On the Representation of Solutions of Delayed Differential Equations via Laplace Transform

In this paper, the unilateral Laplace transform is used to derive a closed-form formula for a solution of a system of nonhomogeneous linear differential equations with any finite number of constant delays and linear parts given by pairwise permutable matrices. This unifies the recent results on the representation of such solutions.

Formula (1.3) was used in [8] to derive stability results.So it is usable for theoretical purposes.However, it does not seem to be very suitable for practical calculation of a solution, as the multi-delayed matrix exponential is built up inductively.
In the present paper, we provide another representation of solutions of linear nonhomogeneous differential equations with any finite number of delays in the sense of the following definition.
To obtain the representation we involve the unilateral Laplace transform.Of course, the idea to apply the Laplace transform to delay differential equations is not a new one.For instance in [2], the Laplace transform of a solution of a linear delayed differential equation is expressed using a Laplace transform of its fundamental solution.Rather than focusing on the Laplace image of a solution, in this paper we make use of properties of the Laplace transform and its inverse [4,10], in particular of the uniqueness of the inverse on the set of continuous functions.So we obtain a closed-form formula for the solution.
The paper is organized as follows.The next section concludes some known and basic results on the Laplace transform.Section 3 contains our main results on the representation of a solution of (1.2), (1.1) which is another extension of Theorem 1.1 to the case of multiple delays (clearly equivalent to Theorem 1.2).Here we consider also the equation with the initial condition (1.1), and derive the representation of its solution (see [6] for the case of one delay).This section is enclosed by an example.
In the whole paper we shall denote | • | the norm of a vector without any respect to its dimension.Further, N and N 0 denote the set of all positive and nonnegative integers, respectively.We also assume the property of an empty sum, ∑ i∈∅ z(i) = 0 for any function z.

Preliminary results
The main tool we use in our computations is the unilateral Laplace transform defined as for Re p > a and an exponentially bounded function f such that | f (t)| ≤ ce at for all t ≥ 0 and some constants a, c ∈ R. For the case of brevity we sometimes adopt the notation Note that here the preimage is assumed to vanish on (−∞, 0), which we emphasize by L −1 {F(p)} = f (t)σ(t), when needed.Recall σ is the Heaviside step function defined as Moreover, we apply L (and L −1 ) to each coordinate when considering the Laplace transform (or its inverse) of a vector.The next lemma concludes some of properties of the Laplace transform (see e.g.[4,10]).
Lemma 2.1.The following equalities hold true for sufficiently large Re p and appropriate functions f , g: Note that due to the arguments preceding the above lemma, point (3) of the lemma can be written as The next two lemmas are corollaries of the latter one.
Lemma 2.2.The following identities hold true for sufficiently large Re p: Proof.
(1) If n = 2, the statement coincides with Lemma 2.1.3.On suppose that the statement holds for n = k, using Lemma 2.1, one obtains (2) If n = 1 the statement becomes Lemma 2.1.2.Now, suppose that it holds for n = k.Lemma 2.1 yields what was to be proved.
Proof.We shall prove the statement by mathematical induction with respect to n.For n = 1, (2.1) is obtained from Lemma 2.1.5and Lemma 2.2.2.Now, suppose that the statement holds with n = l.For simplicity we denote L n the left-hand side of (2.1).Then we expand as in the proof of Lemma 2.2.2, Using the inductive hypothesis and Lemma 2.2.2, we subsequently consider four cases. If by the sifting property of δ function [3].Similarly, if ∈ N, it remains to rewrite the right-hand side of (2.2) as integral Now, take the substitution where B(•, •) is the Euler beta function.Rewriting the beta function using gamma functions, B(u, v) = Γ(u)Γ(v) Γ(u+v) for any u, v > 0, and since Γ(k) = (k − 1)! for any k ∈ N, one obtains The proof is finished.
Remark 2.4.If B 1 , . . ., B n are N × N matrices and w is an N-dimensional vector, then the latter lemma yields Next, we recall an estimation of the multi-delayed matrix exponential from [8].
As a corollary we get a sufficient condition for x(t) of (1.3) to be exponentially bounded.
Lemma 2.6.Let the assumptions of Theorem 1.2 be fulfilled and the function f be exponentially bounded.Then the solution x(t) of (1.2), (1.1) is exponentially bounded.

Main results
This section is devoted to main results of the present paper.First we suppose that the function f is exponentially bounded.
Proof.By Lemma 2.6, the solution is exponentially bounded and we can apply the Laplace transform on the studied equation (1.2).By Lemma 2.1.4,we obtain for ψ(t) given by (1.4).Therefrom, we get From theory of matrices we know (see e.g.[11,Proposition 7.5]) that, on suppose that p is sufficiently large, or more precisely, p is such that is invertible and it holds where Now, applying Lemma 2.1, we get Consequently, by multinomial theorem [1], is the multinomial coefficient.Finally, by Lemma 2.3 and Remark 2.4, For each j = 1, . . ., n we apply Lemma 2.1, and Lemma 2.3, The double sum from the right-hand side of the above identity can be written as Hence, for each j = 1, . . ., n.Note that the above integral can be shrink to τ j 0 when t > τ j , along with ψ(s − τ j ) → ϕ(s − τ j ).On the other side, if t < τ j , it can be extended to τ j 0 , since τ j t = 0 because of the empty sum property.Therefore, Finally, as for A j we derive   The solution of this problem is found using Theorem 3.3 and is illustrated in Figure 3.1.We added a more detailed view at the interval [2,3], as one may get an impression from the first part of Figure 3.1 that the solution is not differentiable at some point.
f ≡ Θ and considering equation (1.2) as a matrix equation, one can see that A(t) of (3.2) is a matrix solution of this equation and initial condition, i.e.