Abstract:
We consider a system of two hybrid vector equations containing linear difference (defined on a discrete set) and functional differential (defined on a half-axis) parts. To study it, a model system of two vector equations is chosen, one of which is linear difference with aftereffect (LDEA), and the other is a linear functional differential with aftereffect (LFDEA). Two equivalent representations of this system are shown: the first representation in the form of LFDEA, the second — in the form of LDEA.
This allows us to study the stability issues of the system under consideration using the well-known results on the stability of LFDEA and LDEA.
Using the results of the article
[Gusarenko S. A. On the stability of a system of two linear differential equations with delayed argument // Boundary value problems. Interuniversity collection of scientific papers. Perm: PPI, 1989. P. 3–9], two examples are shown when a joint system of four equations will be stable with respect to the right side. In the first example, we use the LFDEA for which sufficient conditions for the sign-definiteness of the elements of the $ 2 \times 2 $ Cauchy matrix function are known (in terms of the LFDEA coefficients).
In the second example, LFDEA is given such that LFDEA is a system of linear ordinary differential equations (LODE) of the second order. In both cases, estimates of the components of the Cauchy matrix function are known. An exponential estimate with a negative exponent is given for the components of the Cauchy matrix function of LDEA.
Keywords:hybrid linear system of functional differential equations, linear difference equation with aftereffect, linear functional differential equation with aftereffect, Cauchy formula, stability with respect to the right side, Volterra reversibility, evaluation of operator norm.
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