Stability of switched linear hyperbolic systems by Lyapunov techniques (full version)
11 Stability of switched linear hyperbolic systemsby Lyapunov techniques (full version)
Christophe Prieur, Antoine Girard, and Emmanuel Witrant
Abstract
Switched linear hyperbolic partial differential equations are considered in this paper. They modelinfinite dimensional systems of conservation laws and balance laws, which are potentially affected by adistributed source or sink term. The dynamics and the boundary conditions are subject to abrupt changesgiven by a switching signal, modeled as a piecewise constant function and possibly a dwell time. Bymeans of Lyapunov techniques some sufficient conditions are obtained for the exponential stability ofthe switching system, uniformly for all switching signals. Different cases are considered with or withouta dwell time assumption on the switching signals, and on the number of positive characteristic velocities(which may also depend on the switching signal). Some numerical simulations are also given to illustratesome main results, and to motivate this study.
I. I
NTRODUCTION
Lyapunov techniques are commonly used for the stability analysis of dynamical systems, suchas those modeled by partial differential equations (PDEs). The present paper focuses on a classof one-dimensional hyperbolic equations that describe, for example, systems of conservationlaws or balance laws (with a source term), see [5].A switching behavior occurs for many control applications when the evolution processesinvolve logical decisions, see [7] for the case where a stabilizing feedback is designed by meansof Lyapunov techniques applied to a discretization of switched parabolic PDE; see also [10],
C. Prieur and E. Witrant are from the Department of Automatic Control, Gipsa-lab, Universit´e de Grenoble,11 rue des Math´ematiques, BP 46, 38402 Saint-Matin d’H`eres Cedex, France. Email: { christophe.prieur,emmanuel.witrant } @gipsa-lab.fr and A. Girard is with Laboratoire Jean Kuntzmann, Universit´e de Grenoble,BP 53, 38041 Grenoble, France, [email protected] . This work is partly supported by HYCON2 Network ofExcellence Highly-Complex and Networked Control Systems, grant agreement 257462. November 5, 2018 DRAFT a r X i v : . [ m a t h . O C ] J u l where the well-posed issue and the dependence of the solutions on the data of a network ofhyperbolic equations with switching as a control are considered. Switching can indeed be anefficient control strategy for many infinite dimensional systems such as the wave equation ([8]),the heat equation ([21]) or other infinite dimensional systems written in abstract form (as in [9]).The exponential stabilizability of such systems is often proved by means of a Lyapunovfunction, as illustrated by the contributions from [13], [18] where different control problemsare solved for particular hyperbolic equations. For more general nonlinear hyperbolic equations,the knowledge of Lyapunov functions can be useful for the stability analysis of a system ofconservation laws (see [4]), or even for the design of exponentially stabilizing boundary controls(see [3]). Other control techniques may be useful, such as Linear Quadratic regulation [1] orsemigroup theory [17, Chap. 6].In this paper, the class of hyperbolic systems of balance laws is first considered without anyswitching rule and we state sufficient conditions to derive a Lyapunov function for this class ofsystems. It allows us to relax [5] where the Lyapunov stability for hyperbolic systems of balancelaws has been first tackled (see also [4]). Then, switched systems are considered and sufficientconditions for the asymptotic stability of a class of linear hyperbolic systems with switcheddynamics and switched boundary conditions are stated. Some stability conditions depend onthe average dwell time of the switching signals (if such a positive dwell time does exist). Thestability property depends on the classes of the switching rules applied to the dynamics (as in[15] for finite dimensional systems). The present paper is also related to [19] where unswitchedtime-varying hyperbolic systems are considered.In [2], the condition of [14] is employed. It allows analyzing the stability of hyperbolicsystems, assuming a stronger hypothesis on the boundary conditions. More precisely, our ap-proach generalizes the condition of [4], which is known to be strictly weaker than the oneof [14]. Therefore our stability conditions are strictly weaker than the ones of [2]. Moreoverthe technique in [2] is trajectory-based via the method of characteristics, while our approach isbased on Lyapunov functions, allowing for numerically tractable conditions. Indeed, the obtainedsufficient conditions are written in terms of matrix inequalities, which can be solved numerically.Furthermore the estimated speed of exponential convergence is provided and can be optimized.See Section V for the use of line search algorithms to numerically compute the variables inour stability conditions, and thus to compute Lyapunov functions. The main results and the November 5, 2018 DRAFT computational aspects are illustrated on two examples of switched linear hyperbolic systems.The paper is organized as follows. The class of switched linear hyperbolic systems of balancelaws considered in this paper is given in Section II and a first stability condition is proven.Switched systems of balance laws are presented in Section III. In Section IV our main resultsare derived for the stability of switched hyperbolic systems. The conditions depend on the classof piecewise constant switching signals that is considered (with and without a sufficiently largedwell time). The stability conditions may also differ if the number of positive characteristicvelocities does not depend on the switching signal (see Section IV-A) or if this number is afunction of this signal (see Section IV-B). Section V collects the discussions on computationalaspects. It deals in particular with the numerical check of our stability conditions, and thenumerical computations of the considered Lyapunov functions. In Section VI two examplesillustrate the main results and motivate the class of Lyapunov functions considered in this paper.
