c_0\in \mathbb{R}^n$.

Under a solution of the system is understood a vector-function $x:I\to \mathbb{R}^n$ 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
\begin{gather*}
\frac{dx}{dt} =P(t)x+q(t),\;\;\;x(\tau_l+)-x(\tau_l-)=G(\tau_l) x(\tau_l)+u(\tau_l)\quad (l=1,2,\dots),
\end{gather*}
where $P$ and $q$ are, respectively, the matrix- and the vector-functions with Lebesgue local integrable components, $\tau_l$ $(l=1,2,\dots)$ are the points of impulse actions, and $G(\tau_l)$ $(l=1,2,\dots)$ and $u(\tau_l)$ $(l=1,2,\dots)$ 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.
 

Mathematics Subject Classification: 34A12, 34A30, 34A37, 34D20, 34K06, 34K07, 34K20

Key words and phrases: Generalized linear ordinary differential equations in the Kurzwei sense, initial problem, well-posedness, the Liapunov stability, linear impulsive differential equations, linear ordinary differential equations, numerical solvability, convergence of difference schemes, effective necessary and sufficient conditions.