Research Article
Solving the Linear 1D Thermoelasticity Equations with Pure Delay
Denys Ya. Khusainov
1and Michael Pokojovy
21Department of Cybernetics, Kyiv National Taras Shevchenko University, 64 Volodymyrska Street, Kyiv 01601, Ukraine
2Department of Mathematics and Statistics, University of Konstanz, Universit¨atsstraße 10, 78457 Konstanz, Germany
Correspondence should be addressed to Michael Pokojovy; michael.pokojovy@uni-konstanz.de Received 27 October 2014; Accepted 3 January 2015
Academic Editor: Harvinder S. Sidhu
Copyright © 2015 D. Ya. Khusainov and M. Pokojovy. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We propose a system of partial differential equations with a single constant delay𝜏 > 0describing the behavior of a one-dimensional thermoelastic solid occupying a bounded interval ofR1. For an initial-boundary value problem associated with this system, we prove a well-posedness result in a certain topology under appropriate regularity conditions on the data. Further, we show the solution of our delayed model to converge to the solution of the classical equations of thermoelasticity as𝜏 → 0. Finally, we deduce an explicit solution representation for the delay problem.
1. Introduction
Over the past half-century, the equations of thermoelas- ticity have drawn a lot of attention from the side of both mathematical and physical communities. Starting with the late 50s and early 60s of the last century, the necessity of a rational physical description for elastic deformations of solid bodies accompanied by thermal stresses motivated the more prominent mathematicians, physicists, and engineers to focus on this problem (see, e.g., [1,2]). As a consequence, many theories emerged, mainly in the cross-section of (nonlinear) field theory and thermodynamics, making it possible for the equations of thermoelasticity to be interpreted as an anelastic modification of the equations of elasticity (cf. [3] and the references therein). Both linear and nonlinear models and solution theories were proposed.
An initial-boundary value problem for the general linear equations of classical thermoelasticity in a bounded smooth domainΩ ⊂R𝑛
𝜌𝜕𝑡𝑡𝑢𝑖= (𝐶𝑖𝑗𝑘𝑙𝑢𝑘,𝑙),𝑗− (𝑚𝑖𝑗𝑇),𝑗+ 𝜌𝑓𝑖 inΩ × (0, ∞) , (1) 𝜌𝑐𝐷𝜕𝑡𝑇 + 𝑚𝑖𝑗𝜃0𝜕𝑡𝑢𝑖,𝑗= (𝐾𝑖𝑗𝑇,𝑗),𝑖+ 𝜌𝑐𝐷𝑟 inΩ × (0, ∞)
(2)
was studied by Dafermos in [4]. Here, [𝑢𝑖] and 𝑇denote the (unknown) displacement vector field and the absolute temperature, respectively. Further, 𝜌 > 0 is the material density,𝜃0is a reference temperature rendering the body free of thermal stresses, 𝑐𝐷 is the specific heat capacity, [𝐶𝑖𝑗𝑘𝑙] stands for Hooke’s tensor, [𝑚𝑖𝑗] is the stress-temperature tensor,[𝐾𝑗]is the heat conductivity tensor,[𝑓𝑖]represents the specific external body force, and𝑟is the external heat supply.
Under usual initial conditions, appropriate normalization conditions to rule out the rigid motion as a trivial solution, and general boundary conditions
𝑢𝑖= 0 inΓ1× (0, ∞) ,
(𝐶𝑖𝑗𝑘𝑙𝑢𝑘,𝑙− 𝑚𝑖𝑗𝑇) 𝑛𝑗+ 𝐴𝑖𝑗𝑢𝑗= 0 in int(Γ1𝑐) × (0, ∞) , 𝑇 = 0 inΓ2× (0, ∞) ,
(𝐾𝑖𝑗𝑇,𝑗) 𝑛𝑖+ 𝐵𝑇 = 0 in int(Γ2𝑐) × (0, ∞) , (3) whereΓ1, Γ2⊂ 𝜕Ωare relatively open,[𝐴𝑖𝑗]denotes the “elas- ticity” modulus, and𝐵is heat transfer coefficient, Dafermos proved the global existence and uniqueness of finite energy solutions and studied their regularity as well as asymptotics
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International Journal of Mathematics and Mathematical Sciences Volume 2015, Article ID 479267, 11 pages
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as𝑡 → ∞. In 1D, even an exponential stability result for(1)- (2)under all “reasonable” boundary conditions was shown by Hansen in [5].
In his work [6], Slemrod studied the nonlinear equations of 1D thermoelasticity in the Lagrangian coordinates
𝜕𝑡𝑡𝑢 = ̂𝜓𝐹𝐹(𝜕𝑥𝑢 + 1, 𝜃 + 𝑇0) 𝜕𝑥𝑥𝑢
+ ̂𝜓𝐹𝑇(𝜕𝑢𝑥+ 1, 𝜃 + 𝑇0) 𝜕𝑥𝜃 in (0, 1) × (0, ∞) , 𝜌 (𝜃 + 𝑇0) ( ̂𝜓𝑇𝑇(𝜕𝑥𝑢 + 1, 𝜃 + 𝑇0) 𝜕𝑡𝜃
+ ̂𝜓𝐹𝑇(𝜕𝑥𝑢 + 1, 𝜃 + 𝑇0) 𝜕𝑥𝑡𝑢)
= ̂𝑞(𝜕𝑥𝜃) 𝜕𝑥𝑥𝜃 in (0, 1) × (0, ∞)
(4)
for the unknown functions 𝑢 denoting the displacement of the rod and 𝜃 being a temperature difference to a ref- erence temperature 𝑇0 rendering the body free of thermal stresses. The functions ̂𝜓 and ̂𝑞denote the Helmholtz free energy and the heat flux, respectively, and are assumed to be given. Finally, 𝜌 > 0 is the material density in the references configuration. Under appropriate boundary conditions (when the boundary is free of tractions and is held at a constant temperature or when the body is rigidly clamped and thermally insulated) as well as usual initial conditions for both unknown functions, a local existence theorem for (4) was proved by additionally imposing a regularity and compatibility condition. For sufficiently small initial data, the local classical solution could be globally continued. At the same time, when studying(4)in the whole space, large data are known to lead to a blow-up in final time (cf. [7]).
Racke and Shibata studied in [8](4)under homogeneous Dirichlet boundary conditions for both 𝑢 and 𝜃. Under appropriate smoothness assumptions, they proved the global existence and exponential stability for the classical solutions to the problem. In contrast to Slemrod [6], their method was using spectral analysis rather than ad hoc energy estimates obtained by differentiating the equations with respect to 𝑡 and𝑥. A detailed overview of further recent developments in the field of classical thermoelasticity and corresponding references can be found in the monograph [9] by Jiang and Racke.
