Lyapunov Stability Analysis of a String Equation Coupled with an Ordinary

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To keep linear operator T is dissipative with A + BK non singular, the content clear, u(x, t) is assumed to be a scalar but the then there exists a unique solution (X,u,ut) of system (1) calculations are done as if it was a vector of any dimension. with initial conditions (X0,u0, v0) ∈ D(T ). Moreover, the The measurement we have access to is the state u at x = 1 solution has the following regularity property: (X,u,ut) ∈ which is the right extremity of the string and the control is a C(0, +∞, H). Dirichlet actuation (equation (1c)) because it affects directly Proof : This proof follows the same lines than in [21]. Ap- the state u and not its derivative. Another boundary condition plying Lumer-Phillips theorem (p103 from [28]), as the norm must be added. It is defined at x = 1 by ux(1) = −c0ut(1). is dissipative, it is enough to show that for all λ ∈ (0, λmax) This is a well-known damping condition when c0 > 0 (see with λmax > 0, D(T ) ⊂ R (λI − T ) where R is the range for example [17]). As presented in [4], we find this kind of operator. Let (r,g,h) ∈ H, we want to show that for this systems for instance when modeling a drilling mechanism. The system, there exists (X,u,v) ∈ D(T ) for which the following control is then given at one end and the measurement is done set of equation is verified: at the other end. λX − AX − Bu(1) = r, (2a) More generally, this system can be seen either as the control of the PDE by a finite dimensional dynamic control law λu(x) − v(x)= g(x), (2b) 2 generated by an ODE [8] or on the contrary the robustness λv(x) − c uxx(x)= h(x), (2c) of a linear closed loop system with a control signal conveyed by a damped string equation. On the first scenario, both the for all x ∈ Ω and a given λ> 0. Equations (2b), (2c) give: −1 −1 ODE and the PDE are stable and the stability of the coupled ∀x ∈ (0, 1), u(x)= k1 exp(λc x)+k2 exp(−λc x)+G(x) system is studied. The second case corresponds to an unstable x λg(s)+h(s) λ 2 but stabilizable ODE connected to a stable PDE. To sum up, where G(x) = 0 λc sinh c (s − x) ds ∈ H . k , k ∈ R are constants to be determined. Using the boundary this paper focuses on the stability analysis of closed-loop 1 2 R coupled system (1) with a potentially unstable closed-loop condition u(0) = KX, we get: ODE but a stable PDE. This differs significantly from the − − −1 ∀x ∈ (0, 1), u(x)=2k sinh(λc 1x)+ KXe λc x + G(x), backstepping methodology of [15] which aims at designing 1 an infinite dimensional control law ensuring the stability of a Taking its derivative at the boundary we get: − cascaded ODE-PDE system with a closed-loop stable ODE. −1 −1 −1 −λc 1 ux(1) = 2λc k1cosh(λc ) − λc KXe + G1, A. Existence and regularity of solutions with G1 ∈ R known. We also have ux(1) + c0v(1) = 0, −1 This subsection is dedicated to the existence and regularity leading to u(1) = G2 +KXf(λc ) with G2 ∈ R and f(y)= m − of solutions (X,u,ut) to system (1). We first introduce H = 1 − (cc0 1)sinh(y) e−y. Then using (2a), we get: Rn ×Hm ×Hm−1 for m > 1. We consider the classical norm 2(cosh(y)+cc0sinh(y)) 1 Rn 1 2 −1 on the Hilbert space H = H = × H × L : λIn − (A + BKf(λc )) X = r + BG2. 2 2 2 2 2 2 −1 k(X,u,v)kH = |X|n + kuk + c kuxk + kvk . Since f(λc ) → 1 when λ → 0 and A + BK is non −1 This norm can be seen as the sum of the energy of the ODE singular, there exists λmax > 0 such that A+BKf(λmaxc ) system and the one of the PDE. is non singular and −1 Remark 2: A more natural norm for space H would be ∀λ ∈ (0, λmax), det λIn − (A + BKf(λc )) 6=0. |X|2 +kuk2 +ku k2 +kvk2 which is equivalent to k·k2 . The n x H Rn norm used here makes the calculations easier in the sequel. Then, there is a unique X ∈ for a given (r,f,h ) ∈ H. Once the space is defined, we model system (1) using the We immediately get that (X,u,v) is in D(T ). Then for λ ∈ following linear unbounded operator T : D(T ) → H: (0, λmax), D(T ) ⊂ R(λI − T ). The regularity property falls from Lumer-Phillips theorem. X AX+Bu(1) u v T = 2 and B. Equilibrium point v c uxx 2 An equilibrium xeq = (Xe,ue, ve) ∈ D(T ) of system (1) D(T )= (X,u,v) ∈ H ,u(0) = KX,ux(1) = −c0v(1) . is such that T xeq = (0n,1, 0, 0) = 0H, i.e. it verifies the This operator T is said to be dissipative with respect to a following linear equations: norm if its time-derivative along the trajectories generated by 0= AX + Bu (1), (3a) T is strictly negative. The goal of this paper is then to find e e c2∂ u x , x , , (3b) an equivalent norm to k·kH which allows us to refine the 0= xx e( ) ∈ (0 1) dissipativity analysis of T . This equivalent norm is derived ve(x)=0, x ∈ (0, 1), (3c) from a general formulation of a Lyapunov functional, whose ue(0) = KXe, (3d) parameters are chosen using a semi-definite programming ∂ u (1) = 0. (3e) optimization process. x e Beforehand, from the semi-group theory, we propose the Using equation (3b), we get ue as a first order polynomial in following result on the existence of solutions for (1). x but in accordance to equation (3e), ue is a constant function. 3

