On the Cauchy problem for a linear harmonic oscillator with pure delay

R E S E A R C H Open Access

On the Cauchy problem for a linear harmonic oscillator with pure delay

Denys Ya Khusainov¹, Michael Pokojovy^2*and Elvin I Azizbayov³

*Correspondence:

michael.pokojovy@uni-konstanz.de

2Department of Mathematics and Statistics, University of Konstanz, Konstanz, Germany

Full list of author information is available at the end of the article

Abstract

In the present paper, we consider a Cauchy problem for a linear second order in time abstract differential equation with pure delay. In the absence of delay, this problem, known as the harmonic oscillator, has a two-dimensional eigenspace so that the solution of the homogeneous problem can be written as a linear combination of these two eigenfunctions. As opposed to that, in the presence even of a small delay, the spectrum is infinite and a finite sum representation is not possible. Using a special function referred to as the delay exponential function, we give an explicit solution representation for the Cauchy problem associated with the linear oscillator with pure delay. Finally, the solution asymptotics as the delay parameter goes to zero is studied.

In contrast to earlier works, no positivity conditions are imposed.

MSC: Primary 34K06; 39A06; 39B42; secondary 34K26

Keywords: functional-differential equations; harmonic oscillator; pure delay;

well-posedness; solution representation

1 Introduction

LetXbe a (real or complex) Banach space and letx(t)∈Xdescribe the state of a physical system at timet≥. Witha(t) =x(t) denoting the acceleration of system, Newton’s second¨ law of motion states that

F(t) =Ma(t) fort≥, ()

whereM: D(M)⊂X→Xis a linear, continuously invertible, accretive operator repre- senting the ‘mass’ of the system. When being displaced from its equilibrium situated in the origin, the system is affected by a restoring forceF(t). In classical mechanics, this force is postulated to be proportional to the instantaneous displacement,i.e.,

F(t) =Kx(t) fort≥ ()

for some closed, linear operatorK:D(K)⊂X→X. WhenM^–Kis a bounded linear op- erator, plugging Equation () into (), we arrive at the classical harmonic oscillator model

¨

x(t) =M^–Kx(t) fort≥. ()

©2015 Khusainov et al. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, pro- vided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

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Assuming now that the restoring force is proportional to the value of the system at some past timet–τ, Equation () is replaced with the relation

F(t) =Kx(t–τ) fort≥, ()

whereτ >  is a time delay. Plugging Equation () into () leads then to the linear harmonic oscillator equation with pure delay written as

¨

x(t) =M^–Kx(t–τ) fort≥. ()

Problems similar to Equation () also arise when modeling systems with distributed pa- rameters such as general wave phenomena (cf.[]).

Equations similar to () are often referred to as delay or retarded differential equations.

After being transformed to a first order in time system on a Banach space X, a general equation with constant delay can be written as

˙ u(t) =H

t,u(t),u_t

fort> , u() =u^, u_=ϕ. ()

Here,τ>  is a fixed delay parameter,ut:=u(t+·)∈L^(–τ, ;X),t≥, denotes the his- tory variable,His anX-valued operator defined on a subset of [,∞)×X×L^(–τ, ;X) andu^∈X,ϕ∈L^(–τ, ;X) are appropriate initial data. Equations of type () have been intensively studied in the literature. We refer the reader to the monographs by Els’gol’ts and Norkin [] and Hale and Lunel [] for a detailed treatment of Equation () in finite- dimensional spacesX. In contrast to this, results on Equation () in infinite-dimensional spacesXare less numerous. A good overview can be found in the monograph of Bátkai and Piazzera [].

Khusainovet al.considered in [] Equation () inRⁿwith H

t,u(t),ut

=Au(t) +Au(t–τ) +

u(t)⊗b

u(t) +

u(t)⊗b

u(t–τ) +

u(t–τ)⊗b

u(t–τ)

for symmetric matricesA,A∈R^n×nand column vectorsb,b,b∈Rⁿand proposed a rational Lyapunov function to study the asymptotic stability of solutions to this system.

In their work [], Khusainovet al.studied a modal, or spectrum, control problem for a linear delay equation onRⁿreading as

˙

x(t) =Ax(t) +bu(t) fort>  ()

with a feedback controlu(t) =_m

j=c^T_jx(t–jτ) for some delay timeτ>  and parameter vectorscj∈Rⁿ. For canonical systems, they developed a method to compute the unknown parameters such that the closed-loop system possesses the spectrum prescribed before- hand. Under appropriate ‘concordance’ conditions, they were able to carry over their con- siderations for a rather broad class of non-canonical systems.

