On the necessity of sufficient LMI conditions for time-delay systems arising from Legendre approximation

arXiv:2207.08423v1 [math.OC] 18 Jul 2022 On the necessity of sufficient LMI conditions for time-delay systems arising from Legendre approximation ⋆ Mathieu Bajodek a, Alexandre Seuret a,b, Fr´ed´eric Gouaisbaut a aLAAS-CNRS, Univ de Toulouse, UPS, 7 avenue du Colonel Roche, 31077 Toulouse, France bUniv. de Sevilla, Camino de los Descubrimientos, s/n 41092 Sevilla, Spain Abstract This work is dedicated to the stability analysis of time-delay systems with a single constant delay using the Lyapunov-Krasovskii theorem. This approach has been widely used in the literature and numerous sufficient conditions of stability have been proposed and expressed as linear matrix inequalities (LMI). The main criticism of the method that is often pointed out is that these LMI conditions are only sufficient, and there is a lack of information regarding the reduction of the conservatism. Recently, scalable methods have been investigated using Bessel-Legendre inequality or orthogonal polynomial-based inequalities. The interest of these methods relies on their hierarchical structure with a guarantee of reduction of the level of conservatism. However, the convergence is still an open question that will be answered for the first time in this paper. The objective is to prove that the stability of a time-delay system implies the feasibility of these scalable LMI, at a sufficiently large order of the Legendre polynomials. Moreover, the proposed contribution is even able to provide an analytic estimation of this order, giving rise to a necessary and sufficient LMI for the stability of time-delay systems. Key words: Systems with time-delay; Infinite-dimensional systems; Stability analysis; Lyapunov functionals; LMI; Polynomial approximation. 1 Introduction Time-delay systems represent a wide class of dynamical systems arising in many applications in electronics, biology, transport, etc... Their interest in automatic control is natural since they propose many challenging theoretical problems related to their intrinsic infinitedimensional nature [21,29]. In particular, their stability analysis has been at the heart of many research works for several decades [7,13,17,26,33] and many methods have been developed. Among them, the use of LyapunovKrasovskii theorem remains one of the most popular techniques because of its inherent robustness. In the context of linear time-invariant (LTI) delay systems, several conditions have been obtained over the last two decades by application of the LyapunovKrasovskii theorem leading generally to sum of square constraints [23,24,30] or to linear matrix inequalities (LMI) [6,14,32]. These conditions, combined with semidefinite convex optimization programs, only led to sufficient conditions. Hence, numerous works aimed at ⋆Corresponding author M. Bajodek Email address: mbajodek laas.fr (Mathieu Bajodek). reducing the inherent conservatism and at recovering the necessity using, for instance, discretized functionals [12,13] or delay partitioning methods [3,11]. Among them, an approach based on state extension has been proposed in [28]. The candidate Lyapunov-Krasovskii functional therein depends directly on the projections of the state of the delay system onto Legendre polynomials. This approach led to LMI stability conditions, which benefit from a particular hierarchical structure, arising from the use of the Bessel-Legendre inequality. Similar approaches based on orthogonal polynomials [18,22] have been also considered in the literature. The complexity of these LMI increases with the number of Legendre polynomials ntaken under consideration but their conservatism drastically reduces as nincreases, at least on examples. However, to the best of our knowledge, the proof of convergence of these LMI to necessary and sufficient LMI conditions of stability is still missing. Towards this direction, as for LTI finite-dimensional systems, there exists a converse Lyapunov-Krasovskii theorem for LTI systems with a constant delay. Indeed, it is possible to build a complete Lyapunov-Krasovskii functional for stable LTI delay systems, see for instance [17] for a larger overview on the problem. While this method has been only seen as a theoretical contribuPreprint submitted to Automatica 