Differentiability of Solutions with Respect to the Delay Function in Functional Differential Equations

In this paper we consider a class of functional differential equations with time-dependent delay. We show continuous differentiability of the solution with respect to the time delay function for each fixed time value assuming natural conditions on the delay function. As an application of the differentiability result, we give a numerical study to estimate the time delay function using the quasilinearization method.


Introduction
In this paper we consider a class of functional differential equations (FDEs) with a timedependent delay of the form ẋ(t) = f (t, x(t), x(t − τ(t))), t ≥ 0, (1.1) where the associated initial condition is Here and throughout the manuscript r > 0 is a fixed constant, and 0 ≤ τ(t) ≤ r for all t ≥ 0.
In this paper we consider the delay function τ as parameter in the initial value problem (IVP) (1.1)-(1.2),and we denote the corresponding solution by x(t, τ).The main goal of this paper is to discuss the differentiability of the solution x(t, τ) with respect to (wrt) τ.By differentiability we mean Fréchet-differentiability throughout this paper.Differentiability of solutions of FDEs wrt to other parameters is studied, e.g., in the monograph [6].The first Email: hartung.ferenc@uni-pannon.huF. Hartung paper which discussed and proved the differentiability of solutions of FDEs wrt constant delay was [7].The result was formulated for the class of FDEs of the form ẋ(t) = g(x(t), x(t − η)), (1.3) where g : R n × R n → R n is a continuously differentiable function.It was shown that if α > 0 is such that the solutions x(t, η) of (1.3) are defined for t ∈ [0, α] and η ∈ (δ 1 , δ 2 ) with some 0 < δ 1 < δ 2 , then the map is continuously differentiable.Here W 1,1 ([0, α], R n ) is the space of absolutely continuous functions of finite norm Differentiability of the solution x(t, τ) wrt τ at a fixed time t was an open question, but in many applications this stronger sense of differentiability is needed.This problem was investigated later in [11] and recently in [12].We note that in both papers the proofs are incorrect.
In this paper we prove, under natural conditions, that the solution x(t, τ) of the Equation (1.1) is differentiable wrt the time delay function τ for each fixed time t (see Theorem 4.4 below).The proof uses the method developed in [9] to show differentiability of solutions wrt parameters in FDEs with state-dependent delays.As a consequence of our main result, we get the differentiability of the solutions x(t, η) of (1.3) wrt the constant delay η (see Corollary 4.5).
As an application of the differentiability results, we give a numerical study where we estimate the time delay function using the method of quasilinearization.This method uses point evaluations of the derivatives of the solution wrt the delay function τ.
This paper is organized as follows.Section 2 introduces notations and some preliminary results, Section 3 discusses the well-posedness of the IVP (1.1)-(1.2),Section 4 studies differentiability of the solution wrt the delay function, and Section 5 presents a numerical study for the parameter estimation of the delay function τ using the quasilinearization method.

Notations and preliminaries
In this section we introduce some basic notations which will be used throughout this paper, and recall two results from the literature which will be important in our proofs.
A fixed norm on R n and its induced matrix norm on R n×n are both denoted by the space of Lebesgue-measurable functions which are essentially bounded, where the norm is defined by We note that W 1,∞ is equal to the space of Lipschitzcontinuous functions from [−r, 0] to R n .We also use the notations L(X, Y) denotes the space of bounded linear operators from X to Y, where X and Y are normed linear spaces.An open ball in the normed linear space (X, • X ) centered at a point x ∈ X with radius δ is denoted by B X (x; δ) := {y ∈ X : x − y X < δ}.An open neighbourhood of a set M ⊂ X with radius δ is denoted by B X (M; δ) := {y ∈ X : there exists x ∈ M s.t.x − y X < δ}.
The partial derivatives of a function f : R × R n × R n → R n wrt the second and third variables will be denoted by D 2 f and D 3 f , respectively.Then D i f (t, u, v) ∈ L(R n , R n ) for t ∈ R, u, v ∈ R n and i = 2, 3, which will be identified by its n × n matrix-valued representation.
We recall the following result from [4], which was essential to prove differentiability wrt parameters in SD-DDEs in [9].Note that the second part of the lemma was stated in [4] under the assumption → 0 as k → ∞, but this stronger assumption on the convergence is not needed in the proof.
We recall the following result from [9], which is a simple consequence of Gronwall's lemma.

