Adding Integral Action for Open-Loop Exponentially Stable Semigroups and Application to Boundary Control of PDE Systems A

Home , Exponential stability

Adding integral action for open-loop exponentially stable semigroups and application to boundary control of PDE systems A. Terrand-Jeanne, V. Andrieu, V. Dos Santos Martins, C.-Z. Xu

Abstract—The paper deals with output feedback stabilization of boundary regulation is considered for a general class of of exponentially stable systems by an integral controller. We hyperbolic PDE systems in Section III. The proof of the propose appropriate Lyapunov functionals to prove exponential theorem obtained in the context of hyperbolic systems is given stability of the closed-loop system. An example of parabolic PDE (partial differential equation) systems and an example of in Section IV. hyperbolic systems are worked out to show how exponentially This paper is an extended version of the paper presented in stabilizing integral controllers are designed. The proof is based [16]. Compare to this preliminary version all proofs are given on a novel Lyapunov functional construction which employs the and moreover more general classes of hyperbolic systems are forwarding techniques. considered. Notation: subscripts t, s, tt, ... denote the first or second derivative w.r.t. the variable t or s. For an integer n, I I.INTRODUCTION dn is the identity matrix in Rn×n. Given an operator A over a ∗ The use of integral action to achieve output regulation and Hilbert space, A denotes the adjoint operator. Dn is the set cancel constant disturbances for infinite dimensional systems of diagonal matrix in Rn×n. has been initiated by S. Pojohlainen in [12]. It has been extended in a series of papers by the same author (see [13] II.GENERAL ABSTRACT CAUCHYPROBLEMS for instance) and some other (see [23]) always considering A. Problem description bounded control operator and following a spectral approach Let X be a Hilbert space with scalar product h, i and A : (see also [11]). X D(A) ⊂ X → X be the infinitesimal generator of a C - In the last two decades, Lyapunov approaches have allowed 0 semigroup denoted t 7→ eAt. Let B and C be linear operators, to consider a large class of boundary control problems (see for B from m to X and C from D(C) ⊆ X to m. instance [2]). In this work our aim is to follow a Lyapunov R R In this section, we consider the controlled Cauchy problem approach to solve an output regulation problem. The results with output Σ(A, B, C) in Kalman form, as follows are separated into two parts. In a first part, abstract Cauchy problems are considered. ϕt = Aϕ + Bu + w , y = Cϕ, (1) It is shown how a Lyapunov functional can be constructed where w ∈ X is an unknown constant vector and u : for a linear system in closed loop with an integral controller 7→ m is the controlled input. We consider the following when some bounds are assumed on the control operator and for R+ R exponential stability property for the operator A. an admissible measurement operator. This gives an alternative Assumption 1 (Exponential Stability): The operator A gen- proof to the results of S. Pojohlainen in [12] (and [23]). It erates a C -semigroup which is exponentially stable. In other allows also to give explicit value to the integral gain that solves 0 words, there exist ν and k both positive constants such that,

arXiv:1901.02208v1 [math.AP] 8 Jan 2019 the output regulation problem. ∀ϕ ∈ X and t ∈ In a second part, following the same Lyapunov functional 0 R+ At design procedure, we consider a boundary regulation problem ke ϕ0kX 6 k exp(−νt)kϕ0kX . (2) for a class of hyperbolic PDE systems. This result generalizes many others which have been obtained so far in the regulation We are interested in the regulation problem. More precisely of PDE hyperbolic systems (see for instance [8], [24], [3], [20], we are concerned with the problem of regulation of the output [21], [18]). y via the integral control The paper is organized as follows. Section II is devoted u = kiKiz , zt = y − yref , (3) to the regulation of the measured output for stable abstract m m Cauchy problems. It is given a general procedure, for an where yref ∈ R is a prescribed reference, z ∈ R , Ki ∈ m×m exponentially stable semigroup in open-loop, to construct a R is a full rank matrix and ki a positive real number. Lyapunov functional for the closed loop system obtained with The control law being dynamical, the state space has been an integral controller. Inspired by this procedure, the case extended. Considering the system Σ(A, B, C) in closed loop, with integral control law given in (3) the state space is now m All authors are with LAGEP-CNRS, Universite´ Claude Bernard Lyon1, Xe = X × R which is a Hilbert space with inner product Universite´ de Lyon, Domaine Universitaire de la Doua, 43 bd du 11 Novembre > 1918, 69622 Villeurbanne Cedex, France. hϕea, ϕebiXe = hϕa, ϕbiX + za zb , 2

ϕa ϕb where ϕea = and ϕeb = . The associated norm is On another hand, if one wants to address nonlinear abstract za zb Cauchy problems or unbounded operators, we may need to denoted k · k . Let A :(D(A) ∩ D(C)) × m → X be the Xe e R e follow a Lyapunov approach. For instance in the context of extended operator defined as boundary control, a Lyapunov functional approach has allowed ABK k to tackle feedback stabilization of a large class of PDEs (see A = i i . (4) e C 0 for instance [2] or [5]). It is well known (see for instance [9, Theorem 8.1.3]) The regulation problem to solve can be rephrased as the that exponential stability of the operator A is equivalent to following. existence of a bounded positive and self adjoint operator P in Regulation problem: We wish to find a positive real number L(X) such that m ki and a full rank matrix Ki such that ∀(w, yref ) ∈ X × R : 1) The system (1)-(3) is well-posed. In other words, for hAϕ, PϕiX + hPϕ, AϕiX 6 −µkϕkX , ∀ ϕ ∈ D(A), (8) all ϕ = (ϕ , z ) ∈ X there exists a unique (weak) e0 0 0 e where µ is a positive real number. We assume that this ϕ(t) 0 Lyapunov operator P is given. The first question, we intend solution denoted ϕe(t) = ∈ C (R+, Xe) defined z(t) to solve is the following: Knowing the Lyapunov operator P, ∀t 0 and initial condition ϕe(0) = ϕe0. > is it possible to construct a Lyapunov operator Pe associated 2) There exists an equilibrium point denoted ϕe∞ = to the extended operator Ae? ϕ∞ ∈ Xe, depending on w and yref , which is To answer this question, we first give a construction based z∞ exponentially stable for the system (1)-(3). In other on a well-known technique in the nonlinear finite dimensional the forwarding words, there exist positive real numbers ν and k such control community named (see for instance e e [10], [15] or more recently [4], or [1]). that for all t > 0

