Well-posedness of the Cauchy problem for linear systems of generalized ordinary differential equations on an infinite interval. (English. Russian original) Zbl 1087.34002

Ashordiya, M. T.; Kekeliya, N. A.

Differ. Equ. 40, No. 4, 477-490 (2004); translation from Differ. Uravn. 40, No. 4, 443-454 (2004).

The authors prove sufficient conditions for the well-posedness (continuous dependence of solutions on the right-hand side and initial data) of the problem \[ dx( t) =dA( t) x( t) +df( t) ,\;t\in\mathbb{R}_{+}, \quad x( t_{0}) =c_{0}, \] where \(A:\mathbb{R}_{+}\to\mathbb{R}^{n\times n}\) is a matrix function and \(f:\mathbb{R}_{+}\to\mathbb{R}^{n}\) is a given vector function.
One of the main results asserts that if \(A\in BV( \mathbb{R} _{+};\mathbb{R}^{n\times n}) \) and \(f\in BV( \mathbb{R} _{+};\mathbb{R}^{n}) ,\) then the above problem is well posed.
The authors study also the relationship between the stability of the system and the well-posedness of the Cauchy problem. For example, under some additional hypotheses, if the matrix function \(A\) is \(\xi\)-exponentially asymptotically stable, then the above problem is well posed.
The authors also give specific versions of the preceding results for linear impulsive systems. The case of the corresponding linear finite-difference systems is also treated.

Reviewer: N. C. Apreutesei (Iaşi)

MSC:

34A12	Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
34A37	Ordinary differential equations with impulses
39A10	Additive difference equations

Keywords:

bounded total variation; \(\xi\)-exponentially asymptotical stability; uniform stability; well-posed problem; weakly well-posed problem