In the infinite-dimensional situation, a rather general particular case of () with H(t,v,ψ) =Av+F(ψ), whereAgenerates aC-semigroup (S(t))t≥onXandFis a non- linear operator onL^(–τ, ;X), was studied by Travies and Webb in their work []. Under

appropriate assumptions on F, they proved the integral equation corresponding to the weak formulation of the delay equation given by

u(t) =S(t)ϕ() + _t



S(t–s)F(us) ds fort>  to possess a unique solution inH_loc^ (,∞;X).

Di Blasioet al.addressed in [] a similar problem

˙

u(t) = (A+B)u(t) +Lu(t–r) +Lut fort> , u() =u^, u=ϕ, () whereAgenerates a holomorphicC-semigroup on a Hilbert spaceH,Bis a perturbation ofAandL,Lare appropriate linear operators. Ifu^andϕpossess a certain regularity, they proved the existence of a unique strong solution inH_loc^ (,∞;X)∩L^_loc(,∞;D(A)) by analyzing theC-semigroup inducing the semiflowt →(u(t),ut). These results were elab- orated on by Di Blasioet al.in [] leading to a generalization for the case of weighted and interpolation spaces and including a description of the associated infinitesimal generator.

Finally, the generalL^p-case forp∈(,∞) was investigated by Di Blasio in [].

Diblíket al.[] studied Equation () for the case thatAandBare ×-second order and first order commuting differential operators, respectively, in a bounded interval (,l) ofR andL≡,L≡. Additionally, they allowed for non-homogeneous Dirichlet boundary conditions. For this parabolic system, they proved the existence of solution in a class of classically differentiable functions both with respect to time and space under appropriate regularity conditions.

Recently, in their work [], Khusainovet al.proposed an explicitL^-solution theory for a non-homogeneous initial-boundary value problem for an isotropic heat equation with constant delay

u_t(t,x) =∂_i

a_ij(x)∂_ju(t,x)

+b_i(x)∂_iu(t,x) +c(x)u(t,x) +∂_i

˜

a_ij(x)∂_ju(t–τ,x)

+b˜_i(x)∂_iu(t–τ,x) +˜c(x)u(t–τ,x) +f(t,x) for (t,x)∈(,∞)×,

u(t,x) =γ(t,x) for (t,x)∈(,∞)×∂, u(,x) =u^(x) forx∈,

u(t,x) =ϕ(t,x) for (t,x)∈(–τ, )×,

where⊂R^dis a regular bounded domain and the coefficient functions are appropriate.

Conditions assuring for exponential stability were also given.

Over the past decade, hyperbolic partial differential equations have attracted a con- siderable amount of attention, too. In [], Nicaise and Pignotti studied a homogeneous isotropic wave equation with an internal feedback with and without delay reading as

∂_ttu(t,x) –u(t,x) +a∂_tu(t,x) +a∂tu(t–τ,x) =  for (t,x)∈(,∞)×, u(t,x) =  for (t,x)∈(,∞)×_,

∂u

∂ν(t,x) =  for (t,x)∈(,∞)×_

under the usual initial conditions where_,_⊂∂are relatively open in∂with¯_∩ ¯=∅andνdenotes the outer unit normal vector of a smooth bounded domain⊂R^d. They showed the problem to possess a unique global classical solution and proved the latter to be exponentially stable if a>a>  or instable, otherwise. These results have been carried over by Nicaise and Pignotti [] and Nicaiseet al.[] to the case of time- varying internally distributed or boundary delays.

In [], Khusainovet al.considered a non-homogeneous initial-boundary value problem for a one-dimensional wave equation with constant coefficients and a single constant delay

∂_ttu(t,x) =a^∂_xxu(t–τ,x) +b∂xu(t–τ,x) +cu(t–τ,x) +f(t,x) for (t,x)∈(,T)×(,l),

u(t,x) =γ(t,x) for (t,x)∈(,T)× {, }, u(,x) =u^(x) forx∈(, ),

u(t,x) =ϕ(t,x) fort∈(–τ, ),x∈(, ).

Under appropriate regularity and compatibility assumptions, they proved the problem to possess a uniqueC^-solution for any finiteT> . Their proof was based on extrapolation methods forC_-semigroups and an explicit solution representation formula.

Recently, Khusainov and Pokojovy presented in [] a Hilbert-space treatment of the initial-boundary value problem for the equations of thermoelasticity with pure delay

∂_ttu(x,t) –a∂xxu(x,t–τ) +b∂xθ(x,t–τ) =f(x,t) forx∈,t> ,

∂_tθ(x,t) –c∂xxθ(x,t–τ) +d∂txu(x,t–τ) =g(x,t) forx∈,t> , u(,t) =u(l,t) = , ∂_xθ(,t) =∂_xθ(l,t) =  fort> ,

u(x, ) =u^(x), u(x,t) =u^(x,t) forx∈,t∈(–τ, ),

∂_tu(x, ) =u^(x), ∂_tu(x,t) =u^(x,t) forx∈,t∈(–τ, ), θ(x, ) =θ^(x), θ(x,t) =θ^(x,t) forx∈,t∈(–τ, ).

