On the representation of solutions of delayed differential equations via Laplace transform

On the representation of solutions of delayed differential equations via Laplace transform

Michal Pospíšil

^{B1, 2}

and František Jaroš

²

1Mathematical Institute of Slovak Academy of Sciences, Štefánikova 49, 814 73 Bratislava, Slovakia

2Department of Mathematical Analysis and Numerical Mathematics, Faculty of Mathematics, Physics and Informatics, Comenius University in Bratislava, Mlynská dolina, 842 48 Bratislava, Slovakia

Received 9 May 2016, appeared 13 December 2016 Communicated by Ivan Kiguradze

Abstract. 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.

Keywords: delay equation, multiple delays, matrix polynomial.

2010 Mathematics Subject Classification: 34A30, 34K06.

1 Introduction

Recently, the method of steps [5] was applied in [6] to obtain representation of solutions of differential equations with one delay. We recall this result.

Theorem 1.1. Letτ>0, B be N×N matrix, and ϕ∈C¹([−τ, 0],R^N), f : [0,∞)→_R^N be given functions. Then the solution of the Cauchy problem consisting of the equation

˙

x(t) = Bx(t−τ) + f(t), t≥0 and initial condition

x(t) = ϕ(t), t ∈[−τ, 0] (1.1)

has the form

x(t) =_e^Bt_τ ϕ(−τ) +

Z ₀

−τ

eτ^{B(t−τ−s)}ϕ⁰(s)ds+

Z _t

0 e^Bτ^(t−τ−s)f(s)ds for any t ≥ −τ, wheree^Bt_τ is the delayed matrix exponential defined as

e^Bt_τ =











Θ, t <−τ,

I, −τ≤t<0,

I+Bt+B^2(t−₂^τ)2 +· · ·+B^k(t−(k_k!^−1)τ)k, (k−1)τ≤t <kτ, k∈_N, ΘandIare the N×N zero and identity matrix, respectively.

BCorresponding author. Email: Michal.Pospisil@fmph.uniba.sk

This result was generalized in [8] for the case of n ∈ _N constant delays and C¹ initial function. Nevertheless, we rather recall a simplified result from [9] that was originally pub- lished for equations with variable coefficients, time-dependent delays and only continuous functionϕ.

Theorem 1.2. Let n ∈ _N, 0 < τ₁, . . . ,τn ∈ _R, τ := max{τ₁,τ2, . . . ,τn}, B₁, . . . ,Bn be pairwise permutable N×N matrices, i.e. B_iB_j = B_jB_i for each i,j ∈ {1, . . . ,n}, ϕ ∈ C([−τ, 0],R^N), and f : [0,∞) → _R^N be a given function. Then the solution of the Cauchy problem consisting of the equation

˙

x(t) =B₁x(t−τ₁) +B₂x(t−τ₂) +· · ·+B_nx(t−τ_n) + f(t)_, t≥0 (1.2) and initial condition(1.1)possesses the form

x(t) =











ϕ(t), −τ≤t<0,

X_n(t)ϕ(₀) +Rt

0 X_n(t−s)_∑ⁿ_m=1B_mψ(s−τ_m)ds +Rt

0X_n(t−s)f(s)ds, 0≤t

(1.3)

where

ψ(t) = (

ϕ(t), t∈ [−τ, 0),

θ, t∈/ [−τ, 0), (1.4)

θ is the N-dimensional vector of zeros, and X_n(t) = e^B_τ1¹_,τ^,B₂²_,...,τ^,...,B_n^n(t−τn) is the multi-delayed matrix exponential given by

e^B_τ1¹_,...,τ^,...,B_j^jt =



























Θ, t <−τ_j,

X_j−1(t+τ_j), −τ_j ≤t <0,

X_j−1(t+τ_j) +B_jRt

0 X_j−1(t−s₁)X_j−1(s₁)ds₁+· · ·

· · ·+B^k_j Rt (k−1)τ_j

Rs₁

(k−1)τ_j. . .Rs_k−1

(k−1)τ_jX_j−1(t−s₁)

×_∏^k_i=⁻₁¹X_j−1(s_i−s_i+1)X_j−1(s_k−(k−1)τ_j)ds_k. . .ds₁,

(k−1)τ_j ≤ t<kτ_j, k∈_N

(1.5)

for each j=2, 3, . . . ,n, where X_j−1(t) =e_τ^B₁¹_,...,τ^,...,B_j−^j−₁^{1(t−τj−1)}.