19 July 2022 tions, the authors of [19,20] paved the way to use the converse Lyapunov-Krasovskii theorem to derive not sufficient but necessary stability conditions for timedelay systems through approximation. Among them, the authors of [5] provided a necessary stability conditions for time-delay systems. This method was then extended to various classes of delay systems, see for instance [8,9]. More interestingly, their methods have led to necessary and sufficient stability conditions in [4,10]. The sufficiency part of their necessary condition has been obtained by estimating an upper bound of the approximation error due to the discretization process of the Lyapunov matrix. It is worth noting that these conditions are not LMI but only a test of positive definiteness of a given matrix, which makes their method efficient from computational reasons. However, even though the necessary condition of stability is simple to test, the sufficient condition of stability is still numerically complicated to verify, because the estimated order assessing the sufficiency can be very high. The objective of the present paper follows the same spirit of deriving necessary and sufficient stability conditions of time-delay systems but with a different objective. Indeed, we aim at using the approach developed in [10,12] to prove that the sufficient LMI conditions arising from the Bessel-Legendre inequality are asymptotically necessary, as the degree nof the Legendre polynomial to be considered increases. This corresponds to answer the following questions. •If a time-delay system is stable, does there exist an order N∗for which some LMI stability conditions are necessary satisfied? •Is it possible to estimate this order N∗? The paper is organized as follows. Section 2 formulates the problem of stability analysis of LTI systems subject to a single point-wise constant delay. After presenting a brief summary of the properties of Legendre polynomials, sufficient scalable LMI stability conditions are proposed. Then, Section 3 proves the converse theorem of this first result, by ensuring the existence of an order N∗for which the LMI stability conditions must be verified for stable time-delay systems. Section 4 goes beyond the existence of an order by providing an estimation of the order N∗of the polynomial approximation for which the LMI stability conditions are guaranteed. Finally, the theoretical results are evaluated on academic examples leading to several discussions. The paper ends with a conclusion and several appendices gathering the technical proofs required to derive the main results. Notations : Throughout the paper, N(N∗), Rm×pand Sm(Sm +) denote the sets of natural numbers (excluding zero), real matrices of size m×pand symmetric matrices of size m(positive definite), respectively. For any xin R,⌈x⌉stands for the least integer greater than or equal to x. For any square matrix M∈Rm×m,M(p, q) denotes the entries of Mlocated at the pth row and qth column, M⊤denotes the transpose of M,H(M) stands for M+M⊤and inequality M≻0 means that M∈Sm +. Furthermore, for any Min Sm, its minimal and maximal eigenvalues are denoted σ(M) and ¯σ(M). The 2-norm of matrix Min Rm×pis |M|=p¯σ(M⊤M). The vector u= vec(M) in Rmp×1collocates the columns of matrix Min Rm×pand the inverse of this operation is denoted vec−1and is such that vec−1(vec(M)) = M. Moreover, notation ⊗represents the Kronecker product so that, for any matrices (M1, M2) in Rm1×p1×Rm2×p2,M1⊗M2= "M1(1,1)M2... M1(1,p1)M2 . . ..... . . M1(m1,1)M2... M1(m1,p1)M2#in Rm1m2×p1p2. The set of square-integrable functions from (a, b) to Rm×pis noted L2(a, b;Rm×p). Let finally Cpw(a, b;Rm×p) be the set of piecewise continuous functions with a finite number of discontinuity points. 2 Sufficient LMI stability conditions for timedelay systems 2.1 System data Consider a LTI delay system given by ˙x(t) = Ax(t) + Adx(t−h),∀t≥0, x(t) = ϕ(t)∈ Cpw(−h, 0; Rnx),∀t∈[−h, 0], (1a) (1b) where the single delay h > 0 and matrices A, Adin Rnx×nxare constant and known. Without loss of generality, matrix Adis decomposed into a product BC as Ad=BC, with |C|= 1,(2) with B, CT∈Rnx×nzbeing full column rank matrices, with nzbeing the rank of matrix Ad. Along the paper, the Shimanov notation xt(θ) = x(t+θ), for all (t, θ) in R+×[−h, 0] is adopted. Definition 1 (GES) The trivial solution x(t)≡0of system (1) is globally exponentially stable (GES), if there exist κ≥1and µ > 0 such that the solution to (1) generated by any initial condition ϕ∈ Cpw(−h, 0; Rnx), denoted as x(t, ϕ)verifies |x(t, ϕ)| ≤ κe−µt sup θ∈[−h,0]|ϕ(θ)|,∀t≥0.