Their proof exploited extrapolation techniques for strongly continuous semigroups and an explicit solution representation formula.

In the present paper, we give a Banach space solution theory for Equation () subject to appropriate initial conditions. Our approach is solely based on the step method and does not incorporate any semigroup techniques. In contrast to earlier works by Khusainovet al.

[, , ], we only require the invertibility and not the negativity ofM^–Kin Equation ().

In this sense, our framework is different from that employed by Diblíket al.in [, ], as they required the coefficient matrices to be negative definite. It should though be pointed out that their solution theory accounted for two and more delays, whereas we consider a single delay.

First, we briefly outline some seminal results on second order abstract Cauchy prob- lems. Next, in our main section, we prove the existence and uniqueness of solutions to the Cauchy problem for the delay equation () as well as their continuous dependence on the data. Next, we give an explicit solution representation formula in a closed form based on

the delayed exponential function introduced by Khusainov and Shuklin in []. Finally, we prove the solution of the delay equation to converge to the solution of the original second order abstract differential equation as the delay parameterτ goes to zero.

2 Classical harmonic oscillator

For the sake of completeness, we briefly discuss the initial value problem for the harmonic oscillator being a second order in time abstract differential equation

¨

x(t) –^x(t) =f(t) fort≥ ()

subject to the initial conditions

x() =x∈D(), x() =˙ x∈X. ()

Here, we assume the linear operator:D()⊂X→Xto be continuously invertible and generate aC-group (e^t)t∈R⊂L(X) on a (real or complex) Banach spaceXwith L(X) denoting the space of bounded, linear operators onXequipped with the normAL(X):=

sup{AxX:x∈X,xX≤}. A more rigorous treatment of this problem can be found in [], Section ..

The general solution to the homogeneous equation is known to read as xh(t) =e^tc+e^–tc fort≥

with somec,c∈D(). Vectorsc,ccan be computed using the initial conditions from Equation () leading to a system of linear operator equations

c+c=x, c–c=x. The latter is uniquely solved by

c=

^–(x+x), c=

^–(x–x).

Thus, the unique solution of the homogeneous equation with the initial conditions () is given by

xh(t) = 

^–e^t(x+x) +

^–e^–t(x–x) fort≥ () or, equivalently,

xh(t) = 



e^t+e^–t x+

^–

e^t–e^–t

x fort≥. ()

A particular solution to the non-homogeneous equation with zero initial conditions will be determined in the Cauchy form

xp(t) = _t



K(t,s)f(s) ds fort≥. ()

We refer the reader to Chapter  in [] for the definition of Bochner integrals forX-valued functions. In Equation (), the functionK∈C^([,∞)×[,∞),L(X)) is the Cauchy ker-

nel,i.e., for any fixeds≥, the functionK(·,s) is the solution of the homogeneous problem satisfying the initial conditions

K(t,s)|t=s= L(X), ∂tK(t,s)|t=s= idX. Using the ansatz

K(t,s) =e^tc(s) +e^–tc(s) fort,s≥

for somec_,c_∈C^([,∞),L(X)) and taking into account the initial conditions, we arrive at

K(t,s)|t=s=e^sc(s) +e^–sc(s) = L(X),

∂_tK(t,s)|t=s=e^sc_(s) –e^–sc_(s) = id_X.

Solving this system with generalized Cramer’s rule, we obtain, fors≥,

c(s) =

det_L(X)

e^s e^–s e^s –e^–s

–

det_L(X)

L(X) e^–s idX –e^–s

=

^–e^–s, c(s) =

det_L(X)

e^s e^–s e^s –e^–s

–

det_L(X)

e^s L(X)

e^s idX

=

^–e^s. Thus, the Cauchy kernel is given by

K(t,s) = 

^–

e^(t–s)–e^–(t–s)

fort,s≥,

whereas the particular solution satisfying zero initial conditions reads as xp(t) = 

^–

_t



e^(t–s)–e^–(t–s)

f(s) ds fort≥.

Hence, for x ∈D(), x ∈ X and f ∈L^_loc(,∞;X), the unique mild solution x ∈ W_loc^,(,∞;X) to the Cauchy problem ()-() can be written as

x(t) = 



e^t+e^–t x_+

^–

e^t–e^–t x_

+

^–

t



e^(t–s)–e^–(t–s)

f(s) ds fort≥. ()

If the data additionally satisfyx∈D(^),x∈D() andf ∈W_loc^,(,∞;X)∪C^([,∞), D(^)), then the mild solutionxgiven in Equation () is a classical solution satisfying x∈C^([,∞),X)∩C^([,∞),D())∩C^([,∞),D(^)).