The classical equations of thermoelasticity outlined above, being a hyperbolic-parabolic system, provide a rather good macroscopic description in many real-world applica- tions. At the same time, they sometimes fail when being used to model thermoelastic stresses in some other situations, in particular, in extremely small bodies exposed to heat pulses of large amplitude (see, e.g., [10]) and so forth. To address these issues, a new theory, commonly referred to as the theory of hyperbolic thermoelasticity or second sound thermoelasticity, has emerged. In contrast to the classical thermoelasticity, parabolic equation (2) is replaced with a hyperbolic first-order system
𝜌𝑐𝐷𝜕𝑡𝑇 + 𝑚𝑖𝑗𝜃0𝜕𝑡𝑢𝑖,𝑗= 𝑞𝑖,𝑖+ 𝜌𝑐𝐷𝑟 inΩ × (0, ∞) , (5) 𝜏𝑖𝑗𝜕𝑡𝑞𝑖+ 𝑞𝑖+ 𝐾𝑖𝑗𝑇,𝑗= 0 inΩ × (0, ∞) (6) with[𝑞𝑖]and[𝜏𝑖𝑗]denoting the heat flux and the relaxation tensor, respectively. Both linear and nonlinear versions of
the equations of hyperbolic thermoelasticity(1),(5)-(6)have been studied in the literature. See, for example, [11] by Messaoudi and Said-Houari for a proof of global well- posedness of the 1D system in the whole space or Irmscher’s work [12] for the global well-posedness of nonlinear problem for rotationally symmetric data in a bounded rotationally symmetric domain ofR3. In a bounded 1D domain, a quan- titative stability comparison between the classical and the hyperbolic system was presented by Irmscher and Racke in [13]. For a detailed overview on hyperbolic thermoelasticity, we refer the reader to [14] by Chandrasekharaiah and [15] by Racke.
A unified approach establishing a connection between the classical and hyperbolic thermoelasticity was developed by Tzou in [16,17]. Namely, he proposed to view(6)with𝜏𝑖𝑗≡ 𝜏 as a first-order Taylor approximation of the equation
𝑞𝑖(x, 𝑡 + 𝜏) + 𝐾𝑖𝑗𝑇,𝑗(x, 𝑡) = 0 for (x, 𝑡) ∈ Ω × (−𝜏, ∞) (7) being equivalent to the delay equation
𝑞𝑖(x, 𝑡) + 𝐾𝑖𝑗𝑇,𝑗(x, 𝑡 − 𝜏) = 0 for (x, 𝑡) ∈ Ω × (0, ∞) . (8) More generally, higher-order Taylor expansion to the dual phase lag constitutive equation
𝑞𝑖(x, 𝑡 + 𝜏1) + 𝐾𝑖𝑗𝑇,𝑗(x, 𝑡 + 𝜏2)
= 0 for (x, 𝑡) ∈ Ω × (−max{𝜏1, 𝜏2} , ∞) (9) can be considered. Together with(1)-(2),(5), this leads to the so-called dual phase lag thermoelasticity studied by Racke (cf.
references in [15, page 415]).
If no Taylor expansion with respect to 𝜏 is carried out in(8), it can be shown that the corresponding system is ill- posed when being considered in the same topology as the original system of classical thermoelasticity (cf. [18]); that is, the system is lacking a continuous dependence of solution on the data. Moreover, the delay law (8) can, in general, contradict the second law of thermodynamics as shown in [19].
Nonetheless, it remains desirable to understand the dynamics of equations of thermoelasticity originated from delayed material laws. One of the first attempts to obtain a well-posedness result for a partial differential equation with pure delay is due to Rodrigues et al. In their paper [20], Rodrigues et al. studied a heat equation with pure delay in an appropriate Frech´et space and showed the delayed Laplacian to generate a𝐶0-semigroup on this space. Further, they investigated the spectrum of the infinitesimal generator.
Though their approach can essentially be carried over to the equations of thermoelasticity with pure delay derived in Section 2, we propose a new approach in this paper preserving the Hilbert space structure of the space and thus the connection to the classical equations of thermoelasticity.
To the authors’ best knowledge, no results on thermoelasticity with delay in the highest order terms have been previously published in the literature. At the same time, we refer the
reader to the works by Khusainov et al. [21–24], in which the authors studied the well-posedness and controllability for the heat and/or the wave equation on a finite time horizon. In their recent paper [25], Khusainov et al. exploited the𝐿2-maximum regularity theory to prove a global well- posedness and asymptotic stability results for a regularized heat equation with delay.
The present paper has the following outline. InSection 2, we give a physical model for linear thermoelasticity based on delayed material laws. For the sake of simplicity, we present a 1D model though our approach can easily be carried over to the general multidimensional case. Next, in Section 3, we prove the well-posedness of this model in an appropriate Hilbert space framework and discuss the small parameter asymptotics, that is, the behavior of solutions as 𝜏 → 0. Further, in Section 4, we deduce an explicit solution representation formula. Finally, in the Appendix, we summarize some seminal results on the delayed exponential function and Cauchy problems with pure delay.
2. Model Description
We consider a solid body occupying an axis-aligned rectan- gular domain ofR3. Assuming that the body motion is purely longitudinal with respect to the first space variable𝑥(cf. [6, page 100]), deformation gradient, stress and strain tensors, and so forth are diagonal matrices and a complete rational description of the original 3D body motion can be reduced to studying the 1D projectionΩ = (0, 𝑙),𝑙 > 0, of the body onto the𝑥-axis as displayed inFigure 1. Hence, in the following, we restrict ourselves to considering the relevant physical values only in𝑥-direction.
Let the functions 𝑢 : Ω × [0, ∞) → R and 𝜃 : Ω × [0, ∞) → R denote the body displacement and its relative temperature measured with respect to a reference temperature 𝜃0 > 0 rendering the body free of thermal stresses, respectively. We restrict ourselves to the Lagrangian coordinates and write 𝜎, 𝜀, 𝑆, 𝑞 : Ω × [0, ∞) → R for the stress field, strain field, entropy field, or the heat flux, respectively. With𝜌 > 0denoting the material density, the momentum conservation law as well as the linearized entropy balance law reads as
𝜌𝜕𝑡𝑡𝑢 (𝑥, 𝑡) + 𝜕𝑥𝜎 (𝑥, 𝑡) = 𝜌𝑟 (𝑥, 𝑡) for𝑥 ∈ Ω, 𝑡 > 0, 𝜃0𝜕𝑡𝑆 (𝑥, 𝑡) + 𝜕𝑥𝑞 (𝑥, 𝑡) = ℎ (𝑥, 𝑡) for𝑥 ∈ Ω, 𝑡 > 0, (10) where𝑟 : Ω × [0, ∞) → Randℎ : Ω × [0, ∞) → Rare a known volume force acting on the body and an internal heat source.
Assuming physical linearity for the strain field, the strain can be decomposed into elastic strain𝜀𝑒and thermal stress𝜀𝑡. Further, assuming|𝜃(𝑡, 𝑥)/𝜃0| ≪ 1uniformly with respect to 𝑥 ∈ Ω,𝑡 ≥ 0, we can postulate
𝜀𝑡(𝑥, 𝑡) = 𝛼𝜃 (𝑥, 𝑡) for𝑥 ∈ Ω, 𝑡 > 0, (11)
x
y z
l
Figure 1: 3D rectangular solid body.
where 𝛼 > 0 denotes the thermal expansion coefficient.