Then, using equation (3d), we get ue = KXe. That leads to: The ODE dynamic can then be enriched by considering an (A + BK) Xe =0. We obtain the following proposition: extended system where X0 is viewed as a new dynamical state: Proposition 2: An equilibrium (Xe,ue, ve) ∈ H of system A+BK B˜ 0n,2 X˙ 0 = X + (χ(1) − χ(0)) , (6) (1) verifies (A + BK)Xe =0, ue = KXe, ve =0. Moreover, 02,n 02 0 cI2 if A + BK is not singular, system (1) admits a unique h ⊤ ⊤ ⊤i h i with X0 = [X X ] . Hence, associated to the original equilibrium (X ,u , v )=(0 , 0, 0)=0H. 0 e e e n,1 system (1), we propose a set of equation (4)-(6). They are linked to system (1) but enriched by extra dynamics aiming at III. AFIRST STABILITY ANALYSIS BASED ON MODIFIED representing the interconnection between the extended finite RIEMANN COORDINATES dimensional system and the two transport equations. Never- theless, these two systems are not equivalent. The transport This part is dedicated to the construction of a Lyapunov equation gives trajectories of ut and ux but u can be defined functional. We introduce therefore a new structure based on within a constant. The second set of equations just induces variables directly related to the states of system (1). a formulation for a Lyapunov functional candidate which is developed in the subsection below. A. Modified Riemann coordinates B. Lyapunov functional and stability analysis The PDE considered in system (1) is of second order in The main idea is to rely on the auxiliary variables satisfying time. As we want to use some tools already designed for first equations (4) and (6) to define a Lyapunov functional for the order systems, we propose to define some new states using original system (1). The associated Lyapunov function of ODE modified Riemann coordinates, which satisfy a set of coupled ⊤ (6) is a simple quadratic term on the state X0 P0X0, with P0 ∈ first order PDEs and diagonalize the operator. Let us introduce Sn+2 + . It introduces automatically a cross-term between the these coordinates, defined as follows: ODE and the original PDE through X0. Hence, the auxiliary + ut(x)+ cux(x) χ (x) equations of the previous paragraph shows a coupling between χ(x)= = . a finite dimensional LTI system and a transport PDE. For the u (1 − x) − cu (1 − x) χ−(1 − x) t x latter, inspired from the literature on time-delay systems [3], The introduction of such variables is not new and the reader [10], we provide a Lyapunov functional: can refer to articles [24], [4] or [10] and references therein 1 ⊤ about Riemann invariants. χ+ and χ− are eigenfunctions of V(u)= χ (x) (S + xR) χ(x)dx, equation (1b) associated respectively to the eigenvalues c and Z0 − S2 −c. Therefore, using χ (1 − x), the previous equation leads with S, R ∈ +. The use of the modified Riemann coordinates to a transport PDE: enables us to consider full matrices S and R. As the transport described by the variable χ is going backward, R is multiplied