3 The linear oscillator with pure delay

In this section, we consider a Cauchy problem for the linear oscillator with a single pure delay

¨

x(t) –^x(t– τ) =f(t) fort≥ ()

subject to the initial condition

x(t) =ϕ(t) fort∈[–τ, ]. ()

Here,Xis a Banach space,∈L(X) is a bounded, linear operator andϕ∈C^([–τ, ],X), f ∈L^_loc(,∞;X) are given functions. In contrast to the previous section, the boundedness ofis indispensable here. Indeed, Dreheret al.proved in [] that Equations ()-() are ill-posed even ifXis a Hilbert space andpossesses a sequence of eigenvalues (λ_n)_n∈N⊂ Rwithλn→ ∞orλn→–∞asn→ ∞. The necessity forbeing bounded has also been pointed out by Rodrigueset al.in [] when treating a linear heat equation with pure delay.

Definition  A functionx∈C^([–τ,∞),X)∩C^([–τ, ],X)∩C^([,∞),X) satisfying Equations ()-() pointwise is called a classical solution to the Cauchy problem ()-().

A mild formulation of ()-() is given by

˙

x(t) =x() +˙ ^ _t



x(s– τ) ds+ _t



f(s) ds fort≥, ()

x(t) =ϕ(t) fort∈[–τ, ]. ()

Definition  A functionx∈C^([–τ,∞),X) satisfying Equations ()-() is called a mild solution to the Cauchy problem ()-().

By the virtue of fundamental theorem of calculus, any mild solutionxto ()-() with x∈C^([–τ,∞),X)∩C^([–τ, ],X)∩C^([,∞),X) is also a classical solution. Obvi- ously, for the problem ()-() to possess a classical solution, one necessarily requires ϕ∈C^([–τ, ],X).

In the following subsection, we want to study the existence and uniqueness of mild and classical solutions to the Cauchy problem ()-() as well as their continuous dependence on the data.

3.1 Existence and uniqueness

Rather than using the semigroup approach (cf.[], Chapter ), we decided to use the more straightforward step method here reducing ()-() to a difference equation on the func- tional vector spaceCˆ^_τ(N,X) defined as follows.

Definition  LetXbe a Banach space,τ>  ands∈N. We introduce the metric vector space

Cˆ_τ^s(N,X) :=l^∞_loc N,C^s

[–τ, ],X :=

x= (xn)_n∈N xn∈C^s

[–τ, ],X

forn∈N, d^j

dt^jxn(–τ) = d^j

dt^jxn–() forj= , . . . ,s– ,n∈N

equipped with the distance function d_Cˆτ^s_(N,X)(x,y) :=

n∈N

^–n max_k=,...,nx_k–y_kC^s([–τ,],X)

 +max_k=,...,nxk–ykC^s([–τ,],X)

forx,y∈ ˆC^s_τ(N,X).

Obviously,Cˆ_τ^s(N,X) is a complete metric space which is isometrically isomorphic to the metric spaceC^s_τ([–τ,∞),X) :=C^s([–τ,∞),X) equipped with the distance

d_Cτ^s_{([,∞),X)}(x,y) :=

n∈N

^–n x–yC^s([–τ,τn],X)

 +x–yC^s([–τ,τn],X)

forx,y∈C^s

[–τ,∞),X .

For anyx: [–τ,∞)→X, we define forn∈Nthenth segment ofxvia xn:=x(nτ+s) fors∈[–τ, ].

By induction,xis a mild solution of ()-() if and only if (x_n)_n∈N∈ ˆC_τ^ (N,X) solves

˙

xn(s) =x˙n–() +^ _s

–τ

xn–(σ) dσ +

nτ+s

(n–)τ

f(σ) dσ fors∈[–τ, ] andn∈N, ()

x_(s) =ϕ(s) fors∈[–τ, ].

Theorem  Equation()has a unique solution(xn)_n∈N∈ ˆC_τ^ (N,X).Moreover,x con- tinuously depends on the data in sense of the estimate

xnC^([–τ,],X)≤κⁿ

ϕC^([–τ,],X)+fL^(,τn;X)

for any n∈N

withκ:=  + τ( + τ)( +^_L(X)).

