Differentiability of solutions with respect to the delay function in functional differential equations

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1 Electronic Journal of Qualitative Theory of Differential Equations 216, No. 73, 1 16; doi: /ejqtde Differentiability of solutions with respect to the delay function in functional differential equations Dedicated to Professor Tibor Krisztin on the occasion of his 6th birthday Ferenc Hartung University of Pannonia, H-821 Veszprém, P.O. Box 158, Hungary Received 22 June 216, appeared 12 September 216 Communicated by Hans-Otto Walther Abstract. 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. Keywords: delay differential equation, time-dependent delay, differentiability with respect to parameters. 21 Mathematics Subject Classification: 34K5. 1 Introduction In this paper we consider a class of functional differential equations FDEs with a timedependent delay of the form where the associated initial condition is ẋt = f t, xt, xt τt, t, 1.1 xt = ϕt, t [ r, ]. 1.2 Here and throughout the manuscript r > is a fixed constant, and τt r for all t. In this paper we consider the delay function τ as parameter in the initial value problem IVP , and we denote the corresponding solution by xt, τ. The main goal of this paper is to discuss the differentiability of the solution xt, 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 hartung.ferenc@uni-pannon.hu

2 2 F. 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 ẋt = gxt, xt η, 1.3 where g : R n R n R n is a continuously differentiable function. It was shown that if α > is such that the solutions xt, η of 1.3 are defined for t [, α] and η δ 1, δ 2 with some < δ 1 < δ 2, then the map R δ 1, δ 2 η x, η W 1,1 [, α], R n is continuously differentiable. Here W 1,1 [, α], R n is the space of absolutely continuous functions of finite norm ψ W 1,1 [,α],r n = ψs + ψs ds. Differentiability of the solution xt, 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 xt, 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 xt, 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 , 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. 2 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. For a fixed α >, C α := C[ r, α], R n denotes the Banach space of continuous functions ψ : [ r, α] R n equipped with the norm ψ Cα := sup{ ψs : s [ r, α]}. L α := L [ r, α], R n denotes the space of Lebesgue-measurable functions which are essentially bounded, where the norm is defined by ψ L α := ess sup{ ψs : s [ r, α]}. Wα := W [ r, α], R n denotes the Banach space of absolutely continuous functions ψ : [ r, α] R n of finite norm defined by ψ W := max { } ψ Cα, ψ α L α. For α = we use te notations C, L and W instead of C, L and W. We note that W is equal to the space of Lipschitzcontinuous functions from [ r, ] to R n. We also use the notations C := C[, α], R and W := W [, α], R. LX, 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

3 Differentiability of solutions with respect to the delay function 3 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 LR 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 u k u W as k, but this stronger assumption on the convergence is not needed in the proof. Lemma 2.1 [4]. Let p [1,, g L p [ r, α], R n, ε >, and u Aε, where Then Aε := { v W [, α], [ r, α] : vs ε for a.e. s [, α] }. gus p ds 1 ε r gs p ds. Moreover, if the sequence u k Aε is such that u k u C as k, then lim k gu k s gus p ds =. Let R + := [,. We recall the following result from [9], which is a simple consequence of Gronwall s lemma. Lemma 2.2 [9]. Suppose a, b : [, α] R + and g : [ r, α] R n are continuous functions such that gs a for r s, and gt a + bs max gθ ds, s r θ s t [, α]. Then gt max gθ ae α bs ds, t [, α]. t r θ t 3 Well-posedness Consider the nonlinear FDE with time-dependent delay and the corresponding initial condition ẋt = f t, xt, xt τt, t, 3.1 xt = ϕt, t [ r, ]. 3.2 It is known see, e.g., [6] that if f : R + R n R n R n, τ : R + [, r] and ϕ C are continuous functions, and f is Lipschitz-continuous in its second and third variables, then the IVP has a unique noncontinuable solution on an interval [ r, T for some finite T > or for T =. If we want to emphasize the dependence of this solution on the delay function τ, we will use the notation xt, τ.