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 nonhomo- geneous differential equations with any finite number of delays in the sense of the following definition.

Definition 1.3. Let n ∈ _N, 0 < τ₁, . . . ,τ_n ∈ _R, τ := max{τ₁,τ₂, . . . ,τ_n}, ϕ ∈ C([−τ, 0],R^N), B₁, . . . ,Bn be N×N matrices, and f : [0,∞) → _R^N be a given function. The function x : [−τ,∞)→_R^N is a solution of the Cauchy problem (1.2), (1.1), if x ∈C¹([0,∞),R^N)(att = 0 the derivative in equation (1.2) represents the right-hand derivative),x(t)solves equation (1.2) on[0,∞)and satisfies the condition (1.1).

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 Theorem1.2). Here we consider also the equation

˙

x(t) = Ax(t) +B₁x(t−τ₁) +· · ·+B_nx(t−τ_n) + f(t), t ≥0 (1.6) 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,NandN0denote the set of all positive and nonnegative integers, respec- tively. We also assume the property of an empty sum,∑i∈_∅z(i) =0 for any functionz.

2 Preliminary results

The main tool we use in our computations is the unilateral Laplace transform defined as L{f(t)}=

Z _∞

0 e^−ptf(t)dt

for Rep > 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 F(p) = L{f(t)}. Then f(t) = L⁻¹{F(p)}. 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

σ(t) =

(0, t<_0, 1, t≥0.

Moreover, we apply L(andL⁻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 largeRep and appropriate functions f,g:

1. L{a f(t) +bg(t)}=aL{f(t)}+bL{g(t)}for constants a,b∈_R, 2. L^−1ne−_p^pτo=σ(t−τ)forτ≥0,

3. L⁻¹{F(p)G(p)}= (f∗g)(t)for convolution operator∗, 4. L{f⁰(t)}= pL{f(t)} − f(0),

5. L⁻¹{1}=δ(t)whereδ(t)is Dirac delta distribution.

Note that due to the arguments preceding the above lemma, point (3) of the lemma can be written as

L⁻¹{F(p)G(p)}= ((fσ)∗(gσ))(t) =

Z _t

0 f(s)g(t−s)ds.

The next two lemmas are corollaries of the latter one.

Lemma 2.2. The following identities hold true for sufficiently largeRep:

1. L⁻¹{F₁(p)F₂(p). . .F_n(p)}= (f₁∗ f₂∗ · · · ∗ f_n)(t)for n ∈ _{N, n} ≥ 2 and appropriate func- tions f₁, f₂, . . . , f_n,

2. L^−1ne−_p^pτno= ^(t−_(n^nτ₋₁⁾₎ⁿ_!⁻¹σ(t−nτ)forτ>0, n∈_N.

Proof. (1) Ifn =2, the statement coincides with Lemma 2.1.3. On suppose that the statement holds forn= k, using Lemma2.1, one obtains

L⁻¹{F₁(p)F₂(p). . .F_k+1(p)}= (L⁻¹{F₁(p)F₂(p). . .F_k(p)} ∗ f_k+1)(t)

= (f₁∗f2∗ · · · ∗f_k+1)(t).

(2) If n = 1 the statement becomes Lemma 2.1.2. Now, suppose that it holds for n = k.

Lemma2.1 yields L⁻¹

( e^−pτ

p

k+1)

= L⁻¹ (

e^−pτ p

k)

∗ L⁻¹ e^−pτ

p !

(t)

=

Z _t

0

(s−kτ)^k−1

(k−1)! σ(s−kτ)σ(t−s−τ)ds

=

Z _t−τ

kτ

(s−kτ)^k−1

(k−1)! dsσ(t−(k+1)τ) = (t−(k+1)τ)^k

k! σ(t−(k+1)τ) what was to be proved.