∀t ≥ 0, ∀x ∈ Ω, χt(x, t)= cχx(x, t). (4) by x. Thereby, we propose a Lyapunov functional for system (1) expressed with the extended state variable X0: Remark 3: The norm of the modiﬁed state χ can be directly ⊤ related to the norm of the functions ut and ux. Indeed simple V0(X0,u)= X0 P0X0 + V(u). (7) calculations and a change of variables give: This Lyapunov functional is actually made up of three terms: 2 2 2 2 • kχk =2 kutk + c kuxk . (5) A quadratic term in X introduced by the ODE; • A functional V for the stability of the string equation; Remark 4: This manipulation does not aim at providing • A cross-term between X0 and X described by the ex- an equivalent formulation for system (1) but at identifying a tended state X0. manner to build a Lyapunov functional for system (1). The idea is that this last contribution is interesting since we The second step is to understand how the extra-variable χ may consider the stability of system (1) with an unstable ODE, interacts with ODE (1a). Hence using (1c), we notice: stabilized thanks to the string equation. At this stage, a stability theorem can be derived using the Lyapunov functional V . X˙ = AX + B (u(1) − u(0) + KX) , 0 Theorem 1: Consider the system deﬁned in (1) with a given 1 = (A + BK)X + B 0 ux(x)dx. speed c, a viscous damping c0 > 0 with initial conditions 0 0 0 Sn+2 (X ,u , v ) ∈ D(T ). Assume there exist P0 ∈ + and To express the last integral term usingR χ, we note that: S2 S, R ∈ + such that the following LMI holds: 1 1 1 + − ⊤ ˜ ⊤ ⊤ 2c ux(x)dx = χ (x)dx − χ (x)dx. Ψ0 =He Z0 P0F0 −cR0 +c H0 (S + R)H0 − G0 SG0 ≺ 0 Z0 Z0 Z0 (8) This expression allows us to rewrite the ODE system as where 1 ˙ ˜X X ˜ ⊤ X = (A + BK)X + B 0 where 0 := 0 χ(x)dx and B = 1 ⊤ ⊤ B [ 1 −1 ]. The extra-state X follows the dynamics: F0 = In+2 0n+2,2 ,Z0 = N0 c(H0 −G0) , 2c 0 R 1 h i ˜ h i N0 = A+BK B˜ 0n,2 , R0 = diag (0n,R, 02) , X˙ = c χ (x)dx = c [χ(1) − χ(0)] . 0 x h i (9) Z0 4

K 0 1 Using boundary condition (1c) and equality (5), it becomes G0 = 02,n+2 g + 01,n N0, g = 1+cc0 0 , H h i 01,n , h 1−cc0 0 . 2 2 2 2 2 0 = 02,n+2 h + K N0 = 0 −1 V0(X0,u) > ε1 |X|n + kuk + kutk + c kuxk h i 2ε1 2 2 2 2 Then, there exists a unique solution to system (1) and it + c2 kutk + ε1 2kuxk +2|u(0)| −kuk . is exponentially stable in the sense of k·kH i.e. there exist > Then, we apply Lemma 1 to ensure that the last term is γ 1,δ> 0 such that the following estimate holds for t> 0: 2 positive. It follows that V0(X0,u) > ε1 k(X,u,ut)kH, which 2 −δt 0 0 0 2 k(X(t),u(t),ut(t))kH 6 γe k(X ,u , v )kH. (10) ends the proof of existence of ε1. Sn+2 S2 4) Existence of ε2: Since P0 ∈ + and S, R ∈ +, there Remark 5: The LMI Ψ0 ≺ 0 includes a necessary condition ⊤ ⊤ exists ε2 > 0 such that for x ∈ (0, 1): 0 I given by e3 Ψ0e3 ≺ 0, with e3 = [ 2,n+2 2 ] , which is ⊤ ⊤ h S R h g Sg . This inequality is guaranteed if ε2 ( + ) − ≺ 0 P0 diag(ε2In, I2), and only if the matrix g−1h has its eigenvalues inside the unit 4 S + xR S + R ε2 I . cycle of the complex plan, i.e. c0 > 0, which is consistent 4 2 with the result on exponential stability of [14]. From equation (7), we get:

6 2 1 X⊤X 1 2 C. Proof of Theorem 1 V0(X0,u) ε2 |X|n + 4 0 0 + 4 kχk The proof of stability is presented below. 1 ⊤ ε2 + χ (x) S + xR − I2 χ(x)dx (12) 1) Preliminaries: As a first step of this proof, an inequality 0 4 6 2 1 2 on u is presented below. ε2R|X|n + 2kχk 1 Lemma 1: For u ∈ H , the following inequality holds: X⊤X 2 where we have used 0 0 6 kχk , which is a result of 2 2 2 kuk 6 2kuxk +2|u(0)| . Jensen’s inequality. The proof of the existence of ε2 ends by 1 using (5) so that we get: Proof : Since ux ∈ H (Ω), Young and Jensen inequalities imply that for all x ∈ Ω: 2 2 2 2 2 V0(X0,u) 6ε2 |X|n +kutk +c kuxk 6 ε2k(X,u,ut)kH. x 2 x u(x)2 = u (s)ds + u(0) 6 2 u2(s)ds +2|u(0)|2. 5) Existence of ε3: Differentiating V0 in (7) along the s s trajectories of system (1) leads to Z0 Z0 ⊤ X˙ X V˙0(X0,u)= He P0 X + V˙ (u). The proof of Theorem 1 consists in explaining how the LMI X˙ 0 0 condition presented in the statement implies that there exist a h i ˙ functional V and three positive scalars ε1,ε2 and ε3 such that Our goal is to express an upper bound of V0 thanks to the the following inequalities hold: extended vector ξ0 defined as follows: 2 6 6 2 ⊤ ε1k(X,u,ut)kH V (X,u) ε2k(X,u,ut)kH, ⊤ X⊤ (11) ξ0 = X 0 ut(1) cux(0) . (13) ˙ 2 V (X,u) 6 −ε3k(X,u,ut)kH. h i Let us first concentrate on V˙ . Equation (4) yields: The next steps aim at proving (11) in order to obtain the 1 convergence of the state to the equilibrium. ˙ ⊤ V(u)=2c χx (x, t)(S + xR)χ(x, t)dx. (14) 2) Well-posedness: If the conditions of Theorem 1 are 0 ⊤ Z satisfied, then the inequality Ψ0(1, 1) = e1 Ψ0e1 ≺ 0 holds ⊤ Integrating by parts the last expression leads to: where e1 = [ In 0n,4 ] . After some simplifications, we get ⊤ ˙ ⊤ ⊤ He (A + BK) Q ≺ 0, for some matrix Q depending on V(u)= c χ (1)(S + R)χ(1) − χ (0)Sχ(0) R, S and P . This strict inequality requires that A + BK is 0 1 non singular and, in light of Propositions 1 and 2, the problem − χ⊤(x)Rχ(x)dx . (15) is indeed well-posed and 0H is the unique equilibrium point. Z0 Furthermore, note that since Q is not necessarily symmetric, Then we note that X˙ = N0ξ0, X˙ 0 = c(H0 −G0)ξ0, χ(1) = then matrix A BK does not have to be Hurwitz. + H0ξ0, χ(0) = G0ξ0, with ξ0 defined in (13) and the matrices 3) Existence of ε : Conditions P ≻ 0 and S, R ∈ S2 1 0 + above in (9). We get X0 = F0ξ0 and X˙ 0 = Z0ξ0 which results mean that there exists ε1 > 0, such that for all x ∈ Ω: in the following expression for V˙ : ⊤ 0 P0 ε1diag In +2K K, 02 , ⊤ ⊤ ⊤ ⊤ 2 ˙ 2+c V0(X0,u)= ξ0 He Z0 P0F0 +cH0(S+R)H0 −cG0 SG0 ξ0 S + xR S ε1 2 I2. 2c 1 ⊤ These inequalities lead to: − c χ (x)Rχ(x)dx. (16) 0 2 Z > 2 2 2+c 2 Then, using the definition of Ψ given in (8), the previous V0(X0,u) ε1 |X|n + |KX| + 2c2 kχk 0 2 expression can be rewritten as follows: 1 ⊤ 2+c + χ (x) S + xR − ε1 2 I2 χ(x)dx 0 2c 1 ⊤ ⊤ ⊤ 2+c2 ˙ X X > 2 2 2 V0(X0,u)= ξ0 Ψ0ξ0 +c 0 R 0−c χ (x)Rχ(x)dx. (17) ε1R |X|n + |KX| + 2c2 kχk . Z0 5