Exploiting the second law of thermodynamics for irreversible processes, we obtain (cf. [3, page 3])
𝑆 (𝑥, 𝑡) = 𝛼𝐵𝜀𝑒(𝑥, 𝑡) +𝜌𝑐𝜌
𝜃0 𝜃 (𝑥, 𝑡) for𝑥 ∈Ω, 𝑡 > 0 (12) with𝑐𝜌> 0standing for the specific heat capacity and𝐵 ∈R denoting the bulk modulus.
In our further considerations, we depart from the classical material laws and use their delay counterparts. Let𝜏 > 0be a positive time delay. In the sequel, all functions are supposed to be defined onΩ × [−𝜏, ∞). Assuming a delay feedback between the stress and the strain as well as the heat flux and the temperature gradient, Hooke’s law with pure delay reads as (cf. [1])
𝜎 (𝑥, 𝑡) = (𝐵 +4
3𝐺) 𝜀𝑒(𝑥, 𝑡 − 𝜏) + 𝐵𝜀𝑡(𝑥, 𝑡 − 𝜏) for𝑥 ∈Ω, 𝑡 > 0
(13)
with 𝐺 > 0 denoting the shear modulus. Similarly, we consider a delay version of Fourier’s law given as
𝑞 (𝑥, 𝑡) = −𝜅𝜕𝑥𝜃 (𝑥, 𝑡 − 𝜏) for𝑥 ∈ Ω, 𝑡 > 0, (14) where𝜅 > 0stands for the thermal conductivity. Assuming the elastic strain tensor to be equal to the displacement gradient, we have
𝜀𝑒(𝑥, 𝑡) = 𝜕𝑥𝑢 (𝑥, 𝑡) for𝑥 ∈ Ω, 𝑡 > 0. (15) We further postulate the following relation:
𝜕𝑡𝜕𝑥𝑢 (𝑥, 𝑡) = 𝜕𝑥𝜕𝑡𝑢 (𝑥, 𝑡 − 𝜏) for𝑥 ∈Ω, 𝑡 > 0 (16) which is a delayed counterpart of Schwarz’s theorem. Finally, we also modify(12)to introduce a delay feedback between the entropy, the elastic strain tensor, and the temperature
𝑆 (𝑥, 𝑡) = 𝛼𝐵𝜀𝑒(𝑥, 𝑡 − 𝜏) + 𝜌𝑐𝜌
𝜃0 𝜃 (𝑥, 𝑡 − 𝜏) for 𝑥 ∈ Ω, 𝑡 > 0.
(17)
Exploiting now(10),(13)–(17), we obtain 𝜌𝜕𝑡𝑡𝑢 (𝑥, 𝑡) − (𝐵 + 4
3𝐺) 𝜕𝑥𝑥𝑢 (𝑥, 𝑡 − 𝜏) + 𝛼𝐵𝜕𝑥𝜃 (𝑥, 𝑡 − 𝜏)
= 𝑓 (𝑥, 𝑡) for𝑥 ∈ Ω, 𝑡 > 0,
𝜌𝑐𝜌𝜕𝑡𝜃 (𝑥, 𝑡) − 𝜅𝜕𝑥𝑥𝜃 (𝑥, 𝑡 − 𝜏) + 𝛼𝜃0𝐵𝜕𝑡𝑥𝑢 (𝑥, 𝑡 − 𝜏)
= ℎ (𝑥, 𝑡) for𝑥 ∈ Ω, 𝑡 > 0,
𝜕𝑡𝜕𝑥𝑢 (𝑥, 𝑡) − 𝜕𝑥𝜕𝑡𝑢 (𝑥, 𝑡 − 𝜏) = 0 for𝑥 ∈ Ω, 𝑡 > 0.
(18) To close(18), appropriate boundary and initial conditions for 𝑢and𝜃are required. In the following, we prescribe homoge- neous Dirichlet boundary conditions for𝑢and homogeneous Neumann boundary conditions for𝜃given as
𝑢 (0, 𝑡) = 𝑢 (𝑙, 𝑡) = 0,
𝜕𝑥𝜃 (0, 𝑡) = 𝜕𝑥𝜃 (𝑙, 𝑡) = 0 for𝑡 > 0.
(19)
This particular choice of boundary conditions not only turns out to be convenient for our further mathematical considerations but also is a physically relevant one. Similar to the thermoelasticity with second sound, it is one of the combinations typically arising when studying micro- and nanoscopic strings or plates (cf. [13]).
The initial conditions are given over the whole history period(𝜏, 0)and read as
𝑢 (𝑥, 0) = 𝑢0(𝑥) , 𝑢 (𝑥, 𝑡) = 𝑢0𝜏(𝑥, 𝑡) for𝑥 ∈ Ω, 𝑡 ∈ (−𝜏, 0) ,
𝜕𝑡𝑢 (𝑥, 0) = 𝑢1(𝑥) , 𝜕𝑡𝑢 (𝑥, 𝑡) = 𝑢1𝜏(𝑥, 𝑡) for𝑥 ∈ Ω, 𝑡 ∈ (−𝜏, 0) , 𝜃 (𝑥, 0) = 𝜃0(𝑥) , 𝜃 (𝑥, 𝑡) = 𝜃0𝜏(𝑥, 𝑡) for 𝑥 ∈ Ω, 𝑡 ∈ (−𝜏, 0)
(20)
with known𝑢0, 𝑢1, 𝜃0: Ω → Rand𝑢0𝜏, 𝑢1𝜏, 𝜃0𝜏: Ω×(−𝜏, 0) → R.
3. Well-Posedness and Limit 𝜏 → 0
Letting𝑎 := (𝐵+(4/3)𝐺)/𝜌,𝑏 := 𝛼𝐵/𝜌,𝑐 := 𝜅/𝜌𝑐𝜌,𝑑 := 𝛼𝜃0𝐵/
𝜌𝑐𝜌and 𝑓(𝑥, 𝑡) := 𝑟(𝑥, 𝑡), and𝑔(𝑥, 𝑡) := (1/𝜌𝑐𝜌)ℎ(𝑥, 𝑡)for 𝑥 ∈ Ω,𝑡 ≥ 0,(18)can be rewritten as
𝜕𝑡𝑡𝑢 (𝑥, 𝑡) − 𝑎𝜕𝑥𝑥𝑢 (𝑥, 𝑡 − 𝜏) + 𝑏𝜕𝑥𝜃 (𝑥, 𝑡 − 𝜏)
= 𝑓 (𝑥, 𝑡) for 𝑥 ∈ Ω, 𝑡 > 0,
𝜕𝑡𝜃 (𝑥, 𝑡) − 𝑐𝜕𝑥𝑥𝜃 (𝑥, 𝑡 − 𝜏) + 𝑑𝜕𝑡𝑥𝑢 (𝑥, 𝑡 − 𝜏)
= 𝑔 (𝑥, 𝑡) for𝑥 ∈ Ω, 𝑡 > 0,
𝜕𝑥𝑢 (𝑥, 𝑡) − 𝜕𝑥𝑢 (𝑥, 𝑡 − 𝜏) = 0 for𝑥 ∈Ω, 𝑡 > 0 (21)
subject to the boundary conditions from (19) and initial conditions from(20). Introducing a new vector of unknown functions,
V(𝑥, 𝑡) = (𝑉1(𝑥, 𝑡) 𝑉2(𝑥, 𝑡)
𝑉3(𝑥, 𝑡)) := (𝜕𝑡𝑢 (𝑥, 𝑡)
𝜕𝑥𝑢 (𝑥, 𝑡) 𝜃 (𝑥, 𝑡) ) for𝑥 ∈ Ω, 𝑡 ∈ [−𝜏, 𝑇] .