Well-posedness
Consider the nonlinear FDE with time-dependent delay and the corresponding initial condition It is known (see, e.g., [6]) that if f : R + × R n × R n → R n , τ : R + → [0, r] and ϕ ∈ C are continuous functions, and f is Lipschitz-continuous in its second and third variables, then the IVP (3.1)-(3.2) has a unique noncontinuable solution on an interval [−r, T) for some finite T > 0 or for T = ∞.If we want to emphasize the dependence of this solution on the delay function τ, we will use the notation x(t, τ).

F. Hartung
Throughout the rest of the manuscript we assume and it is continuously differentiable wrt its second and third variables, and ϕ ∈ W 1,∞ .

Differentiability with respect to the delay
In this section we study the differentiability of the solution x(t, τ) of the IVP (3.1)-(3.2) wrt the delay function τ.We define the parameter set F. Hartung and for α > 0 and there exists 0 Clearly, if τ ∈ P, then for any α > 0 it follows τ| [0,α] ∈ P α .Next we show that Proof.Let τ ∈ P α .Then for some 0 Let τ ∈ C(R + , (0, r)) be fixed, and x(t) = x(t, τ) be the corresponding solution of the IVP (3.1)-(3.2) for t ∈ [−r, α] for some α > 0. To simplify the notation, we introduce the n × n matrix-valued functions a.e.t ∈ [0, α], (4.4) It is easy to see that the IVP (4.4)-(4.5)has a unique solution on [−r, α], which we denote by z(t, τ, h).Clearly, both maps Proof.Fix α ∈ (0, T), and let the radius δ > 0, the compact set M ⊂ R n and the Lipschitzconstant L be defined by Lemma 3.1, the constants L 1 and N be defined by (3.5)Let A and B defined by (4.3), and introduce hold.
The functions z k and z satisfy We define the function Note that Ω f is well-defined and (4.12) so we have from (4.11) where āk and bk are defined by The assumed continuity of D 2 f and D 3 f yields āk → 0 as k → ∞.We have To estimate the first term of the last inequality, first note that We note that τ + h k ∈ P α for all k ∈ N, so d ds (s − τ(s) − h k (s)) ≥ ε for some ε > 0 and for a.e.s ∈ [0, α].Therefore Lemma 2.1 gives bk → 0 as k → ∞.But then (4.17) gives the continuity of z(t, τ, •) wrt τ.
The continuity of z(t, τ, •) wrt t follows from (4.7), since This concludes the proof.
Next we prove that for any τ ∈ P the solution x(t, τ) of the IVP (3.1)-(3.2) is continuously differentiable wrt to the time delay function τ on any compact time interval and in a small neighbourhood of τ.We denote this derivative by D 2 x(t, τ).

F. Hartung
Theorem 4.4.Suppose (H) and τ ∈ P. Let x(t, τ) be the corresponding unique noncontinuable solution of the IVP (3.1)-(3.2) defined on the interval [−r, T).Then for every finite α ∈ (0, T) there exists an open neighbourhood U ⊂ W 1,∞ α,1 of τ| [0,α] such that the function is well-defined and it is continuously differentiable wrt τ, and where z(t, τ, h) is the solution of the IVP (4.4)-(4.5) Proof.Fix α ∈ (0, T), and let the radius δ > 0, the compact set M ⊂ R n and the Lipschitzconstant L be defined by Lemma 3.1, the constants L 1 and N be defined by (3.5)Then and We have We define The definition of ω f and simple manipulations yield for s ∈ [0, α] where To get (4.19), it is enough to show that (i) Now we prove the first relation of (4.25).We get by using simple manipulations and Fubini's theorem that
(ii) The second relation of (4.25) follows from (4.7), since we have .
(iii) Finally, we show the third relation of (4.25).It follows from the definition of ω f that We define the function Ω f by (4.11).Then (4.12), (4.26) and the definition of which proves the third relation of (4.25), since Therefore all relations of (4.25) hold, hence (4.24) yields that x(t, τ) is differentiable wrt τ, and we get (4.19).The continuity of D 2 x(t, τ) follows from Lemma 4.3.This completes the proof.
We remark that the result of Theorem 4.4 can be easily extended for FDEs with multiple time delays of the form ẋ(t) = f (t, x(t), x(t − τ 1 (t)), . . ., x(t − τ m (t))). (4.27) Now consider the delay equation where 0 < η < r is a constant delay.We associate the initial condition We observe that constant functions belong to the parameter sets P and P α , so Theorem 4.4 has the following consequence.

