Representation of a Solution of the Cauchy Problem for an Oscillating System with Multiple Delays and Pairwise Permutable Matrices

Abstract

Nonhomogeneous system of linear differential equations of second order with multiple different delays and pairwise permutable matrices defining the linear parts is considered. Solution of corresponding initial value problem is represented using matrix polynomials.

Representation of solutions for linear fractional systems with pure delay and multiple delays

“…Although we considered the exact solutions of (1.1) and (1.2) with order 𝛼 ∈ (1, 2), our results can be applied in the case of 𝛼 = 1 and 𝛼 = 2. In other words, if 𝛼 = 1 or 2, then our results coincide with the corresponding results in previous works 8,19,20,27,28 . Moreover, according to these papers, we can conjecture that Theorem 2 holds true even if the function f is not exponentially bounded.…”

On a study of Sobolev‐type fractional functional evolution equations

“…Particularly, time-delay systems with multidelays have a potential to be more suitable for applications in engineering and science. [18][19][20][21] Diblik et al 18 have studied nonhomogeneous system of linear differential equations with multiple different delays and pairwise permutable matrices. Furthermore, Medved and Pospisil 21 have derived sufficient conditions for the asymptotic stability of nonlinear multidelay differential equations using multidelay exponential functions via Pinto's inequality.…”

“…Liang et al 17 have investigated the finite‐time stability of linear delay differential equations system via the delayed matrix cosine and sine of polynomial degrees and extended to the same issue of delayed differential equations system with nonlinearity by virtue of Gronwall's inequality approach. Particularly, time‐delay systems with multidelays have a potential to be more suitable for applications in engineering and science 18–21 . Diblik et al 18 have studied nonhomogeneous system of linear differential equations with multiple different delays and pairwise permutable matrices.…”

On relative controllability of delayed difference equations with multiple control functions

“…[1]) as well as the stepwise integration method [7] for solving the differential equations with a delay. However, only recently the method was used to obtain representations of solutions of delayed differential [8,9,2,3,14] and difference equations [5,10]. The representations have various applications (see papers [2,3,4,6,11,12,13,14,15] and the references therein) in theory of stability, controllability, observability, boundary value problems, etc.…”