Proof By the virtue of fundamental theorem of calculus, Equation () is satisfied if and only if

x_n(s) =x_n–() + (s+ τ)x˙_n–() +^ _s

–τ

σ –τ

x_n–(ξ) dξdσ +

_s

–τ

_nτ+σ

(n–)τ

f(ξ) dξdσ fors∈[–τ, ],n∈N, () xn(–τ) =xn–(), x˙n(–τ) =x˙n–() forn∈N, ()

x_(s) =ϕ(s) fors∈[–τ, ]. ()

By induction, we can easily show that for any n∈Nthere exists a unique local solu- tion (x,x, . . . ,xn)∈(C^([–τ, ],X))^n+to ()-() up to the indexn. Here, we used the Sobolev embedding theorem stating

W^,(,T;X)→C^

[,T],X

for anyT> .

Further, we can estimate x_nC^([–τ,],X)≤

 + τ+ τ^L(X)

x_n–C^([–τ,],X)+ τfL^((n–)τ,nτ;X). ()

Similarly, Equation () yields

˙xnC^([–τ,],X)≤

 + τ^_L(X)

xn–C^([–τ,],X)+fL^((n–)τ,nτ;X). () Equations () and () imply together

xnC^([–τ,],X)≤κ

ϕC^([–τ,],X)+fL^((n–)τ,nτ:X)

.

By induction, we then get, for anyn∈N, x_nC^([–τ,],X)≤κⁿ

ϕC^([–τ,],X)+fL^(,τn,X)

,

which finishes the proof.

Lettingx(t) :=xk(t– kτ) fort≥ andk:=_τ^t ∈N, we obtain the unique mild solu- tionxof Equations ()-().

Corollary  Equations()-()possess a unique mild solution x satisfying,for any T:=

nτ,n∈N,

xC^([–τ,T],X)≤κⁿ

ϕC^([–τ,T],X)+fL^(,T;X)

for any n∈N

withκ:=  + ( + τ)( +^_L(X)).

Theorem  Under an additional condition that ϕ ∈ C^([–τ, ],X) as well as f ∈ C^([,∞),X),the unique mild solution given in Corollaryis a classical solution.

Proof Differentiating Equation () with respect tot, using the assumptions and the fact thatx∈C^([–τ,∞),X), we deduce thatx|[–τ,]≡ϕ∈C^([–τ, ],X) and

¨

x=^x(·– τ) +f∈C^

[,∞),X .

Hence,x∈C^([–τ,∞),X)∩C^([–τ, ],X)∩C^([,∞),X) and is thus a classical solution

of Equations ()-().

3.2 Explicit representation of solutions

Following Khusainov and Shuklin [] and Khusainovet al.[], we define fort∈Rthe operator-valued delayed exponential function

exp_τ(t;) :=

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎩

L(X), –∞<t< –τ,

idX, –τ≤t< ,

id_X+_!^t, ≤t<τ,

idX+_!^t +^{ (t–τ}_!^), τ≤t< τ,

. . . , . . . ,

idX+_!^t +^{ (t–τ}_!^) +· · ·+^{k(t–(k–)τ}_k! ^)k, (k– )τ≤t<kτ,

. . . , . . . .

()

Throughout this section, we additionally assume that: X→X is an isomorphism from the Banach spaceXonto itself.

Theorem  The delayed exponential functionexp_τ(·;)lies in C^([–τ,∞),X)∩C^([,

∞),X)∩C^([τ,∞),X)and solves the Cauchy problem

¨

x(t) –^x(t– τ) = X for t≥τ, ()

x(t) =ϕ(t) for t∈[–τ,τ], ()

where

ϕ(t) =

idX, –τ ≤t< , idX+t, ≤t≤τ.

Proof To prove the smoothness ofx, we first note thatxis an operator-valued polyno- mial and thus analytic on each of the intervals [(k– )τ,kτ] fork∈Z. By the definition of exp_τ(·;), we further find

d^j

dt^jx(kτ– ) = d^j

dt^jx(kτ+ ) forj= , . . . ,k,k∈N. Hence,x∈C^([–τ,∞),X)∩C^([,∞),X)∩C^([τ,∞),X).

Fork∈N,k≥, we have x(t) = id_X+t

!+^(t–τ)^

! +^(t– τ)^

!

+^(t– τ)^

! +· · ·+^k(t– (k– )τ)^k

k! .

Fort≥τ, differentiation yields

˙

x(t) =+^t–τ

! +^(t– τ)^

! +^(t– τ)^

! +· · ·+^k(t– (k– )τ)^k–

(k– )!

=

id_X+t–τ

! +^(t– τ)^

! +^(t– τ)^

! +· · ·+^k–(t– (k– )τ)^k–

(k– )!

=exp_τ(t–τ;) =x(t–τ) and, therefore,

¨

x(t) =^+^t– τ

! +^(t– τ)^

! +· · ·+^k(t– (k– )τ)^k–

(k– )!