(22)
Equations(21)can be transformed to
𝜕𝑡V(𝑥, 𝑡) +BV(𝑥, 𝑡 − 𝜏) =F(𝑥, 𝑡) for𝑥 ∈ Ω, 𝑡 ∈ (0, 𝑇) (23) with the differential matrix operator and the right-hand side
B:= ( 0 −𝑎𝜕𝑥 𝑏𝜕𝑥
−𝜕𝑥 0 0
𝑑𝜕𝑥 0 −𝑐𝜕𝑥𝑥) , F(𝑥, 𝑡) := (𝑓 (𝑥, 𝑡) 𝑔 (𝑥, 𝑡)0 )
for 𝑥 ∈ Ω, 𝑡 > 0, (24) respectively.
Exploiting (19)and the definition of 𝑉, the boundary conditions for𝑉read as
𝑉1(0, 𝑡) = 𝑉1(𝑙, 𝑡) = 0, 𝜕𝑥𝑉3(0, 𝑡) = 𝜕𝑥𝑉3(𝑙, 𝑡) = 0 for 𝑡 > 0,
(25) whereas the initial conditions are given by
V(𝑥, 0) =V0(𝑥) , V(𝑥, 𝑡) =V0𝜏(𝑥, 𝑡) for𝑥 ∈ Ω, 𝑡 ∈ (−𝜏, 0)
(26)
with
V0(𝑥) = (𝑢0 𝑢1
𝜃0) , V0𝜏(𝑥, 𝑡) = (
𝑢1𝜏(𝑥, 𝑡)
𝜕𝑥𝑢0𝜏(𝑥, 𝑡) 𝜃𝜏0(𝑥, 𝑡)
)
for𝑥 ∈ Ω, 𝑡 ∈ [−𝜏, 0] . (27)
Note that(18)–(20),(23),(25), and(26)are equivalent for; if the vector𝑉is known,𝑢and𝜃are uniquely determined by
𝑢 (𝑥, 𝑡) ={{ {{ {
𝑢0(𝑥) + ∫𝑡
0𝑉1(𝑥, 𝑠)d𝑠, for𝑡 ≥ 0, 𝑢0𝜏(𝑥, 𝑡) , for𝑡 ∈ [−𝜏, 0) ,
𝜃 (𝑡, 𝑥) ={ {{
𝑉3(𝑥, 𝑡) , for𝑡 ≥ 0, 𝜃0𝜏(𝑥, 𝑡) , for𝑡 ∈ [−𝜏, 0) .
(28)
Therefore, in the sequel, we consider the following equivalent first-order-in-time problem:
𝜕𝑡V(𝑥, 𝑡) +BV(𝑥, 𝑡 − 𝜏) =F(𝑥, 𝑡) for𝑥 ∈ Ω, 𝑡 > 0, 𝑉1(0, 𝑡) = 𝑉1(𝑙, 𝑡) = 0,
𝜕𝑥𝑉3(0, 𝑡) = 𝜕𝑥𝑉3(𝑙, 𝑡) = 0 for 𝑡 > 0, V(𝑥, 0) =V0(𝑥) , V(𝑥, 𝑡) =V0𝜏(𝑥, 𝑡) for𝑥 ∈ Ω, 𝑡 ∈ (−𝜏, 0) .
(29) For our well-posedness investigations, we need a solution notion for(29). To this end, appropriate functional spaces have to be introduced. We start with the “na¨ıve” approach by using the case𝜏 = 0as a reference situation. We introduce the Hilbert space𝑋 := 𝐿2(Ω) × 𝐿2(Ω) × 𝐿2(Ω)equipped with the dot product
⟨V,W⟩𝑋:= ⟨𝑉1, 𝑊1⟩𝐿2(Ω)+ 𝑎 ⟨𝑉2, 𝑊2⟩𝐿2(Ω) +𝑏
𝑑⟨𝑉3, 𝑊3⟩𝐿2(Ω) for V,W∈ 𝑋
(30)
and define the operator
B: 𝐷 (B) ⊂ 𝑋 → 𝑋, 𝑉 →B𝑉 (31) with the domain
𝐷 (B) := {V∈ 𝐻01(Ω) × 𝐻1(Ω) × 𝐻2(Ω)𝜕𝑥𝑉3𝜕Ω = 0} . (32) See [26, Section 3] for the definition of Sobolev spaces. With this notation,(29)can be written in the equivalent form
𝜕𝑡V(𝑥, 𝑡) +BV(𝑥, 𝑡 − 𝜏) =F(𝑥, 𝑡) for 𝑥 ∈ Ω, 𝑡 > 0, V(𝑥, 0) =V0(𝑥) ,
V(𝑥, 𝑡) =V0𝜏(𝑥, 𝑡) for𝑥 ∈ Ω, 𝑡 ∈ (−𝜏, 0) .
(33) Under a classical solution to(33)on[−𝜏, 𝑇]for any𝑇 > 0, one would naturally understand a functionV ∈ 𝐶0([−𝜏, 𝑇], 𝐷(B)) ∩ 𝐶1([0, 𝑇], 𝑋)satisfying the equations pointwise.
We know from [5] that the linear operatorBis accretive and satisfies 𝐷(B) = 𝐷(B∗). Its spectrum 𝜎(B) only consists of isolated eigenvalues𝜆𝑛 ∈ C, 𝑛 ∈ N0, of finite multiplicity with Re𝜆𝑛≥ 0,𝑛 ∈N, and𝜆𝑛 → ∞as𝑛 → ∞.
The corresponding eigenfunctions(Ψ𝑛)𝑛∈N⊂ 𝐷(B)build an orthonormal basis of𝑋. Unfortunately, from [18, Theorem 1.1] we know that(33)is ill-posed in𝑋. Hence, a different
solution notion should be adopted. As we already mentioned inSection 1, we want to preserve the Hilbert space structure of the problem and thus cannot follow the approach developed by Rodrigues et al. in [20].