−νet kϕe(t) − ϕe∞kXe 6 ke exp kϕe0 − ϕe∞kXe . B. A Lyapunov approach for regulation

3) The output y is regulated toward the reference yref . Inspired by the forwarding techniques, the following result More precisely, can be obtained. Theorem 2 (Forwarding Lyapunov functional): Assume that ∀ϕe0, lim |Cϕ(t) − yref | = 0. (5) t→+∞ all assumptions of Theorem 1 are satisfied and let P in L(X) be a positive self adjoint operator such that (8) holds. Then We know with the work of S. Pohjolainen in [13] that when there exist a bounded operator M : X → Rm and positive ∗ ∗ A generates an exponentially stable analytic semi-group, when real numbers p and ki , such that for all 0 < ki < ki , there B is bounded and when C is A-bounded, with a rank condition, exists µe > 0 such that the operator the regulation may be achieved. This result has been extended P + pM∗M −pM∗ to more general exponentially stable semi-groups in [23]. Pe = (9) −pM p Id Theorem 1 ([23]): Assume that X is separable and that A m satisfies Assumption 1. Assume moreover that : is positive and satisfies ∀ ϕe = (ϕ, z) ∈ D(A) × R 1) the operator B is bounded; 2 2 hAeϕe, PeϕeiXe + hPeϕe, AeϕeiXe 6 −µe(kϕkX + |z| ). 2) the operator C is A-admissible (see [22]), i.e. (10) • it is A-bounded : Proof: The operator A satisfying Assumption 1, 0 is in its |Cϕ| 6 c(kϕkX + kAϕkX ) , ∀ ϕ ∈ D(A), (6) resolvent set and consequently A−1 : X 7→ D(A) is well for some positive real number c; defined and bounded. Let M : X → Rm be defined by M = −1 • there exist T > 0 and cT > 0 such that CA which is well defined due to the fact that D(A) ⊆ D(C) Z T since C is A-bounded. Moreover, with (6) ∀ϕ ∈ X At 2 2 2 |Ce ϕ| dt 6 cT kϕkX , ∀ ϕ ∈ D(A); −1 −1 0 |Mϕ| = |CA ϕ| 6 c kA ϕkX + kϕkX 6 c˜kϕkX , 3) the rank condition holds. In other words operators A, B where c˜ is a positive real number. Hence, M is a bounded and C satisfy linear operator. Moreover, M satisfies the following equation rank{CA−1B} = m; (7) MAϕ = Cϕ , ∀ϕ ∈ D(A) . (11) ∗ K = (CA−1B)−1 then there exists a positive real number ki and a m×m matrix Let i which exists due to the third assump- ∗ Ki, such that for all 0 < ki < ki the operator Ae given in tion of Theorem 1. Note that, (4) is the generator of an exponentially stable C0-semigroup > hϕe, PeϕeiXe = hϕ, PϕiX + p(z − Mϕ) (z − Mϕ), (12) in the extended state space Xe. More precisely, the system (1) in closed loop with (3) is well-posed and the equilibrium is hence Pe is positive. This candidate Lyapunov functional is exponentially stable. Moreover, for all w and yref , equation similar to the one given in [4, Equation (34)]. It is selected (5) holds (i.e the regulation is achieved). following a forwarding approach. 3

Moreover, we have On another hand, taking the the value of a and b given by (17) and (18), one can rewritten them with 0 < β < 1 and

hAeϕe, PeϕeiXe + hPeϕe, AeϕeiXe = 0 < θ < 1 as hAϕ, Pϕi + hPϕ, Aϕi 2 p X X b = β, a = 2(1 − β)θ . (22) > kMk2 α + 2p(z − Mϕ) (Cϕ − MAϕ) + kihϕ, PBKiziX > Then + kihPBKiz, ϕiX − 2p(z − Mϕ) MBKikiz. ab 2(1 − β)βθp = (23) Employing equation (11) and MBKi = Idm, the former pa + b (1 − β)θkMk2p2 + αβ inequality becomes which right expression is a function of p taking its maximum value when hA ϕ , P ϕ i + hP ϕ , A ϕ i = s e e e e Xe e e e e Xe αβ hAϕ, Pϕi + hPϕ, Aϕi + k hϕ, PBK zi p = X X i i X (1 − β)θkMk2 > + kihPBKiz, ϕiX − 2p(z − Mϕ) kiz. (13) then 2 ( ) Let kPBKikX = α which is well deﬁned due to the bound- µp k∗ = sup , edness assumption on B. Given a, b positive constants, the i p p2 (24) a,b,p,such that (17)−(18) + following inequalities hold a b ( ) pβ(1 − β)θ 1 aα = µ sup √ hϕ, PBK zi kϕk2 + |z|2, (14) (25) i X 6 2a X 2 0<θ<1, 0<β<1 αkMk 2 > 1 2 bkMk 2 It is reached for β = 1 and θ = 1 and this yields z Mϕ 6 kϕk + |z| , (15) 2 2b 2 µ µ ∗ √ ki = = −1 −1 −1 . it yields given (8) that 2kMk α 2kCA kkPB(CA B) kX Of course, this optimal value depends on the considered hAeϕe, PeϕeiXe + hPeϕe, AeϕeiXe Lyapunov operator P solution of (8). Note that a possible ki pki 2 solution to this equation with µ = 1 is given for all (ϕ1, ϕ2) 6 −µ + + kϕkX a b in X 2 by (see [9]) 2 2 + ki p(−2 + bkMk ) + aα |z| . (16) Z t As As hϕ1, Pϕ2iX = lim he ϕ1, e ϕ2iX ds. We pick b sufﬁciently small such that t→+∞ 0

2 Due to (2) it is well defined and positive. Note also that we − 2 + bkMk < 0. (17) have k2 a p kPkX . In a second step, we select sufficiently small and suffi- 6 2ν ciently large such that This implies, the following corollary. p(−2 + bkMk2) + aα < 0. (18) Corollary 1 (Explicit integral gain): Given a system Σ(A, B, C) satisfying the assumptions of the Theorem 1, ∗ −1 −1 Finally, picking ki sufficiently small such that points 1), 2), and 3) of Theorem 1 hold with Ki = (CA B) and k∗ pk∗ ν − µ + i + i < 0 (19) k∗ = . (26) a b i kCA−1kk2kB(CA−1B)−1k the result is obtainedn with o An interesting question would now to know in which aspect µ = min µ − ki − pki , p(2 − bkMk2) − aα . 2 e a b this value may be optimal.