=^

idX+t– τ

! +^(t– τ)^

! +· · ·+^k–(t– (k– )τ)^k–

(k– )!

=^exp_τ(t– τ;) =^x(t– τ).

Hence,xsatisfies Equation (). Finally, by definition ofexp_τ(·;),xsatisfies Equation (),

too.

Corollary  The delayed exponential functionexp_τ(·; –)lies in C^([–τ,∞),X)∩C^([,

∞),X)∩C^([τ,∞),X)and solves the Cauchy problem()-()with the initial data

ϕ(t) =

id_X, –τ≤t< , idX–t, ≤t≤τ. We define the functions

x^_τ(t;) :=



exp_τ(t;) +exp_τ(t; –)

fort≥–τ, x^_τ(t;) :=

^–

exp_τ(t;) –exp_τ(t; –)

fort≥–τ.

()

As we already pointed out in the introduction section, in contrast to earlier works by Khusainovet al.[, , ], only the invertibility and not the negativity ofis necessary for our purposes.

From Equation (), we explicitly obtain

x^_τ(t;) =

⎧⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎩

idX, –∞<t<τ,

idX+^{ (t–τ}_!^), τ≤t< τ, idX+^{ (t–τ}_!^) +^{ (t–τ)}_! ^, τ≤t< τ,

. . . , . . . ,

idX+^{ (t–τ}_!^) +· · ·+^k(t–(k–)τ)^k

(k)! , (k– )τ ≤t< (k+ )τ and

x^_τ(t;) =

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎩

L(X), –∞<t< ,

id_X!^t, ≤t< τ,

idXt

!+^{ (t–τ)}_! ^, τ≤t< τ,

idXt

!+^{ (t–τ)}_! ^ +^{ (t–τ)}_! ^, τ≤t< τ,

. . . , . . . ,

idXt

!+^{ (t–τ)}_! ^ +· · ·+^{k(t–(k)τ)}_(k+)!^k+, kτ≤t< (k+ )τ.

Obviously,x^_τ andx^_τ are even functions with respect to. Figures  and  display the functionsx^_τ(·;) andx^_τ(·;) for various values ofτ and.

Theorem  The functions x^_τ(·;), x^_τ(·;) have the following regularity properties:

x^_τ(·;),x^_τ(·;)∈C^([–τ,∞),X)∩C^([–τ, ],X)∩C^([τ,∞),X). Further, x^_τ(·;) and x^_τ(·;) are solutions to the Cauchy problem ()-()with the initial dataϕ(t) = id_X, –τ≤t≤τ,andϕ(t) = L(X), –τ≤t≤τ,respectively.

First, assumingf ≡X, Equations ()-() reduce to

¨

x(t) –^x(t– τ) =  fort≥, ()

x(t) =ϕ(t) fort∈[–τ, ]. ()

Figure 1 Plot ofx_τ¹(·;) function.

Figure 2 Plot ofx_τ²(·;) function.

Theorem  Letϕ∈C^([–τ, ],X).Then the unique classical solution x to the Cauchy problem()-()is given by

x(t) =x^_τ(t+τ;)ϕ(–τ) +x^_τ(t+ τ;)ϕ(–τ˙ ) +

_

–τ

x^_τ(t–s;)ϕ(s) ds.¨

Proof To solve Equations ()-(), we use the ansatz

x(t) =x^_τ(t+τ;)c+x^_τ(t+ τ;)c

+ _

–τ

x^_τ(t–s;)¨c(s) ds ()

for somec,c∈Xandc∈C^([–τ, ],X).

Plugging the ansatz from Equation () into Equation (), we obtain, fort≥, d^

dt^

x^_τ(t+τ;)c_+x^_τ(t+ τ;)c_+ 

–τ

x^_τ(t–s;)¨c(s) ds

–^

x^_τ

(t+τ) – τ; c+x^_τ

(t+ τ) – τ; c

+ _

–τ

x^_τ

(t– τ) –s;

¨ c(s) ds

=  or, equivalently,

d^

dt^x^_τ(t+τ;) –^x^_τ

(t+τ) – τ; c

+ d^

dt^x^_τ(t+ τ;) –^x^_τ

(t+ τ) – τ; c

+ _

–τ

d^

dt^x^_τ(t–s;) –^x^_τ

(t– τ) –s;

¨

c(s) ds≡X.

Sincex^_τ(·;) andx^_τ(·;) solve the homogeneous equation, all three coefficients atc,c

and¨cvanish implying that the functionxin Equation () is a solution of Equation ().