For𝑇 > 0, we define the space𝑋𝑇:= {𝑉 ∈ ⋂∞𝑘=0𝐷(B𝑘) |
‖𝑉‖𝑋𝑇< ∞}equipped with the scalar product induced by the norm
‖V‖𝑋𝑇 := 𝑒𝑇|B|V𝐷(B2)
= (∑∞
𝑛=0(1 + 𝜆𝑛2+ 𝜆𝑛4)
⋅ exp(2𝑇 𝜆𝑛)⟨V, Ψ𝑛⟩𝑋2)
1/2
forV∈ 𝑋𝑇.
(34)
Hence,𝑋𝑇is closed subspace of𝑋and thus a Hilbert space.
Moreover,𝑋𝑇is dense in𝑋since(Φ𝑛)𝑛 ⊂ 𝑋𝑇. Indeed, for 𝑛 ∈N, we haveΨ𝑛∈ ⋂∞𝑘=0𝐷(B𝑘)and‖Ψ𝑛‖2𝑋𝑇= (1 + |𝜆𝑛|2+
|𝜆𝑛|4)exp(2𝑇|𝜆𝑛|) < ∞.
RestrictingBto the closed subspace𝑋𝑇of𝑋, we trivially obtain a bounded linear operatorB𝑇: 𝑋𝑇 → 𝐷(B2)since forV∈ 𝑋𝑇
B𝑇V2𝐷(B2)= B𝑇V2𝑋+ BB𝑇V2𝑋+ B2B𝑇V2𝑋
=∑∞
𝑛=1(1 + 𝜆𝑛2+ 𝜆𝑛4) 𝜆𝑛2⟨V, Ψ𝑛⟩𝑋2
≤ 1
2𝑇2
∑∞
𝑛=1(1 + 𝜆𝑛2+ 𝜆𝑛4) exp(2𝑇 𝜆𝑛)
⋅ ⟨V, Ψ𝑛⟩𝑋2≤ 1
2𝑇2‖V‖2𝑋𝑇. (35) Now, restricting(33)to𝑋𝑇, we obtain
𝜕𝑡V(𝑥, 𝑡) +B𝑇V(𝑥, 𝑡 − 𝜏) =F(𝑥, 𝑡) for𝑥 ∈ Ω, 𝑡 ∈ (0, 𝑇) , V(𝑥, 0) =V0(𝑥) , V(𝑥, 𝑡) =V0𝜏(𝑥, 𝑡) for 𝑥 ∈ Ω, 𝑡 ∈ (−𝜏, 0) .
(36)
Following the approach in [21], we introduce for𝑡 ∈ Rthe delayed exponential function
exp𝜏(B𝑇, 𝑡) :=
{{ {{ {{ {{ {
0𝐿(𝑋), 𝑡 < −𝜏,
id𝑋+⌊𝑡/𝜏⌋+1∑
𝑘=1
(𝑡 − (𝑘 − 1) 𝜏)𝑘
𝑘! B𝑘𝑇, 𝑡 ≥ −𝜏.
(37)
0 0.5 1 1.5 2 0
1 2 3 4 5 6 7
−0.5
−1 0 0 0.5 1 1.5 2
1 2 3 4 5 6 7
−0.5
−1 0 0 0.5 1 1.5 2
1 2 3 4 5 6 7
−0.5
−1
𝜏 = 0.01 𝜏 = 0.1 𝜏 = 0.3
𝜏 = 0.01 𝜏 = 0.1 𝜏 = 0.3
𝜏 = 0.01 𝜏 = 0.1 𝜏 = 0.3
exp𝜏(−1,t) exp𝜏(0,t) exp𝜏(1,t)
t t t
Figure 2: Delayed exponential function.
Figure 2 displays the delayed exponential function for the case whereB𝑇is a real number.
Obviously, for any𝑡 ∈ R, exp𝜏(−B𝑇, 𝑡) ∈ ⋂∞𝑘=0𝐷(B𝑘).
Moreover, we have
exp𝜏(−B𝑇, 𝑡)𝐿(𝑋𝑇,𝐷(B2))≤ 1 for𝑡 ∈ (−∞, 𝑇] (38) uniformly in𝜏 > 0since
exp𝜏(−B𝑇, 𝑡)V2𝐷(B2)
= exp𝜏(−B𝑇, 𝑡)V2𝑋
+ Bexp𝜏(−B𝑇, 𝑡)V2𝑋+ B2exp𝜏(−B𝑇, 𝑡)V2𝑋
=∑∞
𝑛=0(1 + 𝜆𝑛2+ 𝜆𝑛4) exp𝜏(−𝜆𝑛, 𝑡)2⟨V, Ψ𝑛⟩𝑋2
≤∑∞
𝑛=0(1 + 𝜆𝑛2+ 𝜆𝑛4)
⋅ (1 +⌊𝑡/𝜏⌋+1∑
𝑘=1
(𝑡 − (𝑘 − 1) 𝜏)𝑘 𝑘! 𝜆𝑛𝑘)
2
⋅ ⟨V, Ψ𝑛⟩𝑋2
≤∑∞
𝑛=0(1 + 𝜆𝑛2+ 𝜆𝑛4)exp(2𝑇 𝜆𝑛)
⋅ ⟨V, Ψ𝑛⟩𝑋2= ‖V‖2𝑋𝑇 for V∈ 𝑋𝑇.
(39)
Here,𝐿(𝑋, 𝑌)denotes the space of bounded, linear operators from𝑋to𝑌equipped with the standard operator topology.
Now, we can prove the following well-posedness result.
Theorem 1. For𝑇 > 0, letV0 ∈ 𝑋𝑇,V0𝜏 ∈ 𝐶0([−𝜏, 0], 𝑋𝑇) with V0𝜏(⋅, 0) = V0 and let F ∈ 𝐶0([0, 𝑇], 𝑋𝑇). Then (36) possess a unique classical solutionV ∈ 𝐶0([−𝜏, 𝑇], 𝐷(B)) ∩ 𝐶1([0, 𝑇], 𝑋)explicitly given by
V(⋅, 𝑡)
= {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {
V0𝜏(⋅, 𝑡) , 𝑡 ∈ [−𝜏, 0) ,
V0, 𝑡 = 0,
exp𝜏(−B𝑇, 𝑡 − 𝜏)V0
−B𝑇∫0
−𝜏exp𝜏(−B𝑇, 𝑡 − 2𝜏 − 𝑠)
⋅V0𝜏(𝑠) 𝑑𝑠 +∫𝑡
0exp𝜏(−B𝑇, 𝑡 − 𝜏 − 𝑠)F(⋅, 𝑠) 𝑑𝑠, 𝑡 ∈ (0, 𝑇] . (40) Proof. First, we prove uniqueness of classical solutions.
AssumingV and W to be classical solutions of(36), we con- clude that their differenceZ:=V−W∈ 𝐶0([−𝜏, 𝑇], 𝐷(B)) ∩ 𝐶1([0, 𝑇], 𝑋)is a classical solution to
𝜕𝑡Z(𝑥, 𝑡) +B𝑇Z(𝑥, 𝑡 − 𝜏) =0 for𝑥 ∈ Ω, 𝑡 ∈ (0, 𝑇) , Z(𝑥, 0) =0, Z(𝑥, 𝑡) =0 for𝑥 ∈ Ω, 𝑡 ∈ (−𝜏, 0) .