D. Illustration on a parabolic systems C. Discussion on the result Consider the problem of heating a bar of length L = 10 A direct interest of the Lyapunov approach given in Theo- with both endpoints at temperature zero. We control the heat ∗ ﬂow in and out around the points s = 2, 5, and 7 and measure rem 2, is that it allows to give an explicit value for ki which appears in Theorem 1. We may compute the largest value of the temperature at points 3, 6, and 8. The problem is to ﬁnd ∗ an integral controller such that the measurements at s = 3, 6, ki following this route. First of all, from (19) and 8 are regulated to (for instance) 1, 3, and 2, respectively. ( ) µp Thus the control system is governed by the following PDE k∗ = sup , i p p2 (20) a,b,p,such that (17)−(18) a + b 1 1 φt(s, t) = φss(s, t) + [ 3 , 5 ](s)u1(t) + [ 9 , 11 ](s)u2(t) ab 2 2 2 2 1 = µ sup (21) + [ 13 , 15 ](s)u3(t), (s, t) ∈ (0, 10) × (0, ∞) (27) a,b,p,such that (17)−(18) pa + b 2 2 4 where φ : [0, +∞) × [0, 10] → R with boundary conditions which gives kCA−1k ≤ 6.2466. We have   φ(0, t) = φ(10, t) = 0 −1.250 1.500 −1.125 Ki =  0.500 −2.000 2.250  . φ(s, 0) = φ0(s), (28) 0 1.000 −2.000 For the open-loop system, consider the Lyapunov operator where 1 : [0, 10] → denotes the characteristic function 2 [a,b] R P = I . Then the growth rate may be taken as µ = π . It on the interval [a, b], i.e., d √ 50 is easy to see that kKik = 4.2433, and kBk ≤ 3. Putting 1 ∀ s ∈ [a, b], together the numerical values into the formula (26) allows to 1 (s) = [a,b] 0 ∀ s 6∈ [a, b]. estimate the tuning parameter

∗ ω −3 The output and the reference are given as ki = −1 ≈ 2.1498 ∗ 10 . 2kBKik kCA k φ(t, 3) 1 With Corollary 1, the integral controller (3) with 0 < ki < y(t) = φ(t, 6) , yref = 3 . 2.1498 ∗ 10−3 stabilizes exponentially the equilibrium along φ(t, 8) 2 solutions of the closed-loop system and drives asymptotically the measured temperatures to the reference values for any Let the state space be the Hilbert space X = L2((0, 10), ) R initial condition. with usual inner product, and let the input space and the output space be equal to 3. Clearly, from (28), we get the semigroup R III.CASE OF BOUNDARY REGULATION FOR HYPERBOLLIC generator A : D(A) → X , the input operator B : 3 → X R PDES and the output operator C : D(A) → R3 as follows: In the following section we adapt this framework to hyper- D(A) = {ϕ ∈ H2(0, 10) | ϕ(0) = ϕ(10) = 0}, bolic PDE systems with boundary control. and A. System description Aϕ = ϕ ∀ ϕ ∈ D(A), ss To illustrate the former abstract theory, we consider the case 1 1 1 of hyperbolic partial differential equations as studied in [6]. Bu = [ 3 , 5 ]u1 + [ 9 , 11 ]u2 + [ 13 , 15 ]u3, 2 2 2 2 2 2 More precisely, the system is given by a one dimensional n×n and hyperbolic system ϕ(3) Cϕ = ϕ(6) . φt(s, t) + Λ0(s)φs(s, t) + Λ1(s)φ(s, t) = 0 ϕ(8) s ∈ (0, 1), t ∈ [0, +∞), (29)

Moreover, note that with Sobolev embedding, an integration where φ : [0, +∞) × [0, 1] → Rn by part and by completing the square, we have for all ϕ in Λ (s) = diag{λ (s), . . . , λ (s)} D(A) 0 1 n λi(s) > 0 ∀i ∈ {1, . . . , `} Z 10 Z 10 2 2 sup |ϕ(s)| ≤ c ϕ(s) ds + c ϕs(s) ds λi(s) < 0 ∀i ∈ {` + 1, . . . , n}, s∈(0,10) 0 0 1 10 where the maps Λ0 is in C ([0, 1]; Dn) and Λ1 is in Z 1 n×n ≤ ckϕkX + c |ϕ(s)ϕss(s)|ds C ([0, 1]; R ) with the initial condition φ(0, s) = φ0(s) n 0 for s in [0, 1] where φ0 : [0, 1] → R and with the boundary 3 1 conditions ≤ ckϕkX + ckϕsskX . 2 2 φ+(t, 0) φ+(t, 1) = K + Bu(t) + wb (30) Hence C is A-bounded. φ−(t, 1) φ−(t, 0) Moreover, by direct computation we ﬁnd that K11 K12 φ+(t, 1) B1 = + u(t) + wb (31) 14 15 9  K21 K22 φ−(t, 0) B2 −1 −1 CA B =  8 20 18 . 10 φ+ + ` − n−` 4 10 14 where φ = with φ in R , φ in R and where φ− w p u(t) It is easy to see that the above matrix is regular. Consequently b in R is an unknown disturbance, is a control input m K B all Assumptions of Theorem 1 hold. With Corollary 1, it is taking values in R and , are matrices of appropriate possible to compute explicitly the integral controller gain. By dimensions. The output to be regulated to a prescribed value denoted direct computation we have for all ϕ in X by yref , is given as a disturbed linear combination of the  3 R 10 R 3  boundary conditions. Namely, the outputs to regulate are in 10 0 (s − 10)ϕ(s)ds + 0 (3 − s)ϕ(s)ds Rm given as CA−1ϕ =  3 R 10 R 6  ,  5 0 (s − 10)ϕ(s)ds + 0 (6 − s)ϕ(s)ds    φ+(t, 0) φ+(t, 1) 4 R 10 R 8 y(t) = L1 + L2 + wy, (32) 5 0 (s − 10)ϕ(s)ds + 0 (8 − s)ϕ(s)ds φ−(t, 1) φ−(t, 0) 5