Now, we show that selectingc_:=ϕ(–τ),c_:=ϕ(–τ˙ ) andc:=ϕ, the functionxin Equa- tion () satisfies the initial condition (). Letting, fort∈[–τ, ],

[Iϕ](t) :=

_

–τ

x^_τ(t–s;)ϕ(s) ds¨

and performing a change of variablesσ:=t–s, we find [Iϕ](t) = –

_t

t+τ

x^_τ(σ;)ϕ(t¨ –σ) dσ= _t+τ

t

x^_τ(σ;)ϕ(t¨ –σ) dσ.

Exploiting the fact thatx^_τ vanishes on [–τ, ], we get [Iϕ](t) =

_t+τ



x^_τ(σ;)ϕ(t¨ –σ) dσ.

Integrating by parts, we further get [Iϕ](t) =

t+τ



x^_τ(σ;)ϕ(t¨ –σ) dσ

= – _t+τ



x^_τ(σ;)d dt

ϕ(t˙ –σ) dσ

= –x^_τ(σ;)ϕ(t˙ –σ)|^σσ^=t+τ= + _t+τ



˙

x^_τ(σ;)ϕ(t˙ –σ) dσ. Now, taking into account

x^_τ(t;) =tidX, ≤t≤τ, ()

we obtain

[Iϕ](t) = –x^_τ(t+ τ;)ϕ(–τ˙ ) + t+τ



˙

x^_τ(σ;)ϕ(t˙ –σ) dσ.

Again, using Equation () and x^_τ(t;) = idX, –τ≤t≤τ, we compute

[Iϕ](t) = –x^_τ(t+ τ;)ϕ(–τ˙ ) + _t+τ



˙

ϕ(t–σ) dσ

= –x^_τ(t+ τ;)ϕ(–τ˙ ) –ϕ(t–σ)|^σσ^=t+τ=

= –x^_τ(t+ τ;)ϕ(–τ˙ ) –x^_τ(t+τ;)ϕ(–τ) +ϕ(t).

Hence, fort∈[–τ, ], we have

x(t) =x^_τ(t+τ;)ϕ(–τ) +x^_τ(t+ τ;)ϕ(–τ˙ ) + _

–τ

x^_τ(t–s;)ϕ(s) ds¨ =ϕ(t)

as claimed.

Next, we consider Equations ()-() for the trivial initial data,i.e.,

¨

x(t) –^x(t– τ) =f(t) fort≥, ()

x(t) =  fort∈[–τ, ]. ()

Theorem  Let f ∈C^([,∞),X).The unique classical solution x to the Cauchy problem ()-()is given by

x(t) = _t



x^_τ(t–s;)f(s) ds.

Proof To find an explicit solution representation, we use the ansatz x(t) =

_t



x^_τ(t–s;)c(s) ds fort≥τ

for some functionc∈C^([,∞),X). Differentiating this expression with respect totand exploiting the initial conditions forx^_τ(·;), we get

˙ x(t) =

_t



˙

x^_τ(t–s;)c(s) ds+x^_τ(t–s;)c(s)|s=t

= t



˙

x^_τ(t–s;)c(s) ds+x^_τ()c(t)

= _t



˙

x^_τ(t–s;)c(s) ds.

Differentiating again, we find

¨ x(t) =

_t



¨

x^_τ(t–s;)c(s) ds+x˙^_τ(t–s;)c(s)|s=t

= _t



¨

x^_τ(t–s;)c(s) ds+x˙^_τ(+;)c(t)

= t



¨

x^_τ(t–s;)c(s) ds+c(t).

Plugging this into Equation () and recalling thatx^_τ(·;) is a solution of the homogeneous equation, we get

c(t) + t



x¨^_τ(t–s;) –^x^_τ(t– τ–s;)

c(s) ds=f(t)

and thereforec≡f.

As a consequence from Theorems  and , we obtain using the linearity property of Equations ()-() the following.

Theorem  Letϕ∈C^([–τ, ],X)and f ∈C^([,∞),X).The unique classical solution to Equations()-()is given by

x(t) =x^_τ(t+τ;)ϕ(–τ) +x^_τ(t+ τ;)ϕ(–τ˙ ) + 

–τ

x^_τ(t–s;)ϕ(s) ds¨ +

, t∈[–τ, ),

_t

x^_τ(t–s;)f(s) ds, t≥ for t∈[–τ,∞).

Finally, after a partial integration, we get the following.

Theorem  Letϕ∈C^([–τ, ],X)and f ∈L^_loc(,∞;X).The unique mild solution to Equations()-()is given by

x(t) =x^_τ(t+τ;)ϕ(–τ) +x^_τ(t;)ϕ() –˙ 

–τ

˙

x^_τ(t–s;)ϕ(s) ds˙ +

, t∈[–τ, ),

_t

x^_τ(t–s;)f(s) ds, t≥ for t∈[–τ,∞).