k k! Since R ≻ 0 and Ψ0 ≺ 0, there exists ε3 > 0 such that: l = l!(k−l)! , form an orthogonal family with respect to the 2 ε 2+ c2 L inner product (see [11] for more details). R 3 I , (18a) 2c c2 2 2+ c2 Ψ − ε diag I +2K⊤K, I , 0 . (18b) A. Preliminaries 0 3 n 2c2 2 2 The previous discussion leads to the definition of the Using (18b) and the boundary condition u(0) = KX, equation 2 projection of any function χ in L on the family {L } N: (17) becomes: k k∈ 1 2 ˙ 2 2 2+c 2 ∀k ∈ N, Xk := χ(x)Lk(x)dx, V0(X0,u) 6 −ε3 |X|n +2|u(0)| + 2 kχk 2c 0 2 Z X⊤ ε3 2+c X N +c 0 R − 2c c2 I2 0 An augmented vector XN is naturally derived for any N ∈ : 1 2 ⊤ c χ⊤ x R ε3 2+ c I χ x dx, − 0 ( ) − 2c c2 2 ( ) ⊤ X⊤ X⊤ (20) XN = X 0 ··· N . so that we get by applicationR of Jensen’s inequality: Following the sameh methodology as ini Theorem 1, this 2+ c2 specific structure suggests to introduce a new Lyapunov func- V˙ X ,u 6 ε X 2 u 2 χ 2 . (19) 0( 0 ) − 3 | |n +2| (0)| + 2 k k n+2(N+1) 2c tional, inspired from (7), with PN ∈ S : + ⊤ The most important part of the proof lies in the following VN (XN ,u)= XN PN XN + V(u). (21) trick. Since (5) holds, we get: In order to follow the same procedure, several technical ˙ 6 2 2 2 extensions are required. Indeed, the stability conditions issued V0(X0,u) −ε3k(X,u,ut)kH − ε3 c2 kutk 2 2 2 from the functional V0 are proved using Jensen’s inequality −ε3 2|u(0)| +2kuxk −kuk . and an explicit expression of the time derivative of X0. Therefore, it is necessary to provide an extended version Moreover, Lemma 1 ensures that the last term of the pre- of Jensen’s inequality and of this differentiation rule. These vious expression is negative so that we have V˙ X ,u 6 0( 0 ) technicals steps are summarized in the two following lemmas. −ε k(X,u,u )k2 , which concludes this proof of existence. 3 t H Lemma 2: For any function χ ∈ L2 and symmetric positive 6) Conclusion: Finally, there exist ε ,ε ,ε > such 1 2 3 0 matrix R ∈ S2 , the following Bessel-like integral inequality that (11) holds for a functional V . Hence V defines + 0 0(·) holds for all N ∈ N: an equivalent norm to k·kH and is dissipative. It means, 1 N according to Propositions 1 and 2, that there exists a unique ⊤ X⊤ X χ (x)Rχ(x)dx > (2k + 1) k R k. (22) solution to system (1) in H. Equation (11) also brings: 0 ε3 Z k=0 V˙0(X0,u)+ V0(X0,u) 6 0 and X ε2 This inequality includes Jensen’s inequality as the particular ε − ε3 2 2 ε t 0 0 0 2 ∀t> 0, k(X(t),u(t),ut(t))kH 6 e 2 k(X ,u , v )kH, case N = 0, suggesting that this lemma is an appropriate ε1 extension and should help to address the stability analysis which shows the exponential convergence of all the trajectories using the new Lyapunov functional (21) with the augmented of system (1) to the unique equilibrium 0H. In other words, state XN defined in (20). the solution to system (1) is exponentially stable. The proof of Lemma 2 is based on the expansion of the pos- Remark 6: It is also worth noting that LMI (8) can be 1/2 2 N itive scalar kR χN k where χN (x) = χ(x) − k=0(2k + transformed to extend this theorem to uncertain ODE systems 1)XkLk(x) can be interpreted as the approximation error subject to polytopic-type uncertainties for instance. between χ and its orthogonal projection overP the family {Lk}k6N . The next lemma is concerned by the differentiation of X . IV. EXTENDED STABILITY ANALYSIS k Lemma 3: For any function χ ∈ L2, the following expres- In the previous analysis, we have proposed an auxiliary sion holds for any N in N: system presented in (4)-(6) helping us to define a new Lya- X˙ 0 X0 punov functional for system (1). The notable aspect is that . . . = c1N χ(1) − c1¯N χ(0) − cLN . , the term X = 1 χ(x)dx appears naturally in the dynamics . . 0 0 " X˙ # " XN # of system (1). In light of the previous work on integral N inequalities in [26],R this term can also be interpreted as the where ··· projection of the modified state χ over the set of constant ℓ0,0I2 02 I2 I2 . . . 1 . 1¯ . functions in the sense of the canonical inner product in L2. LN = . .. . , N = . , N = . , (23) " ··· # " # " − N # One may therefore enrich (6) by additional projections of χ ℓN,0I2 ℓN,N I2 I2 ( 1) I2 j+k over the higher order Legendre polynomials, as one can read with ℓk,j = (2j + 1)(1 − (−1) ) if j 6 k and 0 otherwise. in [26], [3] in the context of time-delay systems. The family of Proof : The proof of this lemma is presented in appendix shifted Legendre polynomials, denoted {Lk}k∈N and defined because of its technical nature. k k l k k+l l over [0, 1] by Lk(x) = (−1) l=0(−1) l l x with P 6