(3) Several ways of assessing GES of LTI systems have been provided in the literature as mentioned in the introduction. Here, the contribution focuses on conditions arising from the application of the Lyapunov-Krasovskii theorem [13], adapted to the LTI case. 2 Theorem 1 (Lyapunov-Krasovskii theorem) If there exist positive scalars ε1,ε2and ε3and a continuous and differentiable functional Vdefined from Cpw(−h, 0; Rnx)to Rsuch that, for any ϕin Cpw(−h, 0; Rnx), the following inequalities hold (i) ε1|ϕ(0)|2≤ V(ϕ)≤ε2sup θ∈[−h,0]|ϕ(θ)|2and (ii) ˙ V(ϕ)≤ −ε3|ϕ(0)|2, where ˙ Vdenotes here the derivative of Valong the trajectories of (1), then, the trivial solution of system (1) is GES. The main underlying idea of this theorem is to determine a positive definite functional V, such that its derivative with respect to time along the trajectories of the system (1) is negative definite. The main problem within the application of this theorem is to design a suitable functional and then to provide some conditions that guarantee its positive definiteness and the negative definiteness of its derivative. The derivation of stability conditions using the Lyapunov-Krasovskii theorem usually involves quite elaborate developments. To give an idea of the procedure involved in this approach and to provide a glimpse of its technical flavor, we present here some basics on the procedure to follow in order to derive asymptotic stability criteria for time-delay systems expressed in terms of LMI. The procedure follows three steps. Step 1. To propose a candidate Lyapunov-Krasovskii functional V, based on the structure of the system. Step 2. To compute the derivative of the functional along the trajectories of the system. Step 3. To apply over-approximation technique to ˙ Vand derive a stability condition expressed in terms of LMI. Among the numerous methods employed in the literature, we will focus here on the method based on the application of the Bessel-Legendre integral inequality [27,28], which is scalable with respect to the degree nof the Legendre polynomial to be considered. Interestingly, this method leads to LMI admitting a hierarchical structure with respect to n. Before stating this theorem, let us first recall the main definitions and some characteristics about these polynomials. 2.2 Definition of Legendre polynomials Definition 2 The Legendre polynomials are defined over the normalized interval [0,1] as ∀k∈N, lk(θ) = k X j=0 (k j)( k+j j)(θ−1)j,(4) where (k j)stands for the binomial coefficients. The Legendre polynomials have been widely used in the polynomial approximation theory [2] because they form an orthogonal sequence with respect to the inner product R1 0φ⊤(θ)ψ(θ)dθ, for any φ, ψ in L2(0,1; Rm). Before going into the details, for any order nand min N∗and θ∈[0,1], let us introduce the following notations: ℓm n(θ)=hl0(θ)l1(θ). . . ln−1(θ)i⊤ ⊗Im∈Rnm×m,(5) Im n=I1 n⊗Im∈Rnm×nm,(6) I1 n(p, q)=(2p−1 if p=q, 0 otherwise, ∀(p, q)∈ {1,...,n}2. To ease the reading of the paper and where no confusion is possible, we will omit the upper script m=nzand only use notations ℓnand In. The main motivation for employing these polynomials comes also from an efficient integral inequality, known as Bessel-Legendre inequality, which is stated here. Lemma 1 (Bessel-Legendre inequality) Let z∈ L2(−h, 0; Rnz)and S∈Snz +a positive definite matrix. The integral inequality Z0 −h z⊤(θ)Sz(θ)dθ≥1 hζ⊤ n(z)(I1 n⊗S)ζn(z),(7) hold, for all n∈N∗, where ζn(z) = R0 −hℓnθ+h hz(θ)dθ. Moreover, if zis a polynomial function of degree n−1 over (−h, 0), the equality case holds Z0 −h z⊤(θ)Sz(θ)dθ=1 hζn(z)T(I1 n⊗S)ζn(z).