Lemma 2.3. Let n∈_N,0<τ₁,τ₂, . . . ,τ_n∈_R, k₁,k₂, . . . ,k_n∈_N0. Then L⁻¹

( n m

∏

=1

e^−pτm p

km)

=







δ(t), k₁=k2 =· · ·=kn =0,

(t−_∑ⁿ_m=1kmτ_m)^{∑nm=1km−1}

(_∑ⁿ_m=1km−1)! σ t−

∑

n m=₁

k_mτ_m

!

, k₁+k₂+· · ·+k_n∈_N. ^(2.1) Proof. We shall prove the statement by mathematical induction with respect ton. For n = 1, (2.1) is obtained from Lemma2.1.5and Lemma2.2.2. Now, suppose that the statement holds withn =l. For simplicity we denote Ln the left-hand side of (2.1). Then we expand as in the proof of Lemma2.2.2,

L_l+1 = L_l∗ L⁻¹ (

e^−pτl+1 p

k_l+1)!

(t). (2.2)

Using the inductive hypothesis and Lemma2.2.2, we subsequently consider four cases.

If k₁ = k₂ = · · · = k_l+₁ = 0, we get L_l+₁ = (δ∗δ)(t) = δ(t). If k₁ = k₂ = · · · = k_l = 0, k_l+1 ∈N, (2.2) gives

L_l+1= (t−k_l+1τ_l+1)^kl+1−1

(k_l+1−1)! σ(t−k_l+1τ_l+1)

by the sifting property of δ function [3]. Similarly, if k₁+k₂+· · ·+k_l ∈ _N, k_l+1 = 0, then L_l+1 =L_l.

Finally, ifk₁+k2+· · ·+k_l ∈_Nandk_l+1 ∈N, it remains to rewrite the right-hand side of (2.2) as integral

L_l+₁=

Z _t

0

s−_∑^l_m=1k_mτ_m

_∑^l_m=1km−1

∑^lm=1k_m−1

!

σ s−

∑

l m=1

k_mτ_m

!

×(t−s−k_l+1τ_l+1)^kl+1−1

(k_l+1−1)! σ(t−s−k_l+1τ_l+1)ds

=

Z _{t−kl+1τl+1}

∑^lm=1kmτm

s−_∑^l_m=1kmτm

_∑^l_m=1km−1

∑^lm=1km−1

!

(t−s−k_l+1τ_l+1)^kl+1−1

(k_l+1−1)! dsσ t−

l+1 m

∑

=1

kmτm

! . Now, take the substitution

s =

∑

l m=1

k_mτ_m+ξ t−

l+1 m

∑

=1

k_mτ_m

!

to obtain

L_l+1=

t−_∑^l_m⁺₌¹₁kmτ_m

_∑^l_m⁺₌¹₁km−1

σ

t−_∑^l_m⁺₌¹₁kmτ_m ∑^lm=1km−1

!(k_l+1−1)!

B

∑

l m=1

km,k_l+1

!

where B(·,·) is the Euler beta function. Rewriting the beta function using gamma functions, B(u,v) = ^Γ_Γ⁽₍^u_u⁾₊^Γ(_v^v₎⁾ for anyu,v>0, and sinceΓ(k) = (k−1)! for anyk∈N, one obtains

L_l+1=

t−_∑^l_m⁺₌¹₁kmτ_m

_∑^l_m⁺₌¹₁km−1

σ

t−_∑^l_m⁺₌¹₁kmτ_m ∑^l_m⁺=¹1km−1

!

. The proof is finished.

Remark 2.4. IfB₁, . . . ,B_nareN×Nmatrices andwis anN-dimensional vector, then the latter lemma yields

L⁻¹ ( n

m

∏

=₁

B_me^−pτm p

km! w

)

=L⁻¹ ( n

m

∏

=₁

e^−pτm p

km!

∏

n m=₁

B_m^km

! w

)

=L⁻¹ ( n

m

∏

=1

e^−pτm p

km)

∏

n m=1

B_m^km

! w

=















δ(t)w, k₁= k2 =· · ·= kn =0, (t−_∑ⁿ_m=1k_mτ_m)^{∑nm=1km−1}_∏ⁿ_m=1B_m^km

w

(_∑ⁿ_m=1k_m−1)! σ t−

∑

n m=1

k_mτ_m

! ,

k₁+k2+· · ·+kn ∈_N.