4 4 F. Hartung Throughout the rest of the manuscript we assume H f CR + R n R n, R n, and it is continuously differentiable wrt its second and third variables, and ϕ W. The next result shows that, assuming the condition H and < τt < r for t, the solution xt, τ depends Lipschitz-continuously on τ. Lemma 3.1. Suppose H. Then for every ˆτ CR +,, r there exists a unique noncontinuable solution xt, ˆτ of the IVP defined on the interval [ r, T for some T > or T =. Then for every α, T there exist a radius ˆδ >, a compact set M R n, and a Lipschitz-constant L > such that ϕt M for t [ r, ], and for every τ B C ˆτ [,α] ; ˆδ a unique solution xt, τ of the IVP exists for t [ r, α], and < τt < r and xt, τ M for t [, α], 3.3 and xt, τ xt, τ L τ τ C for t [, α] and τ, τ B C ˆτ [,α] ; ˆδ. 3.4 Proof. Let ˆτ CR +,, r be fixed, and let ˆxt := xt, ˆτ be the unique noncontinuable solution of the corresponding IVP on [ r, T, where T is possibly equal to. Let < α < T be fixed, and define the set M := { ˆxt : t [, α]} {ϕt : t [ r, ]}. Clearly, M is a compact subset of R n. Fix ρ >, and let B R nm, ρ be the neighbourhood of M with radius ρ, and its closure is denoted by M := B R nm, ρ. Define the constant L 1 by { } L 1 := max max{ D i f t, u, v : t [, α], u, v M}. 3.5 i=2,3 Then the Mean Value Theorem yields f t, u, v f t, ū, v L 1 u ū + v v, t [, α], u, v, ū, v M. 3.6 The constants m 1 := min{ ˆτt : t [, α]} and m 2 := max{ ˆτt : t [, α]} satisfy < m 1 m 2 < r. Define δ 1 := min{m 1, r m 2 }. Then for τ B C ˆτ; δ 1 it follows < τt < r for t [, α]. Let τ B C ˆτ [,α] ; δ 1, and let xt := xt, τ be the corresponding unique noncontinuable solution of the IVP which is defined on the interval [ r, T τ for some T τ, α, or on [ r, T τ ] with T τ = α. Since ϕ M, it follows that xt M for small positive t. We introduce { } β τ := sup t, T τ : xs M and xs τs M for s [, t]. We note that xβ τ M, since M is compact. Then < β τ α. We show that β τ = α if τ is close enough to ˆτ. We have xt = ϕ + ˆxt = ϕ + f s, xs, xs τs ds, t [, β τ ] f s, ˆxs, ˆxs ˆτs ds, t [, α].

5 Differentiability of solutions with respect to the delay function 5 Hence, for t [, β τ ], 3.6 implies xt ˆxt f s, xs, xs τs f s, ˆxs, ˆxs ˆτs ds L 1 xs ˆxs + xs τs ˆxs ˆτs ds L 1 xs ˆxs + xs τs ˆxs τs + ˆxs τs ˆxs ˆτs ds. 3.7 We define { } N := max ϕ L, max{ f t, u, v : t [, α], u, v M}. 3.8 Then 3.1, 3.8 and the Mean Value Theorem yield ˆxs τs ˆxs ˆτs N τs ˆτs N τ ˆτ C, s [, β τ ]. 3.9 Therefore, it follows from 3.7 that xt ˆxt αl 1 N τ ˆτ C + 2L 1 Hence, Lemma 2.2 gives where L := αl 1 Ne 2L1α. Fix < ρ 1 < ρ, and define { ˆδ := min max xθ ˆxθ ds, t [, β τ]. 3.1 s r θ s xt ˆxt L τ ˆτ C, t [, β τ ], 3.11 δ 1, ρ 1 L Then 3.11 implies xt ˆxt L ˆδ ρ 1 < ρ for t [, β τ ] and τ B C ˆτ [,α] ; ˆδ Suppose β τ < α. Then xβ τ is in the interior of M, and hence x has a continuation to the right of β τ with values in M. This contradicts to the definition of β τ, hence β τ = α holds for τ B C ˆτ [,α] ; ˆδ. Let τ, τ B C ˆτ [,α] ; ˆδ. Then, similarly to 3.1, we get }.. xt xt αl 1 N τ τ C + 2L 1 Therefore Lemma 2.2 yields 3.4. max xθ xθ ds, s r θ s t [, α]. 4 Differentiability with respect to the delay In this section we study the differentiability of the solution xt, τ of the IVP wrt the delay function τ. We define the parameter set { P := τ W R +, R : < τt < r, t R +, and for every α > } there exists κ < 1 s.t. τt κ for a.e. t [, α], 4.1