m×n n×n where L1 and L2 are two matrices in R and wy is an Assumption 3 (Rank condition 1): The matrix in R m unknown disturbance in R . We wish to find a positive real Φ−(1) − KΦ+(1) is full rank and so is the matrix T defined number ki and a full rank matrix Ki such that as −1 u(t) = kiKiz(t) , zt(t) = y(t) − yref , z(0) = z0 (33) T1 = (L1Φ−(1) + L2Φ+(1)) (Φ−(1) − KΦ+(1)) B. m m (38) where z(t) takes value in R and z0 ∈ R solves the m regulation problem ∀yref ∈ R . Another rank condition has to be introduced. This one is The state space denoted by Xe of the system (29)-(30) in used when solving the forwarding equation. Let Ψ : [0, 1] 7→ closed loop with the control law (33) is the Hilbert space Rn×n be the matrix function solution to the system defined as: −1 2 n m Ψs(s) = Ψ(s) (Λ1(s) − Λ0s(s)) Λ0(s) , Xe = (L (0, 1), R ) × R , (39) Ψ(0) = Idn . equipped with the norm defined for ϕe = (φ, z) in Xe as: Assumption 4 (Rank condition 2): The matrix in Rn×n 2 m kvkXe = kφkL ((0,1),R ) + |z|. Ψ(1)Λ (1)K − Λ (0)K (40) We introduce also a smoother state space defined as: 0 + 0 − 1 n m is full rank and so is the matrix Xe1 = (H (0, 1), R ) × R . B 0 T = −L B + M Λ (0) 1 − Ψ(1)Λ (1) 2 1 0 0 0 B B. Output regulation result 2 In this section, we give a set of sufficient conditions where allowing to solve the regulation problem as described in the −1 M = (L1K + L2) (Λ0(0)K− − Ψ(1)Λ0(1)K+) . (41) introduction. Our approach follows what we have done in the former section. Following [2, Proposition 5.1, p161] we With these assumptions, the following result may be obtained. consider the following assumption. Theorem 3 (Regulation for hyperbolic PDE systems): As- Assumption 2 (Input-to-State Exponential Stability): There sume that Assumptions 2, 3 and 4 are satisfied then with 1 −1 ∗ ∗ exist a C function P : [0, 1] → Dn, a real numbers µ > 0, Ki = T2 there exists ki > 0 such that for all 0 < ki < ki P , P and a positive definite matrix S in Rn×n such that the output regulation is obtained. More precisely, for all p m m > (wb, wy, yref ) in R × R × R , the following holds. (P (s)Λ0(s))s − P (s)Λ1(s) − Λ1 (s)P (s) 1) For all (φ0, z0) in Xe (resp. X1e) which satisfies the 6 −µP (s), (34) boundary conditions (30) (resp. the C1 compatibil- P Idn 6 P (s) 6 P Idn , ∀s ∈ [0, 1], (35) ity condition), there exists a unique weak solution to (29)-(30)-(33) that we denote v and which belongs to and 0 C ([0, +∞); Xe) (Respectively, strong solution in: > > − K+ P (1)Λ0(1)K+ + K− P (0)Λ0(0)K− 6 −S. (36) 0 1 C ([0, +∞); Xe1) ∩ C ([0, +∞); Xe)). (42) where 2) There exists an equilibrium state denoted v∞ in Xe Id 0 K11 K12 K = ` ,K = (37) which is globally exponentially stable in Xe for system + K K − 0 I 21 22 dn−` (29)-(30)-(33). More precisely, we have for all t > 0:

As it will be seen in the following section, this assump- kv(t) − v∞kXe 6 k exp(−νt)kv0 − v∞kXe . (43) tion is a sufficient condition for exponential stability of the 3) Moreover, if v satisfies the C1-compatibility condition equilibrium of the open loop system. It can be found in [2] 0 and is in X , the regulation is achieved, i.e. in the case in which S may be semi-definite positive. The 1e positive definiteness of S is fundamental to get an input-to- lim |y(t) − yref | = 0. (44) state stability (ISS) property of the open loop system with t→+∞ respect to the disturbances on the boundary. More general The next section is devoted to the proof of this result. results are given in [14]. The second assumption is related to the rank condition. Let Φ : [0, 1] → Rn×n be the matrix function solution to the C. About this result system The first assumption needed in Theorem 3 is Assumption 2. −1 Φs(s) = Λ0(s) Λ1(s)Φ(s), When considering only integral control laws, there is no hope

Φ(0) = Idn . to obtain the result without assuming exponential stability of the open loop system. Assumption 2 is slightly more restrictive Φ (s)Φ (s) We denote Φ(s) = 11 12 and than exponential stability since it requires an ISS property with Φ21(s)Φ22(s) respect to the input u. In the case in which this assumption is not satisﬁed for a given hyperbolic system, a possibility is Φ11(1) Φ12(1) Id` 0 Φ+(1) = , Φ−(1) = to modify the boundary condition via a static output feedback 0 Idn−` Φ21(1) Φ22(1) 6