Next, we recall an estimation of the multi-delayed matrix exponential from [8].

Lemma 2.5. Let n∈_N,0<τ₁, . . . ,τn ∈_{R, B1}, . . . ,Bnbe pairwise permutable N×N matrices and e^B_τ1¹_,...,τ^,...,B_n^ntbe given by(1.5). Ifα₁, . . . ,α_n∈_Rare such thatkB_ik ≤α_ie^αiτi for each i=1, . . . ,n, then

e^B_τ1¹_,...,τ^,...,B_n^nt

≤e^{(α1+···+αn)(t+τn)} for any t∈_R.

As a corollary we get a sufficient condition forx(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.

Proof. Let c₁,c₂ ∈ _R be such that |f(t)| ≤ c₁e^c2t for all t ≥ 0, and α₁, . . . ,α_n ∈ _R such that kB_ik ≤ α_ie^αiτi for each i = 1, . . . ,n. We can suppose that c2 > 0 (otherwise take c2 >

0). By Lemma 2.5, kX_n(t)k ≤ e^αt for any t ∈ _R where α = _∑ⁿ_i=1α_i. Then denoting ϕ := max_{t∈[−τ,0]}|ϕ(t)|, fort≥0 we obtain

|x(t)| ≤ ϕe^αt+

∑

n m=1

kB_mkϕ Z _t

0 e^α(t−s)ds+c₁ Z _t

0 e^{α(t−s)+c2s}ds

≤ ϕ 1+

∑

n m=1

kB_mk α

!

e^αt+ ^c1e

(α+c2)t

c₂ ≤Ce^(α+c2)t for a constantC.

3 Main results

This section is devoted to main results of the present paper. First we suppose that the function f is exponentially bounded.

Theorem 3.1. Let n ∈ _N, 0 < τ₁, . . . ,τ_n ∈ _R, τ := max{τ₁,τ₂, . . . ,τ_n}, B₁, . . . ,B_n be pairwise permutable N×N matrices, i.e. B_iB_j = B_jB_i for each i,j ∈ {1, . . . ,n}, ϕ ∈ C([−τ, 0],R^N), and f :[0,∞)→_R^N be a given exponentially bounded function, i.e., there exist constants c₁,c2 ∈_Rsuch that|f(t)| ≤c₁e^c2tfor all t ≥0. Then the solution of the Cauchy problem(1.2),(1.1)has the form

x(t) = (

ϕ(t), −τ≤ t<0,

A(t)ϕ(0) +_∑ⁿ_j=1B_jRτ_j

0 A(t−s)ϕ(s−τ_j)ds+Rt

0A(t−s)f(s)ds, 0≤t (3.1) where

A(t) =

∑

∑ⁿm=₁kmτm≤t k1,...,kn≥0

(t−_∑ⁿ_m=1kmτ_m)^∑nm=1km k₁! . . .kn!

∏

n m=1

B^k_m^m (3.2)

for any t∈_R.

Proof. By Lemma 2.6, the solution is exponentially bounded and we can apply the Laplace transform on the studied equation (1.2). By Lemma2.1.4, we obtain

pL{x(t)} −ϕ(0) =

∑

n i=1

B_i Z _∞

0

e^−psx(s−τ_i)ds+L{f(t)}

=

∑

n i=1

B_i Z _τi

0 e^−psϕ(s−τ_i)ds+

Z _∞

τi

e^−psx(s−τ_i)ds

+F(p)

=

∑

n i=1

B_{i Z ∞}

0 e^−psψ(s−τ_i)ds+e^−pτi Z _∞

0 e^−psx(s)ds

+F(p)

=

∑

n i=1

B_iL{ψ(t−τ_i)}+B_ie^−pτiL{x(t)}+F(p) forψ(t)given by (1.4). Therefrom, we get

pI−

∑

n i=1

B_ie^−pτi

!

L{x(t)}= ϕ(0) +

∑

n i=1

B_iL{ψ(t−τ_i)}+F(p).