6 6 F. Hartung and for α > { P α := τ W : < τt < r, t [, α], and there exists κ < 1 s.t. } τt κ for a.e. t [, α]. 4.2 Clearly, if τ P, then for any α > it follows τ [,α] P α. Next we show that P α is an open subset of W. Lemma 4.1. P α is an open subset of W. Proof. Let τ P α. Then for some κ < 1 it follows τt κ for a.e. t [, α]. Let γ 1 := min{ τt : t [, α]}, γ 2 := max{ τt : t [, α]}, and fix κ < κ < 1. Let δ := min{γ 1, r γ 2, κ κ}. Then for τ B W τ; δ it follows γ 1 δ < τt = τt + τt τt < γ 2 + δ r, t [, α], and τt τt + τt τt κ + δ κ < 1 for a.e. t [, α], hence τ P α. Let τ CR +,, r be fixed, and xt = xt, τ be the corresponding solution of the IVP for t [ r, α] for some α >. To simplify the notation, we introduce the n n matrix-valued functions At := D 2 f t, xt, xt τt and Bt := D 3 f t, xt, xt τt, t [, α]. 4.3 Then for h C we define the variational equation associated to x = x, τ as żt = Atzt + Btzt τt Btẋt τtht, a.e. t [, α], 4.4 zt =, t [ r, ]. 4.5 It is easy to see that the IVP has a unique solution on [ r, α], which we denote by zt, τ, h. Clearly, both maps C h z, τ, h C α W h z, τ, h C α are linear. Part i of the next lemma yields that both maps are also bounded. Lemma 4.2. Assume H and ˆτ P, and let xt, ˆτ be the corresponding noncontinuable solution of the IVP defined on the interval [ r, T. Fix any α, T, and let the radius δ > be defined by Lemma 3.1. For τ B C ˆτ [,α] ; δ and h C let zt, τ, h be the corresponding solution of the IVP for t [ r, α]. Then i there exists N 1 such that zt, τ, h N 1 h C, t [, α], τ B C ˆτ [,α] ; δ, h C ; 4.6 ii there exists N 2 such that żt, τ, h N 2 h C, τ B C ˆτ [,α] ; δ, h C, and a.e. t [, α]. 4.7

7 Differentiability of solutions with respect to the delay function 7 Proof. i Let τ B C ˆτ [,α] ; δ and h C, and let xt = xt, τ and zt = zt, τ, h be the corresponding solutions of the IVP and , respectively, for t [ r, α], and let A and B defined by 4.3. Let the compact set M R n be defined by Lemma 3.1, and L 1 and N be defined by 3.5 and 3.8, respectively, corresponding to α and M. Then we get At L 1 and Bt L 1, t [, α]. 4.8 Hence 4.4, 4.5 and 4.8 yield zt As zs + Bs zs τs + Bs ẋs τs hs ds L 1 Nα h C + 2L 1 max zθ ds, s r θ s Therefore Lemma 2.2 implies 4.6 with N 1 := L 1 Nαe 2L 1α. ii To prove 4.7, we use 3.8, 4.4, 4.6 and 4.8 to get żt 2L 1 N 1 + L 1 N h C, t [, α]. for a.e. t [, α]. Next we show that the map zt, τ, is continuous in t and τ. Lemma 4.3. Suppose H and ˆτ P. Let xt, ˆτ be the corresponding unique noncontinuable solution of the IVP defined on the interval [ r, T. Then for every finite α, T there exists an open neighbourhood U W of ˆτ [,α] such that the map is continuous. R W [, α] U t, τ zt, τ, LC, R n Proof. Fix α, T, and let the radius ˆδ >, 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 and 3.8, respectively. Then the IVP has a unique solution on [ r, α] for any τ B C ˆτ [,α] ; ˆδ, ˆτ [,α] ; δ 1 P α. and 3.3, 3.4 and 3.6 hold. Let < δ 1 ˆδ be such that U := B W Fix τ U, and let δ > be such that B W sequence with < h k W δ for k N and h k W τ; δ U, and h k W k N be a as k. Fix h C, and let xt = xt, τ, x k t = xt, τ + h k, zt = zt, τ, h and z k t := zt, τ + h k, h be the corresponding solutions of the IVP and , respectively, for t [ r, α]. Let A and B defined by 4.3, and introduce A k t := D 2 f t, x k t, x k t τt h k t and B k t := D 3 f t, x k t, x k t τt h k t for t [, α]. Then 4.8 and A k t L 1 and B k t L 1, t [, α], k N 4.9 hold. The functions z k and z satisfy z k t = A k sz k s + B k sz k s τs h k s zt = B k sẋ k s τs h k shs ds, t [, α], Aszs + Bszs τs Bsẋs τshs ds, t [, α].

