On the Cauchy Problem for a Linear Harmonic Oscillator with Pure Delay

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


Introduction
Let X be a (real or complex) Banach space and let x(t) ∈ X describe the state of a physical system at time t ≥ 0. With a(t) =ẍ(t) denoting the acceleration of system, the Newton's second law of motion states that F (t) = Ma(t) for t ≥ 0, (1.1) where M : D(M) ⊂ X → X is a linear, continuously invertible, accretive operator representing the "mass" of the system. When being displaced from its equilibrium situated in the origin, the system is affected by a restoring force F (t). In classical mechanics, this force is postulated to be proportional to the instantaneous displacement, i.e., for some closed, linear operator K : D(K) ⊂ X → X. When M −1 K is a bounded linear operator, plugging Equation (1.2) into (1.1), we arrive at the classical harmonic oscillator modelẍ (t) = M −1 Kx(t) for t ≥ 0. (1.3) Assuming now that the restoring force is proportional to the value of the system at some past time t − τ , Equation (1.2) is replaced with the relation where τ > 0 is a time delay. Plugging Equation (1.4) into (1.1) leads then to the linear harmonic oscillator equation with pure delay written as Problems similar to Equation (1.5) also arise when modeling systems with distributed parameters such as general wave phenomena (cf. [14]).
Equations similar to (1.5) are often referred to as delay or retarted 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 aṡ u(t) = H(t, u(t), u t ) for t > 0, u(0) = u 0 , u 0 = ϕ. (1.6) Here, τ > 0 is a fixed delay parameter, u t := u(t + ·) ∈ L 1 (−τ, 0; X), t ≥ 0, denotes the history variable, H is an X-valued operator defined on a subset of [0, ∞)×X×L 1 (−τ, 0; X) and u 0 ∈ X, ϕ ∈ L 1 (−τ, 0; X) are appropriate initial data. Equations of type (1.6) have been intensively studied in the literature. We refer the reader to the monographs by Els'gol'ts & Norkin [7] and Hale & Lunel [8] for a detailed treatment of Equations (1.6) in finite-dimensional spaces X. In contrast to this, results on Equation (1.6) in infinitedimensional spaces X are less numerous. A good overview can be found in the monograph of Bátkai & Piazzera [2].
Khusainov et al. considered in [9] Equation (1.6) in R n with for symmetric matrices A 1 , A 2 ∈ R n×n and column vectors b 1 , b 2 , b 3 ∈ R n and proposed a rational Lyapunov function to study the asymptotic stability of solutions to this system.
In their work [10], Khusainov, Agarwal et al. studied a modal, or spectrum, control problem for a linear delay equation on R n reading aṡ with a feedback control u(t) = m j=0 c T j x(t − jτ ) for some delay time τ > 0 and parameter vectors c j ∈ R n . For canonical systems, they developed a method to compute the unknown parameters such that the closed-loop system possesses the spectrum prescribed beforehand. Under appropriate "concordance" conditions, they were able to carry over their considerations for a rather broad class of non-canonical systems.
In the infite-dimensional situation, a rather general particular case of (1.6) with H(t, v, ψ) = Av+F (ψ) where A generates a C 0 -semigroup (S(t)) t≥0 on X and F is a nonlinear operator on L 2 (−τ, 0; X) was studied by Travies & Webb in their work [21]. Under appropriate assumptions on F , they proved the integral equation corresponding to the weak formulation of the delay equation given by to possess a unique solution in H 1 loc (0, ∞; X). Di Blasio et al. addressed in [4] a similar probleṁ where A generates a holomorphic C 0 -semigroup on a Hilbert space H, B is a perturbation of A and L 1 , L 2 are appropriate linear operators. If u 0 and ϕ possess a certain regularity, they proved the existence of a unique strong solution in H 1 loc (0, ∞; X) ∩ L 2 loc 0, ∞; D(A) by analyzing the C 0 -semigroup inducing the the semiflow t → (u(t), u t ). These results were elaborated on by Di Blasio et al. in [5] leading to a generalization for the case of weighted and interpolation spaces and including a desription of the associated infinitesimal generator. Finally, the general L p -case for p ∈ (0, ∞) was investigated by Di Blasio in [3].
Recently, in their work [15], Khusainov et al. proposed an explicit L 2 -solution theory for a non-homogeneous initial-boundary value problem for an isotropic heat equation with constant delay where Ω ⊂ R d is 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 considerable amound of attention, too. In [17], Nicaise & Pignotti studied a homogeneous isotropic wave equation with an internal feedback with and without delay reading as under usual initial conditions where Γ 0 , Γ 1 ⊂ ∂Ω are relatively open in ∂Ω withΓ 0 ∩Γ 1 = ∅ and ν denotes the outer unit normal vector of a smooth bounded domain Ω ⊂ R d . They showed the problem to possess a unique global classical solution and proved the latter to be exponentially stable if a 0 > a > 0 or instable, otherwise. These results have been carried over by Nicaise & Pignotti [18] and Nicaise et al. [19] to the case time-varying internally distributed or boundary delays.
In [14], Khusainov et al. considered a non-homogeneous initial-boundary value problem for a one-dimensional wave equation with constant coefficients and a single constant delay Under appropriate regularity and compatibility assumptions, they proved the problem to possess a unique C 2 -solution for any finite T > 0. Their proof was based on extrapolation methods for C 0 -semigroups and an explicit solution representation formula.
Recently, Khusainov & Pokojovy presented in [13] a Hilbert-space treatment of the initialboundary value problem for the equations of thermoelasticity with pure delay 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 (1.5) 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 Khusainov et al. [11,12,14], we only require the invertibility and not the positivity of M −1 K in Equation (1.5).
In Section 2, we briefly outline some seminal results on second-order abstract Cauchy problems. In our main Section 3, we prove the existence and uniqueness of solutions to the Cauchy problem for the delay equation (1.5) 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 & Shuklin in [16]. 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 parameter τ goes to zero.