From theory of matrices we know (see e.g. [11, Proposition 7.5]) that, on suppose that p is sufficiently large, or more precisely, pis such that

∑

n i=1

B_ie^−pτi

< p

for a fixed induced normk · k, the matrixI−_∑ⁿ_i=1^Bie_p^−pτi is invertible and it holds

I−

∑

n i=₁

B_ie^−pτi p

!−1

=

∑

∞ k=0

∑

n i=₁

B_ie^−pτi p

!k

. Hence

L{x(t)}= ¹ p I−

∑

n i=1

B_ie^−pτi p

!−1"

ϕ(0) +

∑

n i=1

B_iL{ψ(t−τ_i)}+L{f(t)}

# , i.e.

x(t) =A0+

∑

n j=1

B_jA_j+A_f where

A₀ =L⁻¹





 1 p





∑

∞ k=0

∑

n i=1

B_ie^−pτi p

!k

ϕ(0)





 ,

A_j =L⁻¹





 1 p





∑

∞ k=0

∑

n i=1

B_ie^−pτi p

!k

L{ψ(t−τ_j)}







, j=1, . . . ,n,

A_f =L⁻¹





 1 p





∑

∞ k=0

∑

n i=1

B_ie^−pτi p

!k

F(p)





 .

Now, applying Lemma2.1, we get

A₀=

∑

∞ k=0

L⁻¹





 1 p

∑

n i=1

B_ie^−pτi p

!k

ϕ(0)







=

∑

∞ k=0



σ∗ L⁻¹







∑

n i=1

B_ie^−pτi p

!k

ϕ(0)









(t)

= (σ∗δ)(t)ϕ(0) +

∑

∞ k=1



σ∗ L⁻¹







∑

n i=1

B_ie^−pτi p

!k

ϕ(0)









(t). Consequently, by multinomial theorem [1],

A0= σ(t)ϕ(0) +

∑

∞ k=1





σ∗ L⁻¹













∑

k₁+···+kn=k k1,...,kn≥0

k k₁, . . . ,kn

_n

m

∏

=1

B_me^−pτm p

km





ϕ(0)











(t)

= σ(t)ϕ(0) +

∑

∞

k=1

∑

k1+···+kn=k k₁,...,kn≥0

k k₁, . . . ,k_n

σ∗ L⁻¹ ( n

m

∏

=₁

B_me^−pτm p

km! ϕ(0)

)!

(t)

where

k k₁, . . . ,k_n

= ^k!

k₁! . . .k_n!

is the multinomial coefficient. Finally, by Lemma2.3and Remark 2.4, A₀=σ(t)ϕ(0) +

∑

∞

k=1

∑

k1+···+kn=k k1,...,kn≥0

k k₁, . . . ,k_n

_{Z t}

0

(s−_∑ⁿ_m=1kmτm)^{∑nm=1km−1} (_∑ⁿ_m=1k_m−1)!

×

∏

n m=1

B_m^km

!

ϕ(0)σ s−

∑

n m=1

k_mτ_m

!

σ(t−s)ds

=σ(t)ϕ(0) +

∑

∞

k=1

∑

k₁+···+kn=k k1,...,kn≥0

k k₁, . . . ,kn

(t−_∑ⁿ_m=1k_mτ_m)^k k!

×σ t−

∑

n m=₁

k_mτ_m

! n m

∏

=₁

B_m^km

!

ϕ(0) =A(t)ϕ(0). For each j=1, . . . ,nwe apply Lemma2.1,

A_j =

∑

∞ k=0



σ∗ L⁻¹







∑

n i=1

B_ie^−pτi p

!k

L{ψ(t−τ_j)}









(t), multinomial theorem,

A_j = (σ∗ψ(· −τ_j))(t) +

∑

∞

k=₁

∑

k₁+···+kn=k k1,...,kn≥0

k k₁, . . . ,kn

× σ∗ L⁻¹ ( n

m

∏

=₁

e^−pτm p

km)

∗

∏

n m=₁

B^k_m^m

!

ψ(· −τ_j)

! (t),

and Lemma2.3, A_j=

Z _t

0 σ(t−s)ψ(s−τ_j)ds+

∑

∞

k=1

∑

k1+···+kn=k k₁,...,kn≥0

k k₁, . . . ,kn

_n

m

∏

=1

B_m^km

!

× σ∗(· −_∑ⁿ_m=1kmτm)^{∑nm=1km−1}

(_∑ⁿ_m=1k_m−1)! σ · −

∑

n m=1

kmτm

!