Estimation of the time delay function by quasilinearization
In this section we present a numerical study to estimate the delay function in FDEs with the quasilinearization method.This method relies on the computation of the derivative ot the solution wrt the time delay function.
We assume that the parameter τ ∈ P in the IVP (3.1)-(3.2) is unknown, but there are measurements X 0 , X 1 , . . ., X l of the solution at the points t 0 , t 1 , . . ., t l ∈ [0, α].Our goal is to find a parameter value which minimizes the least square cost function (5.1) The method of quasilinearization for parameter estimation was introduced for ODEs in [1] and was applied to estimate finite dimensional parameters in FDEs in [2] and [3], and for FDEs with state-dependent delays in [8] and [10].Following [8], we formulate this method to estimate the delay function in the IVP (3.1)-(3.2).First we take finite dimensional approximation τ N ∈ Γ N of the delay function τ.Here Γ N is a finite-dimensional subspace of C α,1 .In our example below we will use linear spline approximation of the delay function, so Γ N will be the space of N-dimensional linear spline functions with equidistant mesh points defined on the interval [0, α].We consider the corresponding IVP Then we minimize the least square cost function by a gradient-based method.Note that this requires the computation of the derivative of J N with respect to the delay function τ N , for which we have to compute the derivative of the solution wrt the delay function.
The quasilinearization algorithm can be formulated as follows: Fix a basis {e N 1 , . . ., e N N } of the finite dimensional subspace Γ N of C α,1 , and let c = (c 1 , . . ., c N ) T be the coordinates of the F. Hartung parameter τ N ∈ Γ N with respect to this basis, i.e., τ N = ∑ N i=1 c i e N i .Then we identify τ N with the column vector c ∈ R N , and simply write x N (t; c) instead of x N (t; τ N ).We approximate the parameter vector c by the fixed point iteration described by the following equations: ) This is exactly the same scheme that was used in [2] and [3] except that there the parameter space was finite dimensional, and the set {e N 1 , . . ., e N N } was the canonical basis of R N .In our example below we will us the usual hat functions as the basis functions in the space of linear spline functions, i.e., let ∆s := α/(N − 1), s i := (i − 1)∆s for i = 1, . . ., N, and let e N i (s j ) = 1 for j = i and e N i (s j ) = 0 for j = i.In our case D 2 x N is a linear functional defined on C α,1 , and D 2 x N (t; c)e N i denotes the value of the linear functional applied to the function e N i .For the derivation of this method and for the proof of its local convergence we refer to [1] for the finite dimensional case, to [11] for abstract differential equations, and to [10] for FDEs with state-dependent delays.
In our numerical study we used these measurements data, the linear spline approximation of the parameter τ with N = 8 equidistant mesh points, for the initial value we used the constant delay function τ (0) IVP (5.10)- (5.11) and also (5.12)-(5.13)by an Euler-type numerical approximation scheme introduced in [5] using the discretization stepsize h = 0.01.The numerical results can be seen in Figures 5.1-5.4 and in Table 5.1 below.We observe convergence of the method starting from this initial value, and even in the third step the cost function has a value J 8 (τ (3) 8 ) = 0.000031, which indicates that the parameter is close to the "true" parameter τ.Table 5.1 contains the errors ∆ 8 for k = 0, 1, 2, 3.The figures show that the method quickly recovers the shape of the "true" time delay function, and the last spline function is a good approximation of τ.

.14) Relation ( 4 . 7 )
and the initial condition (4.5) imply ) and z k (t) := z(t, τ, h k ) be the corresponding solutions of the IVP (3.1)-(3.2) and (4.4)-(4.5),respectively, for t ∈ [−r, α].We note that the definition of z k here is different from that of used in the proof of Lemma 4.3.

=
| τ(s i ) − τ (k) 8 (s i )| at the mesh points of the spline approximation.In Figures 5.1-5.4 the solid blue curve is the delay function τ, and the dotted red graph is the linear spline function τ (k) Figure 5.1: Step 0