(8) Proof : The proof is derived from [28, Lemma 3] and is postponed to Appendix C.  2.3 Sufficient LMI conditions We are now in position to state a stability theorem for system (1) based on the previous developments. Theorem 2 If there exist an order nin N∗and matrices (P n, R, S)in Snx+nnz×Snz +×Snz +such that the following LMI conditions hold Φ+ n=P n+"0 0 01 h(I1 n⊗S)#≻0,(9a) Φ− n="H(P nAn) + Ψ(R, S)P nBn ∗ −S#≺0,(9b) 3 where Ψ(R, S) = "C⊤(hR +S)C0 0−1 h(I1 n⊗R)#, An="A0 ℓn(1)C−1 hLnz n#,Bn="B −ℓn(0)#, ℓn(1) = "Inz . . . Inz#, ℓn(0) = "Inz . . . (−1)n−1Inz#∈Rnnz, Lnz n=L1 n⊗Inz∈Rnnz×nnz, L1 n(p, q) = ((2q−1)(1 −(−1)p+q)if p≥q, 0otherwise, ∀(p, q)∈ {1,...,n}2, then, the trivial solution of system (1) is GES. Proof : For a given integer nin N∗, consider the functional defined as follows Vn(xt) = "xt(0) ζn(Cxt)#⊤ P n"xt(0) ζn(Cxt)#(10) +Z0 −h x⊤ t(θ)C⊤((θ+h)R+S)Cxt(θ)dθ, where matrices P n,Rand Sare solution to the LMI stated in the theorem and where the augmented vector ζnis defined as follows ζn(Cxt) = Z0 −h ℓn(θ)Cxt(θ)dθ. (11) Applying the Bessel-Legendre inequality (7) yields Vn(xt)≥"xt(0) ζn(Cxt)#⊤ Φ+ n"xt(0) ζn(Cxt)#. Therefore, if condition (9a) is verified, then there exists a sufficiently small ε1>0 such that Vn(xt)≥ε1|xt(0)|2. In addition, since Vis quadratic with respect to xt, selecting ε2= ¯σ(P n) + h2 2¯σ(R) + h¯σ(S) ensures that inequality Vn(xt)≤ε2sup θ∈[−h,0]|xt(θ)|2holds. As in [27], computing the derivative of the functional along the trajectories of the system yields ˙ Vn(xt)=2"xt(0) ζn(Cxt) #⊤ P n"˙xt(0) ˙ ζn(Cxt) #−x⊤ t(−h)C⊤ SCxt(−h) +x⊤ t(0)C⊤ (hR+S)Cxt(0)−Z0 −h x⊤ t(θ)C⊤RCxt(θ)dθ. (12) Then, the dynamics (1a) and an integration by parts of ˙ ζn(Cxt) provide an expression h˙xt(0) ˙ ζn(Cxt)iwith respect to xt(0), ζn(Cxt) and xt(−h). It ensures that "˙xt(0) ˙ ζn(Cxt)#=An"xt(0) ζn(Cxt)#+BnCxt(−h). Re-injecting this expression into (12) and applying again the Bessel-Legendre inequality (7) yields ˙ Vn(xt)≤xt(0) ζn(Cxt) Cxt(−h)⊤ Φ− nxt(0) ζn(Cxt) Cxt(−h). Therefore, if condition (9b) is satisfied, then there exists a sufficiently small ε3such that ˙ Vn(xt)≤ −ε3|xt(0)|2, which concludes the proof by application of the Lyapunov-Krasovskii theorem.  Note that the proposed theorem has already been presented in [27] or in [28]. As in [27] or in [28], the previous stability condition is hierarchical with respect to n. This aspect is presented formally in the following lemma. Lemma 2 If there exists an integer Nin N∗, for which a solution to LMI (9) exists, then there also exists a solution to the same problem for any integer n≥N. Proof : The proof uses similar arguments to the ones provided in [28, Theorem 7]. A glimpse of the proof consists in introducing Pn+1 =P n0 0 0 , so that Vn+1(xt) = Vn(xt) and so that one can exhibit a solution to the LMI problem at order n+ 1 based on the solution at order n. The details of the proof are omitted but strongly relies on the structure of Φ+ nand Φ− n. To sum up the results presented so far, Theorem 2 presents scalable LMI conditions for the stability (GES) of LTI delay systems. These conditions is parameterized by nin N∗, corresponding to the degree of the Legendre polynomials considered in the construction of the candidate Lyapunov-Krasovskii functional. Moreover, it is demonstrated that when nincreases, the conservatism can only be reduced. It is however legitimate to wonder if a converse result can be proven. This direction refers to the possibility of assessing the satisfaction of these LMI for a sufficiently large n, when the system under consideration is known a priori to be GES for a given delay h. Apart the convergence, i.e. the existence of an order N∗for which these LMI conditions are guaranteed, an interesting underlying question is to estimate analytically such an order. The next developments aim at providing a solution to both problems. More specifically, the next section presents a converse theorem with a proof of existence of N∗, while the latter address the problem of estimating order N∗. 4 3 Necessity of LMI stability conditions: Existence of an order In this section, the objective is to prove the converse side of Theorem 2. In other words, we aim at proving that, if the trivial solution of system (1) is GES, then there exists an order N∗in N∗from which LMI conditions in (9) hold. The developments go against the grain of usual approaches and are inspired by the converse LyapunovKrasovskii theorem [?,19], stated below. Theorem 3 (Converse Lyapunov-Krasovskii theorem) If the trivial solution of system (1) is GES, then there exist positive scalars ε1,ε2and ε3and a continuous and differentiable functional Vdefined from Cpw(−h, 0; Rnx) to Rsuch that, for any ϕin Cpw(−h, 0; Rnx), inequalities (i) ε1|ϕ(0)|2≤ V(ϕ)≤ε2sup θ∈[−h,0]|ϕ(θ)|2hold and (ii) ˙ V(ϕ)≤ −ε3|ϕ(0)|2holds, where ˙ Vdenotes here the derivative of Valong the trajectories of (1), First, the so-called complete Lyapunov-Krasovskii functional is introduced. It fulfills the converse LyapunovKrasovskii theorem statement and serves as a target functional. Then, a link is established between such a complete functional and the functional used in the previous section with a particular structure. The two LMI conditions in (9) are consequently obtained. On one side, assuming that system (1) is GES, one ensures that (9a) holds, for any order n. On the other side, the time-derivative of the functional along the trajectories of system (1) becomes negative as the order increases so that (9b) holds, for sufficiently large orders. As a final step, the new converse theorem is stated and proven. Step 1. To introduce the complete Lyapunov-Krasovskii functional which can be expressed analytically with respect to a Lyapunov matrix function. Step 2. To make the link between the functional Vnintroduced in (10) and the Legendre approximation of the complete functional. Step 3. To assess that the condition (9a) holds, assuming that system (1) is GES. Step 4. To assess that the condition (9b) holds, for a sufficiently large order. Step 5. To conclude on the necessity of LMI conditions given in Theorem 2. The section is decomposed in five subsections respectively based on the steps listed above. 3.1 The complete Lyapunov-Krasovskii functional In this subsection, based on [17], we introduce the socalled complete Lyapunov-Krasovskii functional, which will be used in conjonction with Theorem 3. Definition 3 For any (W1, W2, W3)in Snx +×Snz +×Snz +, the complete Lyapunov-Krasovskii functional Vis V(xt)=x⊤ t(0)U(0)xt(0)+2x⊤ t(0) Z0 −h U(θ+h)BCxt(θ)dθ +Z0 −hZ0 −h x⊤ t(θ1)(BC)⊤U(θ2−θ1)BCxt(θ2)dθ1dθ2 +Z0 −h x⊤ t(θ)C⊤((θ+h)W2+W3)Cxt(θ)dθ, (13) where Uis the Lyapunov matrix defined from [−h, h]to Rnx×nxas the solution of the following matrix delayed differential equation      U′(θ) = −U(θ)A−U(θ+h)Ad,∀θ≤0, A⊤U(0)+A⊤ dU(h)+U(0)A+U(−h)Ad=−W, U(θ) = U⊤(−θ),∀θ > 0, (14) where W=W1+C⊤hW2+W3C. (15) Interestingly, using vec(M1M2M3)=(M⊤ 3⊗M2)vec(M2), for any matrices M1, M2, M3in Rnx×nx, a technique developed in [17, Section 2.10] provides an analytic expression of Ugiven by U(θ) = (vec−1[In2 x0]eθMN−10 −vec(W)if θ≤0, U⊤(−θ) if θ > 0, (16) where Wis given in (15) and M=h−A⊤⊗Inx−A⊤ d⊗Inx Inx⊗A⊤ dInx⊗A⊤i,(17) N=hIn2 x0 A⊤⊗Inx+Inx⊗A⊤A⊤ d⊗Inxi+h0−In2 x Inx⊗A⊤ d0ie−hM. It is well-known that the complete functional Vsatisfies the following lemma. Lemma 3 For any (W1, W2, W3)in Snx +×Snz +×Snz +, the time-derivative along the trajectories of system (1) of the complete Lyapunov-Krasovskii functional Vgiven by (13) yields ˙ V(xt)=−1 hZ0 −h xt(0) hCxt(θ) Cxt(−h)⊤ W10 0 ∗1 hW20 ∗ ∗ W3 xt(0) hCxt(θ) Cxt(−h). (18) Proof : The proof is provided in [17, Theorem 2.11].  Remark 1 For any symmetric positive definite matrices (W1, W2, W3)and by assuming that there is no characteristic root of (1) such that its opposite is also characteristic root, it is worth mentioning that this functional exists and is the unique solution of (18). For more details, one can refer to [17, Theorems 2.8 and 2.10]. 5 It is also well-known that the complete funtional Vis positive definite if the trivial solution of system (1) is GES as demonstrated in [13, Theorem 5.19]. In light of paper purpose, a variation of this result is proposed, which will corresponds exactly to the present framework. Lemma 4 Consider (W1, W2, W3)in Snx +×Snz +×Snz +. If the trivial solution of system (1) is GES, there exists a positive scalar ε > 0such that the complete functional Vgiven by (13) satisfies V(ϕ)−Z0 −h (θ+h)ϕ⊤(θ)C⊤W2Cϕ(θ)dθ ≥ε|ϕ(0)|2+Z0 −h|Cϕ(θ)|2dθ, (19) for any ϕin Cpw(−h, 0; Rnx). Proof : The proof is postponed to Appendix D.  Lastly, the Lyapunov matrix Uintroduced in Definition 3 satisfies several properties. The first one is related to its continuity. The second property concerns its derivative U′, which is only continuous over (0, h] and [−h, 0) and has a discontinuity at 0, which is measured as follows. Lemma 5 The Lyapunov matrix Ugiven by (16) verifies the following properties. (i) The Lyapunov matrix Uis continuous on [−h, h]and U(0) = U⊤(0). (ii) The Lyapunov matrix Uis infinitely differentiable on (0, h]and [−h, 0). Its derivative has a discontinuity only at 0so that ∆U′(0) := lim ǫ→0(U′(ǫ)−U′(−ǫ)) = W. Proof : The two items are proved in [17, Lemma 2.4 and Lemma 2.6], respectively.  