∗ψ(· −τ_j)

! (t). The double sum from the right-hand side of the above identity can be written as

∑

∞

k=1

∑

k1+···+kn=k k1,...,kn≥0

k k₁, . . . ,k_n

_n

m

∏

=1

B_m^km

! (· −_∑ⁿ_m=1kmτm)^k

k! σ · −

∑

n m=1

kmτm

!

∗ψ(· −τ_j)

! (t)

=

Z _t

0

∑

∞

k=1

∑

k₁+···+kn=k k1,...,kn≥0

∏ⁿm=1B^k_m^m

k₁! . . .kn! t−s−

∑

n m=1

k_mτ_m

!k

σ t−s−

∑

n m=1

k_mτ_m

!

ψ(s−τ_j)ds.

Hence, A_j =

Z _t

0

∑

∞

k=0

∑

k₁+···+kn=k k1,...,kn≥0

∏ⁿm=1B_m^km

k₁! . . .kn! t−s−

∑

n m=1

kmτ_m

!k

σ t−s−

∑

n m=1

kmτ_m

!

ψ(s−τ_j)ds

=

Z _t

0

∑

∑ⁿm=1kmτm≤t−s k1,...,kn≥0

(t−s−_∑ⁿ_m=1k_mτ_m)^∑nm=1km k₁! . . .k_n!

∏

n m=₁

B^k_m^m

!

ψ(s−τ_j)ds

for each j=1, . . . ,n. Note that the above integral can be shrink toRτ_j

0 whent> 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 Rτ_j

0 , since Rτ_j t = 0 because of the empty sum property. Therefore,

A_j =

Z _τj

0

A(t−s)ϕ(s−τ_j)ds, j=1, . . . ,n.

Finally, as forA_j we derive A_f =

Z _t

0

∑

∑ⁿm=1kmτm≤t−s k₁,...,kn≥0

(t−s−_∑ⁿ_m=1k_mτ_m)^∑nm=1km k₁! . . .kn!

∏

n m=1

B_m^km

! f(s)ds

which is exactlyRt

0 A(t−s)f(s)ds. The statement is proved.

As is shown below, the statement of the above theorem holds for a more general function f. Corollary 3.2. Theorem3.1remains valid if the function f is not exponentially bounded.

Proof. On setting

ϕ(t) =

(Θ, t∈[−τ, 0), I, t=0,

f ≡_Θ and considering equation (1.2) as a matrix equation, one can see thatA(t)of (3.2) is a matrix solution of this equation and initial condition, i.e.

A(˙ t) =B₁A(t−τ₁) +· · ·+B_nA(t−τ_n)_, t ≥₀

considering the right-hand derivative att =0, and A(t) =

(Θ, t∈[−τ, 0), I, t=0.

Lett≥0 be arbitrary and fixed. Denote M⊂ {_{1, . . . ,}n}the (possibly empty) set of all indices such thatt <τ_j if and only ifj∈ M. Then from (3.1) we know that

x(t) =A(t)ϕ(0) +

∑

j∈M

B_j Z _t

0

A(t−s)ϕ(s−τ_j)ds +

∑

n M63j=1

B_j Z _τj

0

A(t−s)ϕ(s−τ_j)ds+

Z _t

0

A(t−s)f(s)ds.

Differentiating we obtain x˙(t) =

∑

n m=1

BmA(t−τm)ϕ(0) +

∑

j∈M

B_j A(0)ϕ(t−τ_j) +

Z _t

0

∑

n m=1

BmA(t−τm−s)ϕ(s−τ_j)ds

!

+

∑

n M63j=1

B_j Z _τj

0

∑

n m=1

B_mA(t−τ_m−s)ϕ(s−τ_j)ds +A(0)f(t) +

Z _t

0

∑

n m=1

BmA(t−τm−s)f(s)ds

=

∑

n M63m=1

B_mA(t−τ_m)ϕ(0) +

∑

j∈M

B_j ϕ(t−τ_j) +

Z _τj

0

∑

n M63m=1

B_mA(t−τ_m−s)ϕ(s−τ_j)ds

!