B. Main result 2) Existence of ε1: It strictly follows the proof in Theo- Taking advantage of the previous lemmas, the following rem 1 and is therefore omitted. extension to Theorem 1 is stated: 3) Existence of ε2: Since PN ,S and R are definite positive Theorem 2: Consider system (1) with a given speed c> 0,a matrices, there exists ε2 > 0 such that: 0 0 0 ε viscous damping c0 > 0 and initial conditions (X ,u , v ) ∈ 2 PN diag ε2In, 4 diag {(2k + 1)In}k∈(0,N) , D(T ). Assume that, for a given integer N ∈ N, there exist ε2 Sn+2(N+1) S2 (S + xR) S + R I2, ∀x ∈ (0, 1). PN ∈ + and S, R ∈ + such that inequality 4 ⊤ ˜ Then, from equation (21), we get: ΨN = He ZN PN FN − cRN ⊤ ⊤ N ε2 ⊤ + c HN (S + R)HN − GN SGN ≺ 0 (24) 2 X X 2 VN (XN ,u) 6 ε2|X|n + (2k+1) k k + kχk 4 ! holds, where Xk=0 2 1 2 ⊤ 6 ε2 |X| + kχk . ⊤ ⊤ n 2 FN = In+2(N+1) 0n+2(N+1),2 ,ZN = N cZ , N N While the first inequality is guaranteed by the constraint h i h i NN = A + BK B˜ 0 , ε2 n,2(N+1) (S+xR) 4 I2, for all x ∈ (0, 1), the second estimate results h ¯ i from the application of Bessel inequality (22). Therefore, ZN = 1N HN −1N GN − 0 L 0 , 2N+2,n N 2N+2,2 following the same procedure as in the proof of Theorem 1 h K i GN = 02,n+2(N+1) g + 01,n NN , after equation (12), there indeed exists ε2 > 0 such that V (X ,u) 6 ε k(X,u,u )k2 . h i 01,n N N 2 t H HN = 02,n+2(N+1) h + NN , K 4) Existence of ε3: Differentiating in time VN defined in h i RÑ = diag (0n,R, 3R, ··· , (2 N + 1)R, 02) , (21) along the trajectories of system (1) leads to: (25) ⊤ X˙ X 1 1¯ X and where matrices LN , N and N are given in (23). X˙ 0 0 V˙N (XN ,u)= He  . PN .  + V˙ (u). Then, the coupled infinite dimensional system (1) is expo-  .   .  2 . . nentially stable in the sense of norm k·k and there exist X˙ XN H  N   γ > 1 and δ > 0 such that energy estimate (10) holds.       ˙ Remark 7: Remark 5 also applies for this theorem and it The goal here is to find an upper bound of VN using ⊤ means that c must be strictly positive. In other words, these ⊤ 0 the following extended state: ξN = X ut(1) cux(0) . theorems cannot ensure the stability of the interconnection if N Using equation (15) and Lemmah 3, we note that XN i = the PDE is undamped. FN ξN , X˙ N = ZN ξN , χ(1) = HN ξN , χ(0) = GN ξN where Also note that Theorem 2 with N = 0 leads exactly to the matrices FN ,ZN ,HN , GN are given in (25). Then we can same conditions as presented in Theorem 1. write: N Remark 8: This methodology introduces a hierarchy in the ⊤ ⊤ V˙ (X ,u)= ξ Ψ ξ + c X (2k + 1)RX stability conditions inspired from what one can read in [26] N N N N N k k k=0 in the case of time-delay systems. More precisely, the sets X 1 − c χ⊤(x)Rχ(x)dx. (26) C = c> 0 s.t. ∃P ∈ Sn+2(N+1),S,R ∈ S2 , Ψ ≺ 0 N N + + N Z0 representingn the parameters c for which the LMI of Theoremo 2 Since R ≻ 0 and ΨN ≺ 0, there exists ε3 > 0 such that: 2 is feasible for a given system (1) and for a given N ∈ N, ε3 2+c R 2c c2 I2, satisfy the following inclusion: CN ⊆CN+1. In other words, if ⊤ there exists a solution to Theorem 2 at an order N0, then there ΨN −ε3diag In + K K, (27) 2 also exists a solution at any order N ≥ N0. The proof is very 2+c 2 diag {I2, 3I2,..., (2N +1)I2}, 02 . similar to the one given in [26]. We can proceed by induction 2c with P PN 0 and a sufficiently small ε> . Then, Using (27) and Bessel’s inequality, equation (26) becomes: N+1 = 0 εI2 0 ΨN ≺ 0 ⇒ ΨN+1 ≺ 0. The calculations are tedious and 2+ c2 technical and we do not intend to give them in this article. V˙ (X ,u) 6 −ε |X|2 +2|u(0)|2 + kχk2 , N N 3 n 2c2 C. Proof of Theorem 2 which is comparable to equation (19). Therefore, similarly, we The proof of dissipativity follows the same line as in 2 obtain V˙N (XN ,u) 6 −ε3 k(X,u,ut)kH. Theorem 1 and consists in proving the existence of positive 5) Conclusion: There exist ε1,ε2 and ε3 positive reals such scalars ε1,ε2 and ε3 such that the functional VN verifies the that inequalities (11) are satisfied and the exponential stability inequalities given in (11). of system (1) is therefore guaranteed. 1) Well-posedness: Using a similar reasoning to Theo- rem 1, a necessary condition for LMI (24) to be verified is V. EXAMPLES that A + BK is non singular. Then, according to Proposi- Three examples of stability for system (1) are provided here. tions 1 and 2, the problem is well-posed and 0H is the unique In each case, A+BK is non singular and therefore, there is a equilibrium. unique equilibrium. The solver used for the LMIs is “sedumi” 7

3.5 example, as N increases, the stability area is converging to

3 the exact one. N=0 N=1 2.5 N=2 B. Problem (1) with A + BK Hurwitz and A not Hurwitz. Freq 2

min Let us consider here, system (1) described by the following c 1.5 matrices: 1 2 1 1 − A = [ 0 1 ] , B = [ 1 ] ,K = [ 10 2 ] . (29) 0.5

0 As A is not Hurwitz, we are studying the stabilization of the 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 c 0 ODE through a communication medium modeled by the wave (a) System (28): A and A + BK Hurwitz equation. For the same reason as before, the wave speed must