These regularity conditions satisfied by the Lyapunov matrix have been used for H2[15] or H∞[16] analysis. They will also be at the heart of the derivation of the convergence results. To summarize, this subsection emphasized the structure of the complete functional whose kernels are defined by the Lyapunov matrix U. This particular structure has the benefits of ensuring by construction that its time-derivative along the trajectories of the system verifies (18) and that the positive definiteness is ensured for GES systems. 3.2 Construction of matrices Pn,Rand S In this subsection, the objective is to understand how to relate the functional defined in (10) and the complete one in (13), for a particular structure of matrices (Pn, R, S). To do so, the main idea is to exploit the terms of (13) that are expressed in U(θ+h)Band B⊤U(θ2−θ1)B. The Legendre polynomial approximation of these functions at any order nin N∗writes U(θ+h)B=U1,nℓnθ+h h+˜ U1,n(θ),∀θ∈[−h, 0], B⊤U(θ)B=U2,nℓnθ+h 2h+˜ U2,n(θ),∀θ∈[−h, h]. (20) In this decomposition, the constant matrices U1,n and U2,n have been selected as the orthogonal projection of U(θ+h)Band B⊤U(θ)B, respectively, on the nfirst Legendre polynomials ℓn. Their expressions are given by U1,n =1 hZ0 −h U(θ+h)Bℓ⊤ nθ+h hdθIn, U2,n =1 2h Zh −h B⊤U(θ)Bℓ⊤ nθ+h 2hdθ!In. (21) Functions ˜ U1,n(θ) and ˜ U2,n(θ) can be interpreted as the approximation errors of the orthogonal projections, and verify Z0 −h ˜ U1,n(θ)ℓ⊤ nθ+h hdθ=Z0 −h U(θ+h)Bℓ⊤ nθ+h hdθ |{z } =U1,n(In/h)−1 −U1,n Z0 −h ℓnθ+h hℓ⊤ nθ+h hdθ |{z } =(In/h)−1 = 0. Similarly, the same calculations ensure that error ˜ U2,n is orthogonal to the nfirst Legendre polynomials considered over [−h, h], i.e. Zh −h ˜ U2,n(θ)ℓ⊤ nθ+h 2hdθ= 0. The next developments aim at demonstrating the uniform convergence of the polynomial approximation. Following the theory of polynomial approximation (see for instance [2] and references therein), it results that ˜ U1,n and ˜ U2,n given in (20) converge to zero in the sense of the L2norm. This is actually a by-product of BesselLegendre inequality. This implies that the approximation errors converges to zero almost everywhere on their domain of definition, as ntends to infinity. Nevertheless, uniform convergence properties can even be obtained using the regularity of the Lyapunov matrix. 6 Lemma 6 Consider the Lyapunov matrix Ufor the time-delay system (1) defined for any (W1, W2, W3)in Snx +×Snz +×Snz +. The following statements hold. (i) The approximated Legendre function U1,nℓnθ+h h converges uniformly to U(θ+h)B, when ntends to infinity, on the closed interval [−h, 0]. More precisely, for any n≥4, the following inequality holds sup θ∈[−h,0]˜ U1,n(θ)≤¯u1,n := 1 √n−3|W|,(22) where 1=√nxπ 23 2eh|M| M2N−1h2|B|,(23) and matrices M,Nand Ware given by (17). (ii) The approximated Legendre function 1 hU1,nℓ′ nθ+h h converges uniformly to U′(θ+h)B, when ntends to infinity, on the interval [−h, 0). More precisely, for any n≥6, the following inequality holds sup θ∈[−h,0) ˜ U′ 1,n(θ)≤¯u2,n := 2 √n−5|W|,(24) where 2=1 2√nxπ 23 2eh|M| M4N−1h3|B|,(25) and matrices M,Nand Ware given by (17). (iii) The approximated Legendre function U2,nℓn(θ+h 2h) converges uniformly to B⊤U(θ)B, when ntends to infinity, on the closed interval [−h, h]. More precisely, for any n≥4, the following inequality holds sup θ∈[−h,h]˜ U2,n(θ)≤¯u3,n := 3 √n−3|W|,(26) where 3=√2π1 + √nxπheh|M| M2N−1h|B|2,(27) and matrices M,Nand Ware given by (17). Proof : The proof follows the arguments provided in [31, Theorem 2.5] and is based on the regularity properties of the Lyapunov matrix Uhighlighted in Lemma 5. The proof is postponed to Appendix E, for the sake of readability.  The previous lemma provides uniform upper bounds on the error done by three approximations related to the Lyapunov matrix U. These uniform convergence results are only presented as the polynomial approximations of Ubut addresses more general problems of functional analysis and approximation of continuous functions. All these upper bounds depend explicitly on the arbitrary symmetric positive definite matrices (W1, W2, W3) through the term |W|. As a final result, Lemma 6 implies the following corollary. Corollary 1 For any positive scalar η > 0, there exists an integer N∗ ηsuch that max (¯u1,n,¯u2,n,¯u3,n)≤η, ∀n≥N∗ η.