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 abstact differential equation subject to the initial conditions Here, we assume the linear operator Ω : D(Ω) ⊂ X → X to be continuously invertible and generate a C 0 -group (e tΩ ) t∈R ⊂ L(X) on a (real or complex) Banach space X with L(X) denoting the space of bounded, linear operators on X equipped with the norm A more rigorous treatment of this problem can be found in [1, Section 3.14].
The general solution to the homogeneous equation is known to read as x h (t) = e Ωt c 1 + e −Ωt c 2 for t ≥ 0 with some c 1 , c 2 ∈ D(Ω). Vectors c 1 , c 2 can be computed using the initial conditions from Equation (2.2) leading to a system of linear operator equations The latter is uniquely solved by . Thus, the unique solution of the homogeneous equation with the initial conditions (2.2) is given by or, equivalently, A particular solution to the non-homogeneous equation with zero initial conditions will be determined in the Cauchy form We refer the reader to [1,Chapter 1] for the definition of Bochner integrals for X-valued functions. In Equation (2.5), the function K ∈ C 0 ([0, ∞) × [0, ∞), L(X)) is the Cauchy kernel, i.e., for any fixed s ≥ 0, the function K(·, s) is the solution of the homogeneous problem satisfying the initial conditions Using the ansatz K(t, s) = e Ωt c 1 (s) + e −Ωt c 2 (s) for t, s ≥ 0 for some c 1 , c 2 ∈ C 1 ([0, ∞), L(X)) and taking into account the initial conditions, we arrive at Solving this system with generalized Cramer's rule, we obtain for s ≥ 0 Thus, the Cauchy kernel is given by ) for t, s ≥ 0, whereas the particular solution satisfying zero initial conditions reads as Hence, for x 0 ∈ D(Ω), x 1 ∈ X and f ∈ L 1 loc (0, ∞; X), the unique mild solution x ∈ W 1,1 loc (0, ∞; X) to the Cauchy problem (2.1)-(2.2) can be written as If the data additionally satisfy

The linear oscillator with pure delay
In this section, we consider a Cauchy problem for the linear oscillator with a single pure subject to the initial condition Here, X is a Banach space, Ω ∈ L(X) is a bounded, linear operator and ϕ ∈ C 1 [−2τ, 0], X , f ∈ L 1 loc (0, ∞; X) are given functions. In contrast to Section 2, the boundedness of Ω is indespensable here. Indeed, Dreher et al. proved in [6] that Equations (3.1)-(3.2) are illposed even if X is a Hilbert space and Ω possesses a sequence of eigenvalues (λ n ) n∈N ⊂ R with λ n → ∞ or λ n → −∞ as n → ∞. The necessity for Ω being bounded has also been pointed out by Rodrigues et al. in [20] when treating a linear heat equation with pure delay.
A mild formulation of (3.1)-(3.2) is given bẏ 3) In the following subsection, we want to study the existence and uniquess of mild and classical solutions to the Cauchy problem (3.1)-(3.2) as well as their continuous dependence on the data.
By induction, we then get for any n ∈ N which finishes the proof.

(3.12)
Throughout this Section, we additionally assume that Ω : X → X is an isomorphism from the Banach space X onto itself.
Next, we consider Equations (3.1)-(3.2) for the trivial initial data, i.e., Proof. To find an explicit solution representation, we use the ansatz for some function c ∈ C 0 [0, ∞), X . Differentiating this expression with respect to t and exploiting the initial conditions for x 2 τ (·; Ω), we geṫ Differentiating again, we find Plugging this into Equation (3.20) and recalling that x 2 τ (Ω; Ω) is a solution of the homogeneous equation, we get and therefore c ≡ f .
As a consequence from Theorems 3.10 and 3.11, we obtain using the linearity property of Equations (3.1)-(3.2): The unique classical solution to Equations (3.1)-(3.2) is given by Finally, we get: Theorem 3.13. Let ϕ ∈ C 1 [−2τ, 0], X and f ∈ L 1 loc (0, ∞; X). The unique mild solution to Equations (3.1)-(3.2) is given by , X , applying Theorem 3.12 to solve the Cauchy problem (3.1)-(3.2) for the right-hand side f and the initial data ϕ n , performing a partial integration for the integral involvingφ n and passing to the limit as n → ∞, the claim follows.

Asymptotic behavior as τ → 0
Again, we assume X to be a Banach space and prove the following generalization of [13,Lemma 4]. Proof. Let τ ∈ (0, τ 0 ]. For t ∈ [0, τ ], the claim easily follows from the mean value theorem for Bochner integration. Next, we want to exploit the mathematical induction to show for any k ∈ N. Indeed, assuming that the claim is true for some k ∈ N, we use the fundamental theorem of calculus and find for t ∈ (kτ, (k + 1)τ ] since α ≥ 1. The claim follows by induction. as we claimed.
Applying Theorem 3.16, we further get Combining these inequalities and using again Theorem Theorem 3.16, we deduce the estimate asserted.