π (or proportional feedback) following the route of [2] in order from the value we get following a frequency approach ( 2 in to satisfy this assumptions. this case). Recently in [7], the Lyapunov functional obtained One interest of our approach is that, part of the exponential in [20] has been modified to reach this optimal value of the stability of the closed loop system, only Assumptions 3 and 4 integral gain. A natural question for future research topic is which are rank conditions involving the boundary conditions to know if it is possible to modify the Lyapunov functional have to be satisfied. In the case in which the two above obtained in Theorem 3 following the methods of [7] to remove mentioned assumptions are not satisfied, we may obtain these the conservatism. properties by adding a proportional feedback and consequently changing the value of K in T1 and T2 to obtain these rank conditions. These Assumptions 3 and 4 are version of Point D. Illustration in a 2 × 2 hyperbolic system 3) in Theorem 1. Theorem 3 generalizes many available results on output In the particular case in which Λ0 is constant and Λ1 = 0, regulation via integral action for hyperbolic PDEs available in the matrix function Φ(s) and Ψ(s) are simply equal to identity the literature. For instance, the case of 2 × 2 linear hyperbolic for all s in [0, 1]. In that case, it yields systems has been considered in [19], [8], (see also [2, Section 2.2.4]). The case of cascade of such systems is also considered T = (L + L )(I −K)−1B, 1 1 2 dn (45) in [21]. Note also that in [17], this procedure is applied on and, a Drilling model which is composed of a hyperbolic PDE coupled with a linear ordinary differential equation. −1 B1 T2 = −L1B + (L1K + L2)(K− − K+) In order to compare the way we improve existing results, −B2 the same example as in [8] is considered. In this context, the −1 K11 − Id` K12 linearized de Saint-Venant equations can be written in the form = − L1B + (L1K + L2) −K21 Idn−` −K22 of (29)-(30). After normalization, one gets : I 0 × d` B c 0 0 − Id Λ (s) = and Λ (s) = 0 , ∀s (47) ` 0 0 −d 1 2×2 = − L1B + (L1K + L2) −1 I 0 K − I K where c > 0 and d > 0 and × d` 11 d` 12 B 0 − I −K I −K d` 21 dn−` 22 0 k0 b0 0 −1 K = and B = , (48) = − L1B − (L1K + L2)(Idn −K) B k1 0 0 b1 −1 = [−L1(Id −K) − (L1K + L2)] (Id −K) B n n with b0 6= 0 and b1 6= 0. For the system (29)-(30) with these −1 = −(L1 + L2)(Idn −K) B. (46) parameters, it is shown in [8] that the output of dimension m = 2 defined in (32) with Hence when Λ0 is constant and Λ1 = 0, Assumption 3 and Assumption 4 are equivalent. c 0 0 d L = c+d L = c+d Also, an interesting aspect of this Lyapunov approach is 1 −1 and 2 1 (49) 0 c+d c+d 0 ∗ that explicit values of the supremum value of the gain ki may be given. For instance, as in [20] consider the very particular can be regulated with an integral control law provided case of a transport equation. In this case the system is simply c |k k | < 1 , |k | < 1 , |k | < . (50) 0 1 0 1 d φt(s, t) + φs(s, t) = 0, s ∈ (0, 1), t ∈ [0, +∞) On another hand, employing ([7]-[8]), Assumptions 2 is φ(t, 0) = u(t) + wb satisfied assuming that |k k | < 1. Moreover, with equations y(t) = φ(1, t) + w 0 1 y (45) and (46), it yields, We can apply Theorem 3 with n = 1, Λ0(s) = −1, Λ1(s) = 0, −1 1 c d 1 −k0 b0 0 K = 0, B = 1, L1 = 0, L2 = 1. This yields Ψ(s) = 1, T1 = −T2 = . c + d 1 −1 −k1 1 0 b1 Ψ(s) = 1, T1 = 1, T2 = −1. Hence, Assumptions 3 and 4 are satisfied. Assumption 2 is satisfied for all µ > 0 with P (s) = This matrix is well defined and full rank if |k0k1| < 1 and −µs −µ e , S = 1, P = 1, P = e . In that case, employing consequently Assumptions 3 and 4 are always satisfied. Hence, ∗ theorem 3, it yields that there exists ki > 0 such that for all employing Theorem 3, both outputs defined in (49) can be ∗ 0 < ki < ki with u(t) = −kiz, z˙ = y, the output regulation is regulated with an integral control law with the only assumption obtained and so the output converges asymptotically to zero. that |k0k1| < 1. Following the proof of Theorem 3, equation (85) gives Then ∗ p −µ ki = µe . −1 Ki = T2 b (1 − k ) b (d + ck ) This bound is better then the one obtained in [20] for the linear = ϑ 1 0 1 0 (51) transport equation (its maximal value is obtained for µ = 1 b0(1 − k1) b0(c + dk1) and is √1 . Note however that similar to the bound of [20], 2 2 −1 −1 e −(c + d) (1 − k0k1) b0 b1 the result obtained with our novel Lyapunov functional is far with ϑ = (52) [(1 − k0)(c + dk1) + (1 − k1)(d + ck0)] 7

and Moreover, with zt = 0, we have √ ∗ µP φ∞+(0) φ∞+(1) ki = q L1 + L2 = yref − wy, −1 φ∞−(1) φ∞−(0) |M|Ψ c T2 (53) Hence,

µ is given in [8], and P the lower bound of the Lyapunov (L1Φ−(1) + L2Φ+(1)) φ∞(0) = yref − wy, (55) can be deduced easily from the expression of the Lyapunov function involved. On another side, boundary conditions (30) gives M has been deﬁned above, with T2. As Ψ is the identity (Φ−(1) − KΦ+(1)) φ∞(0) = BkiKiz∞ + wb (56) matrix, Ψ is 1. Remark that in [8], Ki is diagonal and here is m p full matrix. Note that some other choices of Ki are possible For all wy and yref both in R , wb in R , by Assumption 3 as long as and since the matrix Ki is full rank the former equation and > > T2Ki + Ki T2 > 0. (55) admit a unique solution (z∞, φ∞(0)) given as