(28) Proof : Since the upper bounds of approximation errors tend to 0 as ntends to infinity, the existence of such N∗ η for any η > 0 is guaranteed.  We are now in position to construct matrices (Pn, R, S) for given matrices (W1, W2, W3). The guiding principle is to mimic the complete Lyapunov-Krasovskii functional, replacing U(θ+h)Band B⊤U(θ2−θ1)Bby their Legendre polynomial approximation given by (20) in order to benefit from its properties as formulated in the following lemma. Lemma 7 For any (W1, W2, W3)in Snx +×Snz +×Snz +, define the functional Vnby (10) with matrices P n="U(0) U1,n ∗Tn#, R =W2, S =W3, Tn= 0 0 ZZ −h−h In hℓn θ1+h hU2,n(θ2−θ1 )ℓ⊤ nθ2+h hIn hdθ1dθ2, U2,n(θ)=U2,nℓnθ+h 2h, (29) where U1,n,U2,n are given in (21). The time-derivative of this functional along the trajectories of system (1) yields ˙ Vn(xt) = −1 hZ0 −hxt(0) hCxt(θ) Cxt(−h)⊤ Ψn(θ)xt(0) hCxt(θ) Cxt(−h),(30) where matrix Ψn(θ):=W1 +H(˜ U1,n(0)C) Ψ1 n(θ)−˜ U1,n(−h) ∗1 hW2Ψ2 n(θ) ∗ ∗ W3≻0,(31) is defined for any nin N∗and for all θin [−h, 0] with Ψ1 n(θ) = A⊤˜ U1,n(θ)−˜ U′ 1,n(θ) + C⊤˜ U2,n(θ), Ψ2 n(θ) = B⊤˜ U⊤ 1,n(θ)−˜ U⊤ 2,n(θ+h), and with ˜ U1,n,˜ U′ 1,n and ˜ U2,n being the approximation errors of the Lyapunov matrix Uof system (1). 7 Proof : To begin with, re-injecting the expression of U(θ+h)Band B⊤U(θ)Band using their approximation given by (20) into the complete functional Vleads to V(xt) = x⊤ t(0)U(0)xt(0) +2x⊤ t(0)U1,n Z0 −h ℓnθ+h hCxt(θ)dθ +Z0 −hZ0 −h x⊤ t(θ1)C⊤U2,nℓnθ2−θ1+h 2hCxt(θ2)dθ1dθ2 +Z0 −h x⊤ t(θ)C⊤(W2+ (θ+h)W3)Cxt(θ)dθ +2x⊤ tZ0 −h ˜ U1,n(θ)Cxt(θ)dθ +Z0 −hZ0 −h x⊤ t(θ1)C⊤˜ U2,n(θ2−θ1)Cxt(θ2)dθ1dθ2. (32) Since U2,n(θ2−θ1) = U2,nℓnθ2−θ1+h 2hbelongs to Rnz×nzand is a polynomial function of degree n−1 in both θ1and θ2, it can be decomposed using the basis of Legendre polynomials ℓnθ1+h hand ℓnθ2+h h. Thanks to the orthogonality of Legendre polynomials, this decomposition writes U2,nℓnθ2−θ1+h 2h=ℓ⊤ nθ1+h hTnℓnθ2+h h, where matrix Tnis the symmetric matrix given in (29). Note that the symmetry of Tnis ensured by the symmetry of ˜ U2,n highlighted in Property 2 that has been postponed in Appendix B in order to ease the reading. Hence, using the same augmented vector as in the proof of Theorem 2, i.e. ζn(Cxt) = R0 −hℓnθ+h hCxt(θ)dθ, functional Vreduces to the following expression V(xt) = "xt(0) ζn(Cxt)#⊤"U(0) U1,n ∗Tn#" xt(0) ζn(Cxt)# +Z0 −h x⊤ t(θ)C⊤(W2+ (θ+h)W3)Cxt(θ)dθ +2x⊤ t(0) Z0 −h ˜ U1,n(θ)Cxt(θ)dθ +Z0 −hZ0 −h x⊤ t(θ1)C⊤˜ U2,n(θ2−θ1)Cxt(θ2)dθ1dθ2. (33) Therefore, by selecting P n,Rand Sas in (29), functional Vnin (10) is retrieved and we obtain Vn(xt) = V(xt)−2x⊤ t(0) Z0 −h ˜ U1,n(θ)Cxt(θ)dθ −Z0 −hZ0 −h x⊤ t(θ1)C⊤˜ U2,n(θ2−θ1)Cxt(θ2)dθ1dθ2. Differentiating the previous expression leads to ˙ Vn(xt) = ˙ V(xt)−2 ˙x⊤ t(0) Z0 −h ˜ U1,n(θ)Cxt(θ)dθ −2x⊤ t(0) Z0 −h ˜ U1,n(θ)C˙xt(θ)dθ −Z0 −hZ0 −h ˙x⊤ t(θ1)C⊤˜ U2,n(θ2−θ1)Cxt(θ2)dθ1dθ2. −Z0 −hZ0 −h x⊤ t(θ1)C⊤˜ U2,n(θ2−θ1)C˙xt(θ2)dθ1dθ2. Since Vis the complete functional, its derivative is expressed using W1, W2, W3. Several integrations by parts lead to the following expression of ˙ Vn ˙ Vn(xt) = −1 hZ0 −h xt(0) hCxt(θ) Cxt(−h)⊤ W10 0 ∗1 hW20 ∗ ∗ W3 xt(0) hCxt(θ) Cxt(−h) −2(Axt(0) + BCxt(−h))⊤Z0 −h ˜ U1,n(θ)Cxt(θ)dθ −2x⊤ t(0) ˜ U1,n(0)Cxt(0) −˜ U1,n(−h)Cxt(−h) +2x⊤ t(0) Z0 −h ˜ U′ 1,n(θ)Cxt(θ)dθ −x⊤ t(0) Z0 −h C⊤(˜ U2,n(θ) + ˜ U⊤ 2,n(−θ)) |{z } =2 ˜ U2,n(θ) Cxt(θ)dθ +x⊤ t(−h) Z0 −h C⊤(˜ U2,n(θ+h)+ ˜ U⊤ 2,n(−h−θ)) |{z } =2 ˜ U2,n(θ+h) Cxt(θ)dθ +Z0 −hZ0 −h x⊤ t(θ1)C⊤˜ U′ 2,n(θ2−θ1)Cxt(θ2)dθ1dθ2, −Z0 −hZ0 −h x⊤ t(θ1)C⊤˜ U′ 2,n(θ2−θ1)Cxt(θ2)dθ1dθ2. We first notice that the two last terms of the previous expression are opposite and thus sum up to zero. Moreover, Property 2 given in Appendix B helps reducing the expression of the terms that depend on ˜ U2,n. Re-ordering the previous expression yields ˙ Vn(xt)=−1 hZ0 −hxt(0) hCxt(θ) Cxt(−h)⊤ Ψn(θ) xt(0) hCxt(θ) Cxt(−h)dθ, (34) where Ψnis given in (31) and concludes the proof.  To summarize the previous developments, a candidate functional Vnhas been built to fulfill the structure given in (10) and to follow the time-derivative of the complete functional that is defined for any symmetric positive definite matrices (W1, W2, W3). 