(41)
Multiplying these equations with Ψ𝑛 in the inner product of 𝑋, we further deduce that Z𝑛(𝑡, ⋅) := ⟨Z(𝑡, ⋅), Ψ𝑛⟩𝑋, 𝑡 ∈ [−𝜏, 𝑇], is a classical solution to the scalar delay differen- tial equation
̇Z𝑛(𝑡) + 𝜆𝑛Z𝑛(𝑡 − 𝜏) =0 for𝑡 ∈ (0, 𝑇) ,
Z𝑛(0) =0, Z𝑛(𝑡) =0 for𝑥 ∈ Ω, 𝑡 ∈ (−𝜏, 0) . (42) FromTheorem A.4in the Appendix, the later equations are known to be uniquely solvable byZ𝑛 ≡0. Hence, Z𝑛 ≡0 for all𝑛 ∈N. With(Ψ𝑛)𝑛∈Nbeing a basis of𝑋, this impliesZ≡0, and, therefore,V≡W.
For the existence proof, we show that the function V in (40) is a classical solution to (36). Performing the diagonalization, we obtain for𝑡 ∈ [0, 𝑇]
V(⋅, 𝑡)
=∑∞
𝑛=1
exp𝜏(−𝜆𝑛, 𝑡 − 𝜏) ⟨V0, Ψ𝑛⟩𝑋Ψ𝑛
−∑∞
𝑛=1
𝜆𝑛(∫0
−𝜏exp𝜏(−𝜆𝑛, 𝑡 − 2𝜏 − 𝑠) ⟨V0𝜏(𝑠) , Ψ𝑛⟩𝑋d𝑠) Ψ𝑛 +∑∞
𝑛=1
(∫𝑡
0exp𝜏(−𝜆𝑛, 𝑡 − 𝜏 − 𝑠) ⟨F(⋅, 𝑠) , Ψ𝑛⟩𝑋d𝑠) Ψ𝑛
≡∑∞
𝑛=1
V𝑛(𝑡) Ψ𝑛.
(43) From [25], we knowV𝑛 ∈ 𝐶0([−𝜏, 𝑇],C) ∩ 𝐶1([0, 𝑇],C)for all𝑛 ∈N. Further, by the virtue of(38), the series converges uniformly in𝐷(B)and its derivative converges uniformly in 𝑋. Hence, the limiting function lies in𝐶([−𝜏, 𝑇], 𝐷(B)) ∩ 𝐶1([0, 𝑇], 𝑋)and thus possesses the regularity of a classical solution. Finally, using the properties of scalar delay expo- nential (cf. [25]), we easily verify thatV solves(36).
Taking into account inequality(38)and applying H¨older’s inequality to(40), we obtain the following estimate.
Corollary 2. The solutionVcontinuously depends on the data in sense of the estimate
‖V‖𝐶0([0,𝑇],𝑋)≤ V0𝑋𝑇+ 𝜏 V0𝜏𝐶0([0,𝑇],𝑋𝑇)
+ √𝑇 ‖F‖𝐿2(0,𝑇;𝑋𝑇) for𝑇 > 0. (44) For the rest of this section, we want to study the behavior of system(33)as𝜏 → 0. Let𝜏0 > 0and𝑇 > 0be fixed.
Similar to𝑋𝑇, we consider the Hilbert space𝑌𝑇 := {𝑉 ∈
⋂∞𝑘=0𝐷(B𝑘) | ‖𝑉‖𝑌𝑇 < ∞}equipped with the scalar product induced by the norm
‖V‖𝑌𝑇 := (∑∞
𝑛=0(1 + 𝜆𝑛2)
⋅exp(2 (1 + 𝜆𝑛exp(𝜏0𝜆𝑛))𝑇𝜆𝑛)
⋅ ⟨V, Ψ𝑛⟩𝑋2)
1/2
forV∈ 𝑌𝑇.
(45)
Obviously,𝑌𝑇→ 𝑋𝑇. For simplicity, despite a slight abuse of notation, we letB𝑇now denote the part ofBin𝑌𝑇(namely, [27, page 139]). Formally, the limiting system of(33)as𝜏 → 0 is given by
𝜕𝑡V(𝑥, 𝑡) +B𝑇V(𝑥, 𝑡) =F(𝑥, 𝑡) for𝑥 ∈ Ω, 𝑡 ∈ (0, 𝑇) , V(𝑥, 0) =V0(𝑥) for𝑥 ∈ Ω.
(46) From [27, Corollary 3.3.13], we know that −B𝑇 gener- ates a uniformly bounded𝐶0-semigroup(exp(−B𝑇𝑡))𝑡≥0of bounded linear operators on𝑌𝑇. The unique classical solution to(46)can then be written using Duhamel’s formula as
V(⋅, 𝑡) =exp(−B𝑇𝑡)V0 + ∫𝑡
0exp(−B𝑇(𝑡 − 𝑠))F(𝑠)d𝑠 for𝑡 ∈ [0, 𝑇] . (47) Theorem A.3from the Appendix implies the following.
Lemma 3. There holds
exp𝜏(−B𝑇, 𝑡 − 𝜏) −exp(−B𝑇𝑡)𝐿(𝑌𝑇,𝐷(B))≤ 𝜏
𝑓𝑜𝑟 𝑡 ∈ [0, 𝑇] . (48) Now, we can prove the following.
Theorem 4. Let V0 ∈ 𝑌𝑇, F ∈ 𝐶0([0, ∞), 𝑌𝑇) be fixed.
For 𝜏 > 0, let V0𝜏 ∈ 𝐶0([−𝜏, 0], 𝑌𝑇) with V0𝜏(0) = V0 andlim sup𝜏 → 0‖V0𝜏‖𝐿1(0,𝜏;𝑌𝑇) < ∞. Denoting with𝑉(⋅; 𝜏)the classical solution of(36)corresponding to the initial dataV0, V0𝜏and the right-hand sideF, one has
V(⋅; 𝜏) −V(⋅; 𝜏)𝐶0([0,𝑇],𝑋)= 𝑂 (𝜏) 𝑎𝑠 𝜏 → 0. (49) Proof. Using the representation formulas forV and V, we can estimate for any𝑡 ∈ [0, 𝑇]
V(⋅; 𝜏) −V(⋅; 𝜏)𝑋
≤ exp𝜏(−B𝑇, 𝑡 − 𝜏) −exp(−B𝑇𝑡)𝐿(𝑌𝑇,𝑋)V0𝑌𝑇
+ ∫0
−𝜏B𝑇exp𝜏(−B𝑇, 𝑡 − 2𝜏 − 𝑠)𝐿(𝑌𝑇,𝑋)V0𝜏(𝑠)𝑌𝑇d𝑠 + ∫𝑡
0exp𝜏(−B𝑇, 𝑡 − 𝜏 − 𝑠) −exp(−B𝑇(𝑡 − 𝑠))𝐿(𝑌𝑇,𝑋)
⋅ ‖F(⋅, 𝑠)‖𝑌𝑇d𝑠
≤ 𝜏 V0𝑌𝑇+ 𝜏 (1 + 𝜏)lim sup
𝜏 → 0 V0𝜏𝐿1(−𝜏,0;𝑌𝑇)
+ 𝜏𝑇 ‖F‖𝐿∞(0,𝑇;𝑌𝑇)= 𝑂 (𝜏) as𝜏 → 0.