To conclude, a work is needed to transpose this approach to −1 Ki −1 the global de Saint-Venant equations this is the aim of another z∞ = T1 (yref − wy) ki paper. −1 Ki −1 − T1 (L1Φ−(1) + L2Φ+(1)) ki IV. PROOFOF THEOREM 3 −1 × (Φ−(1) − KΦ+(1)) wb (57) The proof of Theorem 3 is divided into three steps. In a first part, it is shown that with Assumption 3, it can and, be shown that the closed loop system (29)-(30)-(33) admits −1 a steady state. In a second step, it is established that the φ∞(0) = (Φ−(1) − KΦ+(1)) kiBKiz∞ −1 desired regulation is obtained provided the steady state is + (Φ−(1) − KΦ+(1)) wb. (58) exponentially stable. Finally, the construction of an appropriate Lyapunov functional is performed to show the exponential Finally, in that case, we can introduce φ˜(s, t) = φ(s, t) − ˜ stability of the equilibrium. φ∞(s) and z˜(t) = z(t) − z∞. It can be checked that φ, z˜ satisfies the following system: ˜ ˜ ˜ A. Stabilization implies regulation φt(s, t) + Λ0(s)φs(s, t) + Λ1(s)φ(s, t) (59) In this first subsection, we explicitly give the equilibrium = 0 , s ∈ (0, 1), state of the system (29)-(30)-(33). We show also that if we ˜ ˜ assume that ki and Ki are selected such that this equilibrium φ+(t, 0) φ+(t, 1) z˜t = L1 ˜ + L2 ˜ (60) point is exponentially stable along the closed loop, then the φ−(t, 1) φ−(t, 0) regulation is achieved. with the boundary conditions 1) Definition of the equilibrium: The first step of the study ˜ ˜ is to exhibit equilibrium denoted φ∞, z∞ of the disturbed φ+(t, 0) φ+(t, 1) ˜ = K ˜ + Bu(t), (61) hyperbolic PDE in closed loop with the boundary integral φ−(t, 1) φ−(t, 0) control (i.e. system (29)-(33)). u(t) = kiKiz˜(t). (62) We have the following proposition. ˜ Proposition 1: Assumption 3 is a necessary and sufficient As it is shown in [2], for each initial condition v˜0 = (φ0, z˜0) in condition for the existence of an equilibrium of the system Xe which satisfies the boundary conditions (30), there exists a (29)-(30)-(33). Moreover, if Assumption 3 holds then point 1) unique weak solution that we denoted v˜ and which belongs to 0 of Theorem 3 holds. C ([0, +∞); Xe). Moreover, if the initial condition v˜0 satisfies 1 Proof: First of all, equilibria are such that also the C -compatibility condition (see [2] for more details) and lies in Xe1 then the solution lies in the set defined in (42). −1 φ∞s(s) = −Λ0(s) Λ1(s)φ∞(s), 2 for all s in [0, 1]. Hence, 2) Sufficient conditions for Regulation: In the following, φ∞(s) = Φ(s)φ∞(0). (54) we show that the regulation problem can be rephrased as a Hence, stabilization of the equilibrium state introduced previously. Proposition 2: Assume Assumption 3 holds and that there φ∞+(0) exist a functional Ve : Xe → +, and positive real numbers = Φ−(1)φ∞(0), R φ∞−(1) µe and Le such that: φ (1) ∞+ kv − vk2 = Φ+(1)φ∞(0) ∞ Xe 2 φ∞−(0) V (v) L kv − vk . 6 e 6 e ∞ Xe (63) Le 8

Assume moreover that for all v0 in Xe and all t0 in R+ such sufficient to construct a Lyapunov functional Ve which satisfies 1 that the solution v of system (29)-(30)-(33) initiated from v0 (63)-(64) along C -solutions of (29)-(30)-(33) or equivalently 1 1 is C at t = t0, we have: along C -solutions of (59)-(61). This is considered in the next ˙ section following the route of Section II-B. Ve(t) 6 −µeVe(t), (64) where with a slight abuse of notation Ve(t) = Ve(v(t)). Then B. Lyapunov functional construction points 1), 2) and 3) of Theorem 3 hold. 1) Open loop ISS: Inspired by the Lyapunov functional Proof: Point 1) is directly obtained from Proposition 1. The construction introduced in [6] (see also [2]), we know that proof of point 2) is by now standard. Let v0 be in Xe1 and typical Lyapunov functionals allowing to exhibit stability prop- satisfies the C0 and C1-compatibility conditions. It yields that erty for this type of hyperbolic PDE are given as functional 1 V : L2((0, 1), n) → v is C for all t. Consequently, (64) is satisfied for all t > 0. R R+ defined as With Gronwall’s¨ lemma, this implies that: Z 1 > −µet V (ϕ) = ϕ(s) P (s)ϕ(s)ds , (68) Ve(v(t)) 6 e Ve(v0). Hence with (63), this implies that (43) holds with k = L and 0 e 1 µe where P : [0, 1] → Dn is a C function. Typically in [6], ν = 2 for initial conditions in Xe1. Xe1 being dense in Xe, the result holds also with initial condition in X and point 2) these functions are taken as exponential. e ˜ is satisfied. With a slight abuse of notation, we write V (t) = V (φ(·, t)) ˙ On another hand, we have and we denote by V (t) the time derivative of the Lyapunov 1 functional along solutions which are C in time. In our con- φ+(t, 0) φ+(t, 1) y(t) − yref = L1 + L2 + wy − yref , text, with Assumption 2, it yields the following proposition. φ−(t, 1) φ−(t, 0) Proposition 3: If Assumption 2 holds, there exists a positive (65) real number c such that for every solution φ of (29)-(30) ˜ ˜ ˜ φ+(t, 0) φ+(t, 1) initiated from (φ0, z˜0) in Xe which satisfies (61) = L1 + L2 , (66) φ˜ (t, 1) φ˜ (t, 0) ˙ 2 − − V (t) 6 −µV (t) + c|u(t)| . (69) with φ˜(t, x) = φ(t, x) − φ . To show that equation (44) ∞ Proof: First of all, with (29), holds, we need to show that the right hand side of the former equation tends to zero. This may be obtained provided Z 1 ˙ > the initial condition is in X . Indeed, let v be in X and V (t) = − 2φ(t, s) P (s)Λ0(s)φs(t, s)ds 1 0 1 0 1 satisfies C -compatibility conditions. With (42), we know Z 1 > > that vt ∈ C([0, ∞); Xe). Moreover, vt satisfies the dynamics − φ(t, s) P (s)Λ1(s) + Λ1(s) P (s) φ(t, s)ds 0 system (29)-(30)-(33) with wb = 0, wy = 0, yref=0 (simply differentiate with time these equations). Hence, kvt(t)kXe converges exponentially toward 0 and in particular With an integration by part, this implies