8 3.3 Necessary condition for LMI condition (9a) As an extension to Lemma 4, the next developments aim at understanding how the positive definiteness condition on the complete functional Vcan be transferred to the candidate functional Vnand then to LMI (9a) at any order. This is formulated in the next lemma. Lemma 8 If the trivial solution of system (1) is GES, then condition Φ+ n≻0holds with matrices (P n, S, R) given by (29) for all nin N∗. Proof : Assume that system (1) is GES. According to Lemma 4, the complete functional Vin (13) verifies inequality (19) for a sufficiently small scalar ε > 0 for all ϕin Cpw(−h, 0; Rnx). In particular, consider ϕas a function expressed as follows ϕ(θ) = δ0(θ)Inx 1 hC†f⊤ n(θ)Inδ−h(θ)C†hϕ0 ϕn ϕhi,(35) with C†=C⊤(CC⊤)−1in Rnx×nzis the right pseudoinverse of Cand with fn(θ) = ℓn(θ)−ℓn(1)δ0(θ)−ℓn(0)δ−h(θ). In this formulation, vector hϕ0 ϕn ϕhiin Rnx+nz(n+1) is arbitrary and δθ0is zero everywhere except at θ0, where it equals 1, i.e. δθ0(θ) = (1 for θ=θ0, 0 otherwise. Note that such a function ϕhas been selected so that ϕat the boundary of [−h, 0] is given by ϕ(0) = ϕ0, ϕ(−h) = C†ϕh, and in the interval (−h, 0) is given by ϕ(θ) = 1 hC†ℓn(θ)Inϕn, for all θin (−h, 0)., which is a polynomial function of θof degree n−1. Re-injecting the particular function ϕinto the definition of Vin (13), the orthogonality of the Legendre polynomials decomposition (20) ensures that V(ϕ)=ϕ0U(0)ϕ0+2ϕ⊤ 0Z0 −h U(θ+h)Bℓ⊤ nθ+h hdθIn h | {z } U1,n ϕn +ϕ⊤ nIn h 0 0 ZZ −h−h ℓn θ1+h hB⊤U(θ2−θ1)Bℓ⊤ nθ2+h hdθ1dθ2In h |{z } Tn ϕn +Z0 −h x⊤ t(θ)C⊤(W2+ (θ+h)W3)Cxt(θ)dθ=Vn(ϕ), (36) where we recognize the functional Vnbuilt with matrices (P n, S, R) in (29). Hence, applying the Bessel-Legendre equality case (8) in Lemma 1 leads to V(ϕ)−Z0 −h ϕ⊤ (θ)C⊤(θ+h)W2Cϕ(θ)dθ=[ ϕ0 ϕn]⊤ Φ+ n[ϕ0 ϕn]. (37) Moreover,re-injecting (35) into the inequality (19) yields V(ϕ)−Z0 −h ϕ⊤ (θ)C⊤(θ+h)W2Cϕ(θ)dθ ≥ε|ϕ0|2+ϕ⊤ nInϕn≥ε|ϕ0 ϕn|2 . (38) Altogether, (37) and (38) ensure [ ϕ0 ϕn]⊤ Φ+ n[ϕ0 ϕn]≥ε|ϕ0 ϕn|2, for any vector [ ϕ0 ϕn] in Rnx+nnz, which guarantees the positive definiteness of matrix Φ+ n. 3.4 Existence of an order for having LMI condition (9b) As an extension to Lemma 7, the next developments use convergence arguments to asymptotically ensure the negative definiteness of the time-derivative of Vnalong the trajectories of system (1). The following lemma guarantees the satisfaction of LMI (9b), for sufficiently large orders. Lemma 9 There exists an order N∗in N∗such that LMI condition Φ− n≺0holds with matrices (P n, S, R)given by (29) for all n≥N∗. Proof : As a first step to the proof, let emphasize the structure of matrix Ψnin (31). Such a matrix can be decomposed as the sum of a block diagonal positive definite matrices that is independent of nand of a matrix whose entries are all expressed using the approximation errors ˜ Unand ˜ U′ n, which can be made uniformly arbitrarily small in light of convergence properties in Lemma 6. Therefore, we will use the following equivalence results. For any matrix Xin Rp×qsuch that X⊤X |X|2Iq, then inequality h|X|IpX X⊤|X|Iqi0 holds. Using this inequality, the following lower bounds of Ψnis derived Ψn(θ)µ1,nInx0 0 0µ2,nInz0 0 0 µ3,nInz, with µ1,n =σ(W1)−(1+|A|+2|C|)¯u1,n−¯u2,n−|C|¯u3,n, µ2,n =1 hσ(W2)−(|A|+|B|)¯u1,n −¯u2,n−(1+|C|)¯u3,n, µ3,n =σ(W3)−(1+|B|)¯u1,n −¯u3,n, (39) where ¯u1,n, ¯u2,n and ¯u3,n are the upper bounds of the approximation errors of the polynomial approximation given by (22), (24) and (26), respectively. Then, Corollary 1 ensures that all the negative terms 9 we get to Uk 2=−h 2(2k−1)B⊤∆U′(0)B(lk(1/2) −lk−2(1/2)) +h2 (2k−1)Z1 0 B⊤U′′ h(2θ−1)B(lk−2(θ)−lk(θ))dθ +h 2(2k+3)B⊤∆U′(0)B(lk+2(1/2) −lk(1/2)) −h2 (2k+3)Z1 0 B⊤U′′ h(2θ−1)B(lk(θ)−lk+2(θ))dθ. where ∆U′(0) := limǫ→0(U′(ǫ)−U′(−ǫ)). Then, an upper bound of the norm of Uk 2can be derived by the use of property (A.5) for θ∈[0,1] and especially for θ=1 2, yielding Uk 2≤√2πh |B|2 √k−2(2k−1) |∆U′(0)|+h Z1 0U′′ h(2θ−1)  pθ(1 −θ)dθ !. Thanks to Property 5 (ii), upper bound (E.1), inequality (2k−1) ≤2(k−2) and R1 0 dθ √θ(1−θ)=π, we have Uk 2≤√2πh |B|2 2(k−2)3 2 (1+ρπh)|W|=3|W| 2(k−2)3 2 ,∀k≥3. where notation 3:= √2π(1 + ρπh)h|B|2is introduced. For the same reasons stated in the proof of Lemma 6 (i), for any integer N≥nand for all θin [0,1], the following upper bound is obtained  N X k=n Uk 2lkθ+h h≤3|W|1 √n−3−1 √N−3. 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