30 be large enough for the control to be not too much delayed but also with a moderated damping to transfer the state variable 25 N=0 N=1 X through the PDE equation. Then, a c is appears as one Freq 0max 20 can see in Figure 1b. min

c Some numerical simulations have been performed on this 15 example. Figure 1b shows that for system (29) with c0 = 10 0.15, the minimum wave speed is cmin =6.83. The numerical

5 stability can also be seen in Figure 2 and indeed, the system is 0 0.05 0.1 0.15 0.2 0.25 c at the boundary of the stable area in Figure 2b and unstable for 0 (b) System (29): A not Hurwitz and A + BK Hurwitz smaller values of c. The results coming from the exact criterion and Theorem 2 are close even for small N. That means the 1.3 stability area provided with N =1 is a good estimation of the N=1 1.2 N=2 maximum stability set. 1.1 Freq 1 C. Problem (1) with A and A BK not Hurwitz.

min +

c 0.9

0.8 Consider an open loop unstable system deﬁned by:

0.7 0 1 0 1 0 (30) 0.6 A = −2 0.1 , B = [ 1 ] ,K = [ ] .

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 c Gain K has been chosen such that the closed loop is also 0 unstable. Surprisingly, the proposed methodology shows the (c) System (30): A and A + BK not Hurwitz stability for some pairs (c,c0). The results are presented in Fig. 1: Minimum wave speed cmin as a function of c0 for Figure 1c. The LMIs are not feasible for Theorem 2 with system (1) to be stable. The values for A, B and K are given N = 0. For N > 1, there is a stability area and the slope of by equations (28), (29) or (30). the right asymptotic branch is decreasing at each order. Hence, it appears that the introduction of the string equation in the feedback loop helps the stabilization of the closed loop system. with the YALMIP toolbox [19]. The dashed curve denoted For N =1, the stability area is quite far from the maximum “Freq” is obtained using a frequency analysis and displays the one but this gap reduces significantly for higher orders. exact stability area. This exact method is explained in [1] and does not use Lyapunov arguments. VI. CONCLUSION A hierarchy of stability criteria has been provided for the A. Problem (1) with A and A + BK Hurwitz stability of systems described by the interconnection between In this first numerical example, the considered system is a finite dimensional linear system and an infinite dimensional defined as follows: system modeled by a string equation. The proposed methodology relies on an extensive use of Bessel’s inequality, which −2 1 1 − (28) A = 0 −1 , B = [ 1 ] ,K = [ 0 2 ] . allows to design new and accurate Lyapunov functionals. This Matrices A andA + BK are Hurwitz. The ODE and the PDE new methodology encompasses the classical notion of energy are then stable if they are not coupled. As shown in Figure proposed in that case. In particular, the stability of the closed- 1a, the frequency argument shows that there exists a minimum loop or open-loop system is not a requirement anymore. Future works will include the study of robustness of this approach and wave speed called here cmin which is function of the damping the design of a controller. c0 for system (1) to be stable. The phenomenon induced by the coupling can be under- APPENDIX stood as the robustness of the ODE to a disturbance generated A. Proof of Lemma 3 by a wave equation. Intuitively, if the wave speed is large For a given integer k in N, differentiating of Xk along enough, the perturbation tends to 0 fast enough for the ODE X˙ 1 the trajectories of (4) yields k = c 0 χx(x)Lk (x)dx. Then, to keep its stability behavior. Another interesting thing to integrating by parts, we get R notice is the decrease of cmin as N increases (hierarchy of X˙ 1 1 ′ (31) the stability criteria with respect to the order N). For this k = c [χ(x)Lk(x)]0 − 0 χ(x)Lk(x)dx . R 8

(a) c = 10 (b) c = 6.83 (c) c = 6.5 0 KX0 ⊤ 0 Fig. 2: Chart of u for system (29) with the parameters: u (x) = (cos(πx)+1) 2 , X0 = [1 1] , v (x)=0 and c0 =0.15 for 3 values of c. These results are obtained using Euler forward as a numerical scheme.