8 8 F. Hartung Therefore it follows for t [, α] z k t zt A k s As zs + B k s Bs zs τs + ẋs τs hs ds + + B k s zs τs h k s zs τs + ẋ k s τs h k s ẋs τs hs ds A k s z k s zs + B k s z k s τs h k s zs τs h k s ds. 4.1 We define the function { Ω f ε := sup max D 2 f t, u, v D 2 f t, ũ, ṽ, D 3 f t, u, v D 3 f t, ũ, ṽ : } u ũ + v ṽ ε, t [, α], u, ũ, v, ṽ M Note that Ω f is well-defined and Ω f ε as ε, since M is compact, and D 2 f and D 3 f are uniformly continuous on [, α] M M. Relations 3.3, 3.4, 3.8 and the Mean Value Theorem yield so we have from 4.11 x k s xs + x k s τs h k s xs τs x k s xs + x k s τs h k s xs τs h k s + xs τs h k s xs τs 2L + N h k C, s [, α], 4.12 A k s As D2 f t, x k t, x k t τs h k s D 2 f t, xt, xt τs h k s Ω f 2L + N h k C, s [, α] Similarly, we get B k s Bs Ω f 2L + N h k C, s [, α] Relation 4.7 and the initial condition 4.5 imply zs τs h k s zs τs N 2 h k C h C, s [, α] Combining 4.6, 4.9, 4.13, 4.14 and 4.15, we get from 4.1 z k t zt ā k + b k h C + 2L 1 max z kθ zθ ds, t [, α], 4.16 s r θ s where ā k and b k are defined by ā k := Ω f 2L + N h k C 2N 1 + Nα + L 1 N 2 h k C

9 Differentiability of solutions with respect to the delay function 9 and Then Lemma 2.2 gives b k := L 1 ẋ k s τs h k s ẋs τs ds. z k t zt ā k + b k e 2L 1α h C, t [, α] The assumed continuity of D 2 f and D 3 f yields ā k as k. We have ẋ k s τs h k s ẋs τs ẋ k s τs h k s ẋs τs h k s + ẋs τs h k s ẋs τs To estimate the first term of the last inequality, first note that ẋ k s τs h k s ẋs τs h k s =, if s τs h k s and ϕ is differentiable at s τs h k s. Suppose s is such that s τs h k >. Then for such s we have Hence 4.18 yields ẋ k s τs h k s ẋs τs h k s f s τs h k s, x k s τs h k s, x k s 2τs 2h k s f s τs h k s, xs τs h k s, xs 2τs h k s L 1 x k s τs h k s xs τs h k s + x k s 2τs 2h k s xs 2τs h k s L 1 L h k C + x k s 2τs 2h k s xs 2τs 2h k s + xs 2τs 2h k s xs 2τs h k s L 1 2L + N h k C. b k L 2 1 2L + Nα h k C + L 1 ẋs τs h k s ẋs τs ds. We note that τ + h k P α for all k N, so d ds s τs h ks ε for some ε > and for a.e. s [, α]. Therefore Lemma 2.1 gives b k as k. But then 4.17 gives the continuity of zt, τ, wrt τ. The continuity of zt, τ, wrt t follows from 4.7, since zt, τ, h z t, τ, h N 2 t t h C, t, t [, α], τ U, h C. This concludes the proof. Next we prove that for any τ P the solution xt, τ of the IVP 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 xt, τ.