˜ −νt 1 kφt(t, ·)k(L2(0,1), n) ke kvt(0)k. Z R 6 > V˙ (t) = φ(t, s) [(P (s)Λ0(s))s] φ(t, s)ds On another hand, employing (29), it yields: 0 Z 1 ˜ − φ(t, s)>(P (s)Λ (s) + Λ (s)>P (s))φ(t, s)ds kφs(t, ·)kL2((0,1), n) 1 1 R 0 −1 ˜ 2 n > = kΛ0 (φt(t, ·) + Λ1(·)φ(t, ·))kL ((0,1),R ). − φ(t, 1) P (1)Λ0(1)φ(t, 1) > Hence, + φ(t, 0) P (0)Λ0(0)φ(t, 0). With (34), it gives ˜ 2 n kφs(t, ·)kL ((0,1),R ) ˙ > ˜ 2 n ˜ 2 n V (t) −µV (t) − φ(t, 1) P (1)Λ0(1)φ(t, 1) 6 c kφt(t, ·)kL ((0,1),R ) + kφ(t, ·))kL ((0,1),R ) . (67) 6 + φ(t, 0)>P (0)Λ (0)φ(t, 0). where c is a positive constant. Consequently 0 ˜ 2 n With the boundary condition (31) and (36), this implies kφs(t, ·)kL ((0,1),R ) converges also to zero and so is ˜ 1 n kφ(t, ·)kH ((0,1),R ). With Sobolev embedding ˙ > > φ+(1) V (t) 6 −µV (t) − φ+(1) φ−(0) S ˜ ˜ 1 n φ−(0) sup |φ(t, x)| 6 Ckφ(t, ·)kH ((0,1),R ), x∈[0,1] > > > + 2 φ+(1) φ−(0) Qu(t) + u(t) Ru(t), (70) where C is a positive real number. It implies that: where, ˜ ˜ lim |φ(t, 1)| + |φ(t, 0)| = 0. t→+∞ > 0 R = − 0 B2 (P (1)Λ0(1) + Λ0(1)P (1)) Consequently, with (65), it yields that (44) holds and point 3) B2 is satisﬁed. 2 > 0 + B1 0 (P (0)Λ0(0) + Λ0(0)P (0)) , (71) With this proposition in hand, to prove the Theorem 3, it is B1 9

and, Note that along C1 solution to system (59), we have ˜ ˜ ˜ > 0 Mφt(t, ·) = M(−Λ0(·)φs(t, ·) − Λ1(·)φ(t, ·)) Q = −K+ (P (1)Λ0(1) + Λ0(1)P (1)) B2 Z 1 = MΨ(s)(−Λ (s)φ˜ (t, s) − Λ (s)φ˜(t, s))ds B 0 s 1 + K>(P (0)Λ (0) + Λ (0)P (0)) 1 . 0 − 0 0 0 With an integration by part this implies Since S is positive definite, selecting c sufficiently large, it yields Z 1 Mφ˜ (t, ·) = M(Ψ(s)Λ (s)) φ˜(t, s)ds −SQ t 0 s 0 . 0 Q> R − c I 6 Z 1 dm ˜ − MΨ(s)Λ1(s)φ(t, s))ds Consequently, (70) implies that (69) holds. 2 0 ˜ ˜ − M Ψ(1)Λ0(1)φ(t, 1) − Λ0(0)φ(t, 0) . 2) Forwarding approach to deal with the integral part: Following the route of Section II-B, a Lyapunov functional This gives, is designed from V adding some terms to take into account ˜ the state of the integral controller. Let the operator M : Mφt(t, ·) = 1 n m Z 1 L ((0, 1); R ) → R be given as ˜ M (Ψs(s)Λ0(s) + Ψ(s)(Λ0s(s) − Λ1(s))) φ(t, s)ds Z 1 0 Mϕ = MΨ(s)ϕ(s)ds (72) ˜ ˜ − M Ψ(1)Λ0(1)φ(t, 1) − Λ0(0)φ(t, 0) . 0 where Ψ is the matrix function defined in (39), and M is a With the definition of Ψ, it yields matrix in Rm×n defined in (41). ˜ ˜ ˜ Following the Lyapunov functional construction in Theo- Mφt(t, ·) = −M Ψ(1)Λ0(1)φ(t, 1) − Λ0(0)φ(t, 0) . rem 2, we consider the candidate Lyapunov functional Ve : L2((0, 1); Rn) × Rm given as With the boundary condition (61), it yields > ˜ Ve(ϕ, z) = V (ϕ) + p(z − Mϕ) (z − Mϕ). (73) ˜ φ+(t, 1) Mφt(t, ·) = −M (Ψ(1)Λ0(1)K+ − Λ0(0)K−) ˜ φ−(t, 0) In the following theorem, it is shown that by selecting properly 0 B K , k and p, this function is indeed a Lyapunov functional for −MΨ(1)Λ (1) u(t) + MΛ (0) 1 u(t) i i 0 B 0 0 the closed loop system. Again, with a slight abuse of notation, 2 ˜ we write Ve(t) = Ve(φ(·, t), z˜(t)) and we denote by V˙e(t) Hence, with the definition of M, it implies the time derivative of the Lyapunov functional along solutions 1 ˜ which are C in time. ˜ φ+(t, 1) Mφt(t, ·) = (L1K + L2) ˜ Proposition 4: Assume that Assumptions 2 and 3 hold. Then φ−(t, 0) there exists a matrix K in m×m and k∗ > 0 such that for i R i B1 0 all 0 < k < k∗, there exist positive real numbers L and µ + M Λ0(0) − Ψ(1)Λ0(1) u(t) i i e e 0 B2 such that for all (ϕ, z) in Xe On another hand, 1 kϕk2 + |z|2 V (ϕ, z) L kϕk2 + |z|2 , (74) L X 6 e 6 e X φ˜ (t, 1) e z (t) = (L K + L ) + + L Bu(t). t 1 2 φ˜ (t, 0) 1 and along C1 solution of the system (59)-(61)-(62) − ˙ This gives, Ve(t) 6 −µeVe(t) , ∀t ∈ R+. (75) ˜ 2 n Mφt(t, ·) = zt(t) − L1Bu(t) Proof: With (35), it yields for all ϕ in L ((0, 1); R ), B1 0 2 2 + M Λ0(0) − Ψ(1)Λ0(1) u(t). P kϕk 2 n V (ϕ) P kϕk 2 n . (76) 0 B2 L ((0,1);R ) 6 6 L ((0,1);R ) Let Ψ > 0 be such that Hence, it yields Mφ˜ (t, ·) = z (t) + T u(t). (78) |Ψ(s)| 6 Ψ , ∀s ∈ [0, 1]. t t 2 Note that for all ϕ in L2((0, 1); Rn, by Cauchy-Schwartz We recognize here equation (11) when u = 0. This gives with inequality, (69)