Proof ApproximatingϕinC^([–τ, ],X) with (ϕn)n∈N⊂C^([–τ, ],X) andf inL^_loc(,

∞;X) with (f_n)_n∈N ⊂C^([,∞),X), applying Theorem  to solve the Cauchy problem ()-() for the right-hand sidef and the initial dataϕ_n, performing a partial integration for the integral involvingϕ¨_nand passing to the limit asn→ ∞, the claim follows.

3.3 Asymptotic behavior asτ_→0

Again, we assumeXto be a Banach space and prove the following generalization of Lem- ma  in [].

Lemma  Let∈L(X),T> ,τ_> and let α:=  + exp

τ_L(X)

Then,for anyτ∈(,τ_], exp_τ(t–τ;) –exp(t)

L(X)≤τL(X)exp

α(T+τ_)L(X)

for t∈[,T].

Proof First, we want to exploit the mathematical induction to show, for anyk∈N, exp_τ(t–τ;) –exp(t)

L(X)≤τL(X)exp

αkτL(X)

()

fort∈[(k– )τ,kτ]. Letτ∈(,τ_]. Fort∈[,τ], the claim easily follows from the mean value theorem for Bochner integration since

exp_τ(t–τ;) –exp(t)

L(X)

=exp(t) – id_X

L(X)≤τL(X)exp

τL(X)

≤τL(X)exp

ατL(X)

,

where we used the factα≥. Assuming now that inequality () is valid up to somek∈N, we use the fundamental theorem of calculus to estimate, fort∈[kτ, (k+ )τ],

exp_τ(t–τ;) –exp(t)

L(X)

= exp_τ

(k– )τ;

–exp(kτ ) + t

kτ

d ds

exp_τ(s–τ;) –exp(s) ds

L(X)

≤exp_τ

(k– )τ;

–exp(kτ )

L(X)

+ _t

kτ

d ds

exp_τ(s–τ;) –exp(s)

L(X)

ds

≤τL(X)exp

αkτL(X)

+

_(k+)τ

kτ

d

dsexp_τ(s–τ;) – d

dsexp(s) L(X)

ds

≤τL(X)exp

αkτL(X)

+L(X)

(k+)τ kτ

exp_τ(s– τ;) –exp(s)

L(X)ds

≤τL(X)exp

αkτL(X)

+L(X)

_(k+)τ

kτ

exp_τ(s– τ;) –exp

(s–τ)

L(X)ds

+L(X)

(k+)τ kτ

exp(s) –exp

(s–τ)

L(X)ds

≤τL(X)exp

αkτL(X)

+L(X)

_kτ

(k–)τ

exp_τ(s–τ;) –exp(s)

L(X)ds +L(X)

_(k+)τ

kτ

_s

s–τ

d

dσ exp(σ ) L(X)

dσds

≤τL(X)exp

αkτL(X)

+τ^_L(X)exp

αkτL(X)

+τ^_L(X)exp

(k+ )τL(X)

≤τL(X)exp

αkτL(X)

 +τL(X)+τL(X)exp

τL(X)

≤τL(X)exp

αkτL(X)

 + τL(X)exp

τL(X)

≤τL(X)exp

αkτL(X)

exp

τL(X)exp

τL(X)

≤τL(X)exp

α(k+ )τL(X)

.

By induction, we obtain, for anyk∈N, exp_τ(t–τ;) –exp(t)

L(X)≤τL(X)exp

αkτL(X)

()

fort∈((k– )τ,kτ]. Now, taking into account that for anyt∈[,T],τ∈(,τ_] andk∈N such thatt∈[(k– )τ,kτ], we havekτ≤T+τ. This together with () yields the claim.

Corollary  Let the assumptions of Lemmabe satisfied and letγ ≥.Then,for t∈ [,T]andτ∈(,τ_],we have

exp_τ(t+γ;) –e^t

L(X)≤(γ +τ)L(X)exp

α(T+γ +τ_)L(X)

.

Proof Lemma  and the mean value theorem for Bochner integration yield exp_τ(t+γ;) –e^t

L(X)

≤exp_τ(t+γ;) –e^(t+γ+τ)

L(X)+e^(t+γ+τ)–e^t

L(X)

≤τL(X)exp

α(T+γ +τ_)L(X)

+ (γ+τ)L(X)exp

(T+γ+τ)L(X)

≤(γ+τ)L(X)exp

α(T+γ+τ_)L(X)

as we claimed.

LetT> ,τ_> ,x,x∈Xandf ∈L^_loc(,∞;X) be fixed and letx¯∈C^([,∞),X) denote the unique mild solution to the Cauchy problem ()-() from the section on classical harmonic oscillator.