10 1 F. Hartung Theorem 4.4. Suppose H and ˆτ P. Let xt, ˆτ be the corresponding unique noncontinuable solution of the IVP defined on the interval [ r, T. Then for every finite α, T there exists an open neighbourhood U W of ˆτ [,α] such that the function R W [, α] U t, τ xt, τ Rn is well-defined and it is continuously differentiable wrt τ, and D 2 xt, τh = zt, τ, h, t [, α], τ U, h W, 4.19 where zt, τ, h is the solution of the IVP for t [, α], τ U and h W. Proof. Fix α, T, and let the radius ˆδ >, 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 and 3.8, respectively. Then the IVP has a unique solution on [ r, α] for any τ B C ˆτ [,α] ; ˆδ, ˆτ [,α] ; δ 1 P α. and 3.3, 3.4 and 3.6 hold. Let < δ 1 ˆδ be such that U := B W Let τ U and h W, and xt = xt, τ and zt = zt, τ, h be the corresponding solutions of the IVP and , respectively, for t [ r, α]. Let A and B be defined by 4.3. Then 4.8 holds. Let δ > be such that B W τ; δ U, and h k W k N be a sequence with < h k W δ for k N and h k W as k. Let x k t = xt, τ + h k and z k t := zt, τ, h k be the corresponding solutions of the IVP and , 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. Then x k t = ϕ + xt = ϕ + f s, x k s, x k s τs h k s ds, f s, xs, xs τs ds, t [, α], t [, α], and z k t = Asz k s + Bsz k s τs Bsẋs τsh k s ds, t [, α]. We have x k t xt z k t = f s, x k s, x k s τs h k s f s, xs, xs τs Asz k s Bsz k s τs + Bsẋs τsh k s ds. 4.2 We define ω f t, ū, v, u, v := f t, u, v f t, ū, v D 2 f t, ū, vu ū D 3 f t, ū, vv v 4.21

11 Differentiability of solutions with respect to the delay function 11 for t R +, ū, u, v, v R n. The definition of ω f and simple manipulations yield for s [, α] f s, x k s, x k s τs h k s f s, xs, xs τs Asz k s Bsz k s τs + Bsẋs τsh k s = Asx k s xs + Bs x k s τs h k s xs τs + ω f s, xs, xs τs, x k s, x k s τs h k s Asz k s Bsz k s τs + Bsẋt τsh k s = Asx k s xs z k s + Bs x k s τs h k s xs τs h k s z k s τs h k s + Bs xs τs h k s xs τs + ẋt τsh k s + Bs z k s τs h k s z k s τs + ω f s, xs, xs τs, x k s, x k s τs h k s Using 4.8, we get from 4.2 and 4.22 where x k t xt z k t a k + b k + c k + 2L 1 max x kθ xθ z k θ ds, t [, α], 4.23 s r θ s a k := L 1 xs τs h k s xs τs + ẋt τsh k s ds, b k := L 1 z k s τs h k s z k s τs ds, c k := Hence Lemma 2.2 yields ω f s, xs, xs τs, x k s, x k s τs h k s ds. x k t xt z k t a k + b k + c k e 2L 1α, t [, α] To get 4.19, it is enough to show that lim k a k h k W =, lim k b k h k W = and lim k c k h k W = i Now we prove the first relation of We get by using simple manipulations and Fubini s theorem that xs τs h k s xs τs + ẋs τsh k s ds = = s τs hk s s τs 1 h k W = h k W ẋv ẋs τs dv ds ẋs τs νh k s ẋs τs 1 1 h k s dν ds ẋs τs νh k s ẋs τs dν ds ẋs τs νh k s ẋs τs ds dν.

12 12 F. Hartung Lemma 2.1 is applicable since, for a.e. s [, α] and for all ν [, 1], it follows d ds s τs νh k s ε for some ε > and large enough k. Therefore lim k ẋs τs νh k s ẋs τs ds =, ν [, 1], hence we conclude the first relation of 4.25 by using the Lebesgue s Dominated Convergence Theorem. ii The second relation of 4.25 follows from 4.7, since we have b k = L 1 z k s τs h k s z k s τs ds αl 1 N 2 h k 2. W iii Finally, we show the third relation of It follows from the definition of ω f that ω f t, ū, v, u, v = therefore 1 ω f t, ū, v, u, v sup <ν<1 [ D 2 f t, ū + νu ū, v + νv v D 2 f t, ū, v u ū ] + D 3 f t, ū + νu ū, v + νv v D 3 f t, ū, v v v dν, D 2 f t, ū + νu ū, v + νv v D 2 f t, ū, v u ū + D 3 f t, ū + νu ū, v + νv v D 3 f t, ū, v v v 4.26 for t R +, ū, u, v, v R n. We define the function Ω f by Then 4.12, 4.26 and the definition of Ω f imply ω f s, xs, xs τs, x k s, x k s τs h k s ds αω f N + 2L h k C N + 2L h k C, which proves the third relation of 4.25, since Ω f N + 2L h k C as k. Therefore all relations of 4.25 hold, hence 4.24 yields that xt, τ is differentiable wrt τ, and we get The continuity of D 2 xt, 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 ẋt = f t, xt, xt τ 1 t,..., xt τ m t Now consider the delay equation ẋt = f t, xt, xt η, t, 4.28 where < η < r is a constant delay. We associate the initial condition xt = ϕt, t [ r, ] We observe that constant functions belong to the parameter sets P and P α, so Theorem 4.4 has the following consequence.

