Abstract
Solutions of nonhomogeneous systems of linear differential equations with multiple constant delays are explicitly stated without a commutativity assumption on the matrix coefficients. In comparison to recent results, the new formulas are not inductively built, but depend on a sum of noncommutative products in the case of constant coefficients, or on a sum of iterated integrals in the case of time-dependent coefficients. This approach shall be more suitable for applications.Representation of a solution of a Cauchy problem for a system of higher order delay differential equations is also given.
Highlights
Explicit representation of solutions of systems of delay differential equations (DDEs) with constant delays and linear parts given by pairwise permutable constant matrices was first stated in [11] motivated by a pioneering work [6] on DDEs with one constant delay
In the particular case of constant coefficients, the fundamental solution had a form of a matrix polynomial of a degree depending on time
To obtain a representation of a solution, which would be more suitable for applications, Laplace transform was applied in [16] to an initial function problem consisting of a system of DDEs with constant delays and linear parts given by pairwise permutable constants matrices, x (t) = Ax(t) + B1x(t − τ1) + · · · + Bnx(t − τn) + f (t), t ≥ 0, (1.1)
Summary
Explicit representation of solutions of systems of delay differential equations (DDEs) with constant delays and linear parts given by pairwise permutable constant matrices was first stated in [11] motivated by a pioneering work [6] on DDEs with one constant delay. In the particular case of constant coefficients, the fundamental solution had a form of a matrix polynomial of a degree depending on time. This approach is suitable for theoretical problems, such as existence, stability, controllability, etc. To obtain a representation of a solution, which would be more suitable for applications, Laplace transform was applied in [16] to an initial function problem consisting of a system of DDEs with constant delays and linear parts given by pairwise permutable constants matrices, x (t) = Ax(t) + B1x(t − τ1) + · · · + Bnx(t − τn) + f (t), t ≥ 0, (1.1). Using a particular sum of noncommutative matrix products and unilateral Laplace transform, we derive representation of a solution of a DDE with constant coefficients. Θ, Θ and I will stand for the zero vector, the zero matrix and the identity matrix of a suitable dimension, respectively
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