(50) This finishes the proof.
4. Explicit Solution Representation
In this section, we want to deduce an explicit representation of solutions to(36)in the form of a Fourier series with respect to an orthogonal basis(Φ𝑛)𝑛∈N0of𝑋(and thus of𝑋𝑇) given by
Φ𝑛(𝑥)
= {{ {{ {{ {{ {
√ 12𝑙(0, 1, 1)𝑇, if𝑛 = 0,
√ 23𝑙(sin(]𝑛𝑥) ,cos(]𝑛𝑥) ,cos(]𝑛𝑥))𝑇, otherwise for𝑥 ∈ Ω, 𝑛 ∈N0
(51) with
]𝑛:= 𝜋𝑛
𝐿 for 𝑛 ∈N0. (52)
Note that the sequence (Φ𝑛)𝑛∈N0 does not coincide, in general, with the eigenfunctions (Ψ𝑛)𝑛∈N0 but, at the same time, (Φ𝑛)𝑛∈N0 ⊂ 𝐷(B𝑇) constitutes a basis of 𝐷(B𝑇).
To this end, we assume that the conditions of Theorem 1 are satisfied which yields a unique classical solution V ∈ 𝐶0([−𝜏, ∞), 𝐷(B)) ∩ 𝐶1([0, ∞), 𝑋).
DenotingΦ𝑛= (Φ1𝑛, Φ2𝑛, Φ3𝑛)𝑇and computing the compo- nentwise Fourier coefficients
𝑉𝑛0,𝑘= ⟨𝑉0,𝑘, Φ𝑘𝑛⟩𝐿2(Ω),
𝑉𝜏,𝑛0,𝑘(𝑡) = ⟨𝑉𝜏0,𝑘(⋅, 𝑡) , Φ𝑘𝑛⟩𝐿2(Ω) for𝑡 ∈ [−𝜏, 0] , 𝐹𝑛𝑘(𝑡) = ⟨𝐹𝑘(⋅, 𝑡) , Φ𝑘𝑛⟩𝐿2(Ω) for𝑡 ≥ 0
(53)
for 𝑛 ∈ N0 and 𝑘 = 1, 2, 3, we get the following Fourier expansions:
V0=∑∞
𝑛=0
(𝑉𝑛0,1Φ1𝑛, 𝑉𝑛0,2Φ2𝑛, 𝑉𝑛0,3Φ3𝑛) ,
V0𝜏(⋅, 𝑡) =∑∞
𝑛=0
(𝑉𝜏,𝑛0,1Φ1𝑛, 𝑉𝑛0,2Φ2𝑛, 𝑉𝑛0,3Φ3𝑛) for𝑡 ∈ [−𝜏, 0] ,
F(⋅, 𝑡) =∑∞
𝑛=0(𝐹𝑛1Φ1𝑛, 𝐹𝑛2Φ2𝑛, 𝐹𝑛3Φ3𝑛) for 𝑡 ≥ 0
(54) uniformly in Ω(due to the Sobolev embedding theorem).
Similarly, the solutionV can be expanded into Fourier series V(⋅, 𝑡) =∑∞
𝑛=0
(𝑉𝑛1(𝑡) Φ1𝑛, 𝑉𝑛2(𝑡) Φ1𝑛, 𝑉𝑛3(𝑡) Φ1𝑛) (55) for some𝑉𝑛,𝑘 ∈ 𝐶0([−𝜏, ∞),C) ∩ 𝐶1([0, ∞),C),𝑛 ∈ N0,𝑘 = 1, 2, 3, to be determined later. Using this ansatz and letting
B𝑛:= ( 0 𝑎]𝑛 −𝑏]𝑛
−]𝑛 0 0
𝑑]𝑛 0 𝑐]2𝑛 ) , (56)
we observe that(36)decompose into a sequence of ordinary delay differential equations
̇V𝑛(𝑡) = −B𝑛V𝑛(𝑡 − 𝜏) +F𝑛(𝑡) for𝑡 > 0,
V𝑛(0) =V0𝑛, V𝑛(𝑡) =V0𝜏,𝑛(𝑡) for 𝑡 ∈ (−𝜏, 0) . (57) By the virtue ofTheorem A.4, for any𝑛 ∈ N0, the unique solution to(57)is given by
V𝑛(𝑡)
= {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {
V0𝜏,𝑛(𝑡) , 𝑡 ∈ [−𝜏, 0) ,
V0𝑛, 𝑡 = 0,
exp𝜏(−B𝑛, 𝑡 − 𝜏)V0𝑛
−B𝑛∫0
−𝜏exp𝜏(−B𝑛, 𝑡 − 2𝜏 − 𝑠)
⋅V0𝜏,𝑛(𝑠)d𝑠 +∫𝑡
0exp𝜏(−B𝑛, 𝑡 − 𝜏 − 𝑠)F𝑛(𝑠)d𝑠, 𝑡 ∈ (0, 𝑇] . (58) To explicitly compute the function given in(58), we need to diagonalize the matrixB𝑛.
Lemma 5. Let
Δ0= 𝑐2]4𝑛− 3 (𝑎 + 𝑏𝑑)]2𝑛,
Δ1= −2𝑐3]6𝑛+ 9𝑐 (𝑎 + 𝑏𝑑)]4𝑛− 27𝑎𝑐]4𝑛, 𝐶 =√ 13
2(Δ1+ √Δ21− 4Δ30),
(59)
where √⋅ and √⋅3 stand for the main branch of complex square and cubic roots. The spectrum ofB𝑛 consists of three eigenvalues
𝜇𝑛,𝑘={ {{
0, 𝑛 = 0,
1
3(𝑐]2𝑛− 𝐶𝑒2𝑖𝑘𝜋/3− 𝑒−2𝑖𝑘𝜋/3Δ0
𝐶) , 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 (60) for𝑘 = 0, 1, 2with𝑖denoting the imaginary unit.
Proof. For𝑛 = 0, we have ]𝑛 = 0 and therefore B𝑛 = 03×3. Hence,0is the only eigenvalue ofB𝑛with an algebraic multiplicity of3.