+

∑

n M63j=1

B_j Z _τj

0

∑

n M63m=1

BmA(t−τm−s)ϕ(s−τ_j)ds+ f(t) +

Z _t

0

∑

n M63m=1

B_mA(t−τ_m−s)f(s)ds

=

∑

j∈M

B_jϕ(t−τ_j) +

∑

n M63m=1

B_m

A(t−τ_m)ϕ(0) +

∑

n j=1

B_j Z _τj

0

A(t−τ_m−s)ϕ(s−τ_j)ds +

Z _t−τm

0

A(t−τ_m−s)f(s)ds

+ f(t)

=

∑

n j=1

B_jx(t−τ_j) + f(t)

sincex(t−τ_j) = ϕ(t−τ_j)ifj∈ M. This completes the proof.

Taking a simple substitution we obtain the following result for delayed differential equa- tions with a constant non-delayed term. The solution is understood in the sense analogous to Definition1.3.

Theorem 3.3. Let n ∈ _N,0< τ₁, . . . ,τn ∈ _R, τ:=max{τ₁,τ2, . . . ,τn}, A,B₁, . . . ,Bn be pairwise permutable N×N matrices, ϕ ∈ C([−τ, 0],R^N), and f : [0,∞) → _R^N be a given function. Then the solution of the Cauchy problem(1.6),(1.1)has the form

x(t) =











ϕ(t), −τ≤t <0,

B(t)ϕ(₀) +_∑ⁿ_j=1BjRτ_j

0 B(t−s)ϕ(s−τ_j)ds +Rt

0B(t−s)f(s)ds, 0≤t

(3.3)

where

B(t) =e^At

∑

∑ⁿm=1kmτm≤t k1,...,kn≥0

(t−_∑ⁿ_m=1k_mτ_m)^∑nm=1km k₁! . . .k_n!

∏

n m=1

Be_m^km (3.4)

for any t ∈_{R, and}Be_m = B_me^−Aτm for each m=1, . . . ,n.

Proof. Let us sety(t) =e^−Atx(t). Thenysatisfies

˙

y(t) =Be₁y(t−τ₁) +· · ·+Beny(t−τn) + f^˜(t), t ≥0, y(t) =_e^−Atϕ(t) =_: ϕ_e(t)_, −τ≤t ≤₀

for ˜f(t) =e^−Atf(t)for allt ≥0. Application of Theorem3.1yields

y(t) =











ϕe(t), −τ≤ t<0,

A(e t)ϕ_e(0) +_∑ⁿ_j=1Be_jRτj

0 A(e t−s)ϕ_e(s−τ_j)ds +Rt

0 A(e t−s)f^˜(s)ds, 0≤ t

with

A(e t) =

∑

∑ⁿm=1kmτm≤t k1,...,kn≥0

(t−_∑ⁿ_m=1kmτm)^∑nm=1km k₁! . . .k_n!

∏

n m=1

Be^k_m^m. Now, returning back toxand using ϕe(0) = ϕ(0),

e^AtBe_jA(e t−s)ϕ_e(s−τ_j) =B_je^A(t−s)A(e t−s)ϕ(s−τ_j)_, e^AtA(e t−s)f^˜(s) =e^A(t−s)A(e t−s)f(s)

andB(t) =e^AtA(e t), the statement follows immediately.

Finally, we present an application of the results derived on a scalar equation. In practical computations, we rewrite the sum in (3.2) or (3.4) as

∑ⁿm=1

∑

kmτm≤t k1,...,kn≥0

=

j t τ1

k

k

∑

1=0 j_t−k

1τ1 τ2

k

k

∑

2=0

· · ·

t−_∑ⁿ−1 m=1kmτm

τn

k

∑

n=0

.

Now the solution given by (3.1) or (3.3) can be computed by hand or using a software.

Example 3.4. Let us consider the following initial value problem

˙

x(t) =−3.5x(t) +2x(t−1)−x(t−2) +3x(t−2.5) +1, t≥0

x(t) =−t, t∈ [−2.5, 0]. (3.5)

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 Figure3.1that the solution is not differentiable at some point.

Acknowledgements

M. Pospíšil was supported by the Grant VEGA-SAV 2/0153/16.

Figure 3.1: The solution of (3.5) with a delayed view at the interval[_{2, 3}]_.

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