13 Differentiability of solutions with respect to the delay function 13 Corollary 4.5. Suppose H and ˆη, r. Let xt, ˆη be the corresponding unique noncontinuable solution of the IVP defined on the interval [ r, T. Then for every finite α, T there exists δ > such that the solution xt, η of the IVP exists for t [, α] and τ ˆη δ, ˆη + δ, and the function is continuously differentiable wrt η, and R R [, α] ˆη δ, ˆη + δ t, η xt, η R n D 2 xt, ηh = zt, η, h, t [, α], η ˆη δ, ˆη + δ, h R, 4.3 where zt, η, h is the solution of the IVP żt = Atzt + Btzt η Btẋt ηh, a.e. t [, α], 4.31 zt =, t [ r, ] 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 is unknown, but there are measurements X, X 1,..., X l of the solution at the points t, t 1,..., t l [, α]. Our goal is to find a parameter value which minimizes the least square cost function Jτ := l i= xt i, τ X i 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 [1]. Following [8], we formulate this method to estimate the delay function in the IVP 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. 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 [, α]. We consider the corresponding IVP ẋ N t = f t, x N t, x N t τ N t, t [, α] 5.2 x N t = ϕt, t [ r, ]. 5.3 Then we minimize the least square cost function J N τ N := l i= x N t i ; τ N X i 2, τ N Γ N 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,..., en N } of the finite dimensional subspace Γ N of C, and let c = c 1,..., c N T be the coordinates of the

14 14 F. Hartung parameter τ N Γ N with respect to this basis, i.e., τ N = N i=1 c ie 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: c k+1 = gc k, k =, 1,..., 5.4 gc = c Dc 1 bc 5.5 Dc = l M T t i ; cmt i ; c 5.6 i= l bc = M T t i ; cx N t i ; c X i 5.7 i= Mt; c = M 1 t; c,..., M N t; c 5.8 M i t; c = D 2 x N t; ce N i, i = 1,..., N. 5.9 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 {e1 N,..., en N } was the canonical basis of RN. 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 ei N s j = 1 for j = i and ei N s j = for j = i. In our case D 2 x N is a linear functional defined on C, and D 2 x N t; cei N denotes the value of the linear functional applied to the function ei N. 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 [1] for FDEs with state-dependent delays. Next we apply the quasilinearization method for a scalar equation with a single time-dependent delay. Example 5.1. Consider the scalar nonlinear FDE with time-delay ẋt =.2 cos t +.6xt τt.1 sin t +.2x 2 t, t [, 4], 5.1 and the initial condition xt = t, t Here the delay function τ is a parameter in the IVP. We consider τt =.4 sin2t + 2 as the true parameter. Note that τ P 4. We solved the IVP using this parameter value, and generated the measurements by evaluating the solutions at the mesh points t i =.4i, i =,..., 1, i.e., we consider X i = xt i, τ, i =,..., 1. For a fixed h C we associate the variational equation to 5.1: żt =.2 cos t +.6zt τt.2 cos t +.6ẋt τtht.1 sin t +.22xtzt, t [, 4] 5.12 zt =, t Then Theorem 4.4 yields that, in a neighbourhood of τ, the solution zt = zt, τ, h of the IVP satisfies D 2 xt, τh = zt, τ, h. 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 8 t = 1.5, and we generated the first three terms of the quasilinearization sequence defined by In the course of the computation, we solved the

15 Differentiability of solutions with respect to the delay function 15 IVP and also by an Euler-type numerical approximation scheme introduced in [5] using the discretization stepsize h =.1. The numerical results can be seen in Figures 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 =.31, which indicates that the parameter is close to the true parameter τ. Table 5.1 contains the errors k i = τs i τ k 8 s i at the mesh points of the spline approximation. In Figures the solid blue curve is the delay function τ, and the dotted red graph is the linear spline function τ k 8 for k =, 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. 3 τt 3 τt Figure 5.1: Step Figure 5.2: Step 1 3 τt 3 τt Figure 5.3: Step Figure 5.4: Step 3 Acknowledgements This research was partially supported by the Hungarian National Foundation for Scientific Research Grant No. K73274.

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