Now, let us assume𝑛 > 1. To compute the eigenvalues of B𝑛, we consider the characteristic polynomial
𝑃𝑛(𝜇) :=det(B𝑛− 𝜇I3×3) = 𝜇3− 𝑐]2𝑛𝜇2
+ (𝑎 + 𝑏𝑑)]2𝑛𝜇 − 𝑎𝑐]4𝑛 for 𝜇 ∈C. (61) Since the matrix
B𝑛:= ( 1 0 0 0 1
𝑎 0 0 0 𝑏 𝑑
) (
0 𝑎]𝑛 −𝑏]𝑛
−𝑎]𝑛 0 0
𝑏]𝑛 0 𝑐𝑑
𝑏 ]2𝑛) (62)
has real components and is skew-symmetrizable, it has to possess one real and two complex-conjugate eigenvalues.
Thus, introducing the expressions Δ0= 𝑐2]4𝑛− 3 (𝑎 + 𝑏𝑑)]2𝑛,
Δ1= −2𝑐3]6𝑛+ 9𝑐 (𝑎 + 𝑏𝑑)]4𝑛− 27𝑎𝑐]4𝑛, 𝐶 =√ 13
2(Δ1+ √Δ21− 4Δ30),
(63)
we obtain the three roots𝜇𝑛,1,𝜇𝑛,2,𝜇𝑛,3of ̃𝑃𝑛 (cf. [28, page 179])
𝜇𝑛,𝑘= 1
3(𝑐]2𝑛− 𝐶𝑒2𝑖𝑘𝜋/3− 𝑒−2𝑖𝑘𝜋/3Δ0
𝐶) , (64) where√⋅and√⋅3 stand for the main branch of complex square and cubic roots.
Lemma 6. EigenvectorsV𝑛,𝑘,𝑘 = 0, 1, 2, ofB𝑛 corresponding to the eigenvalues𝜇𝑛,𝑘ofB𝑛fromLemma 5are given by
k𝑛,𝑘= {{ {{ {{ {{ {
e𝑘, if𝑛 = 0,
(
−𝑏]𝑛𝜇𝑛,𝑘 𝑏]2𝑛 𝑎]2𝑛+ 𝜇𝑛,𝑘2
) , 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 (65)
withe1= (1, 0, 0)𝑇,e2= (0, 1, 0)𝑇, ande3= (0, 0, 1)𝑇. Proof. Since the first case𝑛 = 0is obvious, we only consider the case𝑛 > 1. For𝑘 ∈ {0, 1, 2}, we consider the matrix
𝜇𝑛,𝑘I3×3−B𝑛= (𝜇𝑛,𝑘 −𝑎]𝑛 𝑏]𝑛 ]𝑛 𝜇𝑛,𝑘 0
−𝑑]𝑛 0 𝜇𝑛,𝑘− 𝑐]2𝑛) . (66) The latter is singular since𝛼𝑛,𝑘is an eigenvalue ofB𝑛. Further, due to the fact that
det(B𝑛) = 𝑎𝑐]4𝑛 > 0, (67) B𝑛 is invertible and, therefore,𝜇𝑛,𝑘 ̸= 0. We want to find a nontrivial vectork𝑛,𝑘∈R3satisfying
(𝛼𝑛,𝑘I3×3−B𝑛)V𝑛,𝑘=03×1. (68) Thus, we can apply a Gauss-Jordan iteration to the former matrix and find
𝜇𝑛,𝑘I3×3−B𝑛∼ (
𝜇𝑛,𝑘 −𝑎]𝑛 𝑏]𝑛
0 𝜇2𝑛,𝑘− 𝑎]2𝑛 −𝑏]2𝑛 0 −𝑎𝑑]2𝑛 𝜇2𝑛,𝑘− 𝑐]2𝑛𝜇𝑛,𝑘+ 𝑏𝑑]2𝑛
) . (69) Since the latter matrix must be singular, the third row must be proportional to the second one. Thus,(68)is equivalent to
(𝜇𝑛,𝑘 −𝑎]𝑛 𝑏]𝑛
0 𝜇2𝑛,𝑘+ 𝑎]2𝑛 −𝑏]2𝑛)k𝑛,𝑘=03×1. (70)
Since the rank of this matrix is 2, the equation above yields only one eigenvector
k𝑛,𝑘= (
−𝑏]𝑛𝜇𝑛,𝑘 𝑏]2𝑛 𝑎]2𝑛+ 𝜇𝑛,𝑘2
) (71)
being determined up to a multiplicative constant.
Note thatk𝑛,1,k𝑛,2, andk𝑛,3are linearly independent, but, in general, not orthonormal.
Letting now
D𝑛:=diag(𝜇𝑛,1, 𝜇𝑛,2, 𝜇𝑛,3) , (72) we obtain a singular value decomposition forB𝑛
B𝑛=S𝑛D𝑛S−1𝑛 (73) with an invertible matrix
S𝑛= (k𝑛,1 k𝑛,2 k𝑛,3)𝑇. (74) Exploiting nowCorollary A.2from Appendix,(58)can finally be written as
V𝑛(𝑡)
= {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {
V0𝜏,𝑛(𝑡) , 𝑡 ∈ [−𝜏, 0) ,
V0𝑛, 𝑡 = 0,
S𝑛exp𝜏(−D𝑛, 𝑡 − 𝜏)S−1𝑛 𝑉𝑛0
−S𝑛D𝑛∫0
−𝜏exp𝜏(−D𝑛, 𝑡 − 2𝜏 − 𝑠)
⋅S−1𝑛 V0𝜏,𝑛(𝑠)d𝑠 +∫𝑡
0S𝑛exp𝜏(−D𝑛, 𝑡 − 𝜏 − 𝑠)
⋅S−1𝑛 F𝑛(𝑠)d𝑠, 𝑡 ∈ (0, 𝑇] , (75)
where the inverse ofS𝑛is given by the Laplace formula S−1𝑛
= (
𝑆22𝑛 𝑆33𝑛 − 𝑆23𝑛 𝑆32𝑛 −𝑆12𝑛 𝑆33𝑛 + 𝑆13𝑛 𝑆32𝑛 𝑆12𝑛 𝑆23𝑛 − 𝑆13𝑛 𝑆22𝑛
−𝑆21𝑛 𝑆33𝑛 + 𝑆𝑛23𝑆𝑛31 𝑆11𝑛 𝑆33𝑛 − 𝑆13𝑛 𝑆31𝑛 −𝑆11𝑛 𝑆23𝑛 + 𝑆13𝑛 𝑆21𝑛 𝑆21𝑛 𝑆32𝑛 − 𝑆22𝑛 𝑆32𝑛 −𝑆11𝑛 𝑆32𝑛 + 𝑆12𝑛 𝑆31𝑛 𝑆11𝑛 𝑆22𝑛 − 𝑆12𝑛 𝑆21𝑛
)
⋅ (𝑆11𝑛 𝑆22𝑛 𝑆33𝑛 + 𝑆12𝑛 𝑆23𝑛 𝑆31𝑛 + 𝑆13𝑛 𝑆21𝑛 𝑆32𝑛 − 𝑆31𝑛 𝑆22𝑛 𝑆13𝑛
− 𝑆32𝑛 𝑆23𝑛 𝑆11𝑛 − 𝑆33𝑛 𝑆21𝑛 𝑆12𝑛 )−1.
(76)