2 |Mϕ| |M| Ψ kϕk 2 n . ˙ 6 L ((0,1);R ) (77) Ve(t) 6 −µV (t) + c|u(t)| > Hence, for each p > 0 equation (74) holds. − 2p(z(t) − Mφ(·, t)) T2u(t). (79) 10

−1 Let now, Ki = T2 . Hence, this gives with u = kiKiz, REFERENCES [1] Daniele Astolfi and Laurent Praly. Integral action in output feedback 2 2 for multi-input multi-output nonlinear systems. IEEE Transactions on V˙e(t) −µV (t) + ck |Kiz(t)| 6 i Automatic Control, 62(4):1559–1574, 2017. 2 > − 2p|z(t)| ki + 2pki(Mφ(·, t)) z(t), (80) [2] Georges Bastin and Jean-Michel Coron. Stability and boundary stabilization of 1-d hyperbolic systems, volume 88. Springer, 2016. [3] Georges Bastin, Jean-Michel Coron, and Simona Oana Tamasoiu. Stabil- With (77), and completing the square it yields for all ϕ in ity of linear density-flow hyperbolic systems under pi boundary control. 2 n m L ((0, 1); R ) and z in R , Automatica, 53:37–42, 2015. [4] S. Benachour, V. Andrieu, L. Praly, and H. Hammouri. Forwarding 2(Mϕ)>z |Mϕ|2 + |z|2, design with prescribed local behavior. IEEE Transactions on Automatic 6 (81) Control, 58(12):3011–3023, Dec 2013. 2 2 2 2 [5] Jean-Michel Coron. Control and nonlinearity. 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In this paper we have Automatica, submitted. shown that is was also possible to construct Lyapunov func- [22] Marius Tucsnak and George Weiss. Observation and control for operator tionals to address the regulation problem in the case in which semigroups. Springer Science & Business Media, 2009. [23] Cheng-Zhong Xu and Hamadi Jerbi. A robust pi-controller for infinite- is used an integral action. This framework allows to explicitly dimensional systems. International Journal of Control, 61(1):33–45, give an integral gain. Moreover, it is no more necessary to 1995. impose boundedness of control or measurement operators to [24] Cheng-Zhong Xu and Gauthier Sallet. Multivariable boundary pi control and regulation of a fluid flow system. Mathematical Control and Related guarantee the regulation. This is applied to PDE hyperbolic Fields, 4(4):501–520, 2014. systems and this allows to generalize many available results in this field. 11

Alexandre Terrand-Jeanne graduated in electrical engineering from ENS Cachan, France, in 2013. After one year in the robotic laboratory ”Centro E.Piaggio” in Pisa, Italy, he is currently a doctoral student in LAGEP, university of Lyon 1. His PhD topic concerns the stability analysis and control laws design for systems involving hyperbolic partial differential equations coupled with nonlinear ordinary differential equations. This work is under the supervision of V. Dos Santos Martins, V. Andrieu and M. Tayakout-Fayolle.

Vincent Andrieu graduated in applied mathematics from INSA de Rouen, France, in 2001. Af- ter working in ONERA (French aerospace research company), he obtained a PhD degree from Ecole des Mines de Paris in 2005. In 2006, he had a research appointment at the Control and Power Group, Dept. EEE, Imperial College London. In 2008, he joined the CNRS-LAAS lab in Toulouse, France, as a CNRS-charge´ de recherche. Since 2010, he has been working in LAGEP-CNRS, Universite´ de Lyon 1, France. In 2014, he joined the functional analysis group from Bergische Universitat¨ Wuppertal in Germany, for two sabbatical years. His main research interests are in the feedback stabilization of controlled dynamical nonlinear systems and state estimation problems. He is also interested in practical application of these theoretical problems, and especially in the ﬁeld of aeronautics and chemical engineering. Since 2018 he is an associate editor of the IEEE Transactions on Automatic Control, System & Control Letters and IEEE Control Systems Letters.

Cheng-Zhong XU received the Ph.D. degree in automatic control and signal processing from Insti- tut National Polytechnique de Grenoble, Grenoble, France, in 1989, and the Habilitation degree in applied mathematics and automatic control from University of Metz, Metz, France, in 1997. From 1991 to 2002, he was a Charge´ de Recherche (Re- search Ofﬁcer) in the Institut National de Recherche en Informatique et en Automatique. Since 2002, he has been a Professor of automatic control at the University of Lyon, Lyon, France. His research interests include control of distributed parameter systems and its applications to mechanical and chemical engineering. He was an associated editor of the IEEE Transactions on Automatic Control, from 1995 to 1998. He was an associated editor of the SIAM Journal on Control and Optimization, from 2011 to 2017.

Valerie´ Dos Santos Martins graduated in Math- ematics from the University of Orleans,´ France in 2001. She received the Ph.D degree in 2004 in Ap- plied Mathematics from the University of Orleans.´ After one year in the laboratory of Mathematics MAPMO in Orleans´ as ATER, she was post-doct in the laboratory CESAME/INMA of the University Catholic of Louvain, Belgium. Currently, she is professor assistant in the laboratory LAGEP, University of Lyon 1. Her current research interests include nonlinear control theory, perturbations theory of operators and semigroup, spectral theory and control of nonlinear partial differential equations.