In order to derive the expression of X˙ k, we use the following [11] R. Courant and D. Hilbert. Methods of mathematical physics. John properties of the Legendre polynomials. On the one hand, the Wiley & Sons, Inc., 1989. [12] B. d’Andréa Novel, F. Boustany, F. Conrad, and B. P. Rao. Feedback values of Legendre polynomials at the boundaries of [0 1] are stabilization of a hybrid PDE-ODE system: Application to an overhead k given by Lk(0) = (−1) and Lk(1) = 1. On the other hand, crane. Mathematics of Control, Signals and Systems, 7(1):1–22, 1994. the Legendre polynomials verifies the following differentiation [13] N. Espitia, A. Girard, N. Marchand, and C. Prieur. Event-based control of linear hyperbolic systems of conservation laws. Automatica, 70:275– rule for k> 0: 287, 2016. − [14] A. Helmicki, C. A. Jacobson, and C. N. Nett. Ill-posed distributed d k 1 L (x)= (2j+1)(1−(−1)j+k)L (x). parameter systems: A control viewpoint. IEEE Trans. on Automatic dx k j Control, 36(9):1053–1057, 1991. j=0 X [15] M. Krstic. Delay compensation for nonlinear, adaptive, and PDE systems. Springer, 2009. Hence, injecting these expressions into (31) leads to: [16] M. Krstic. Dead-time compensation for wave/string PDEs. Journal of X˙ k N X Dynamic Systems, Measurement, and Control, 133(3), 2011. k = c χ(1,t) − (−1) χ(0) − c j=0 ℓk,j j, [17] J. Lagnese. Decay of solutions of wave equations in a bounded region with boundary dissipation. Journal of Differential equations, 50(2):163– where the coefficient ℓk,j are defined inP equation (23). The 182, 1983. end of the proof consists in gathering the previous expression [18] J.-L. Lions. Exact controllability, stabilization and perturbations for from k = 1 to k = N, leading to the definition of matrices distributed systems. SIAM review, 30(1):1–68, 1988. ¯ [19] J. Löfberg. Yalmip: A toolbox for modeling and optimization in matlab. LN , 1N and 1N given in (23). pages 284–289, 2005. [20] Z.-H. Luo, B.-Z. Guo, and O.¨ Morgül. Stability and stabilization of REFERENCES infinite dimensional systems with applications. Springer Science & Business Media, 2012. [1] M. Barreau, F. Gouaisbaut, A. Seuret, and R. Sipahi. Input / output [21] O.¨ Morgül. A dynamic control law for the wave equation. Automatica, stability of a damped string equation coupled with ordinary differential 30(11):1785–1792, 1994. system. Working paper, 2018. [22] O.¨ Morgül. On the stabilization and stability robustness against small [2] L. Baudouin, A. Seuret, F. Gouaisbaut, and M. Dattas. Lyapunov delays of some damped wave equations. IEEE Trans. on Automatic stability analysis of a linear system coupled to a heat equation. IFAC- Control, 40(9):1626–1630, 1995. PapersOnLine, 50(1):11978 – 11983, 2017. 20th IFAC World Congress, [23] O.¨ Morgül. An exponential stability result for the wave equation. Toulouse. Automatica, 38(4):731–735, 2002. [3] L. Baudouin, A. Seuret, and M. Safi. Stability analysis of a system [24] C. Prieur, S. Tarbouriech, and J. M. G. da Silva. Wave equation coupled to a transport equation using integral inequalities. 2016. 2nd with cone-bounded control laws. IEEE Trans. on Automatic Control, IFAC Workshop on CPDE in Italy. 61(11):3452–3463, 2016. [4] D. Bresch-Pietri and M. Krstic. Output-feedback adaptive control of a [25] M. Safi, L. Baudouin, and A. Seuret. Refined exponential stability wave PDE with boundary anti-damping. Automatica, 50(5):1407–1415, analysis of a coupled system. IFAC-PapersOnLine, 50(1):11972 – 11977, 2014. 2017. 20th IFAC World Congress. [5] F. Castillo, E. Witrant, C. Prieur, and L. Dugard. Dynamic Boundary [26] A. Seuret and F. Gouaisbaut. Hierarchy of LMI conditions for the Stabilization of Linear and Quasi-Linear Hyperbolic Systems. In 51st stability analysis of time delay systems. Systems & Control Letters, Annual Conference on Decision and Control (CDC), pages 2952–2957. 81:1–7, 2015. IEEE, 2012. [27] S. Tang and C. Xie. State and output feedback boundary control for a [6] F. Castillo, E. Witrant, C. Prieur, and L. Dugard. Dynamic boundary coupled PDE–ODE system. Systems & Control Letters, 60(8):540–545, stabilization of first order hyperbolic systems. In Recent Results on 2011. Time-Delay Systems, pages 169–190. Springer Int. Publishing, 2016. [28] M. Tucsnak and G. Weiss. Observation and control for operator [7] E. Cerpa and C. Prieur. Effect of time scales on stability of coupled semigroups. Springer Science & Business Media, 2009. systems involving the wave equation. In IEEE Conf. on Dec. and Cont. [29] H.-N. Wu and J.-W. Wang. Static output feedback control via PDE (CDC’17), Melbourne, Australia, 2017. boundary and ODE measurements in linear cascaded ODE-beam sys- [8] G. Chen and J. Zhou. The wave propagation method for the analysis of tems. Automatica, 50(11):2787–2798, 2014. boundary stabilization in vibrating structures. SIAM Journal on Applied Mathematics, 50(5):1254–1283, 1990. [9] J. M. Coron. Control and nonlinearity. Number 136 in Mathematical Surveys and Monographs. American Mathematical Soc., 2007. [10] J.M. Coron, B. d’Andrea Novel, and G. Bastin. A strict Lyapunov function for boundary control of hyperbolic systems of conservation laws. IEEE Trans. on Automatic Control, 52(1):2–11, Jan 2007.