Notation.
The set R + is the set of nonnegative real numbers. Given a matrix G , the transpose matrix of G is denoted as G (cid:62) . When G is invertible, then, to simplify the notation, ( G − ) (cid:62) is denoted as G −(cid:62) . For positiveintegers m and n , I n and n,m are respectively the identity and the null matrix in R n × n and in R n × m . Givensome scalar values ( a , . . . , a n ) , diag ( a , . . . , a n ) is the matrix in R n × n with zero non-diagonal entries, and with ( a , . . . , a n ) on the diagonal. Moreover given two matrices A and B , diag [ A, B ] is the block diagonal matrixformed by A and B (and zero for the other entries). The notation A ≥ B means that A − B is positive semidefinite.The usual Euclidian norm in R n is denoted by | · | and the associated matrix norm is denoted (cid:107) · (cid:107) , whereas the setof all functions φ : (0 , → R n such that (cid:82) | φ ( x ) | < ∞ is denoted by L ((0 , R n ) that is equipped with thenorm (cid:107) · (cid:107) L ((0 , R n ) . Given a topological set S , and an interval I in R + , the set C ( I, S ) is the set of continuousfunctions φ : I → S . II. L
INEAR HYPERBOLIC SYSTEMS
Let us first consider the following linear hyperbolic partial differential equation: ∂ t y ( t, x ) + Λ ∂ x y ( t, x ) = F y ( t, x ) , x ∈ [0 , , t ∈ R + (1)where y : R + × [0 , → R n , F is a matrix in R n × n , Λ is a diagonal matrix in R n × n such that Λ = diag ( λ , . . . , λ n ) , with λ k < for k ∈ { , . . . , m } and λ k > for k ∈ { m + 1 , . . . , n } .We use the notation y = (cid:16) y − y + (cid:17) , where y − : R + × [0 , → R m and y + : R + × [0 , → R n − m . In November 5, 2018 DRAFT addition, we consider the following boundary conditions: (cid:16) y − ( t, y + ( t, (cid:17) = G (cid:16) y − ( t, y + ( t, (cid:17) , t ∈ R + (2)where G is a matrix in R n × n . Let us introduce the matrices G −− in R m × m , G − + in R m × ( n − m ) , G + − in R ( n − m ) × m and G ++ in R ( n − m ) × ( n − m ) such that G = (cid:16) G −− G − + G + − G ++ (cid:17) .We shall consider an initial condition given by y (0 , x ) = y ( x ) , x ∈ (0 , (3)where y ∈ L ((0 , R n ) . Then, it can be shown (see e.g. [5]) that there exists a unique solution y ∈ C ( R + ; L ((0 , R n )) to the initial value problem (1)-(3). As these solutions may not bedifferentiable everywhere, the concept of weak solutions of partial differential equations hasto be used (see again [5] for more details). The linear hyperbolic system (1)-(2) is said tobe globally exponentially stable (GES) if there exist ν > and C > such that, for every y ∈ L ((0 , R n ) ; the solution to the initial value problem (1)-(3) satisfies (cid:107) y ( t, . ) (cid:107) L ((0 , R n ) ≤ Ce − νt (cid:107) y (cid:107) L ((0 , R n ) , ∀ t ∈ R + . (4)Sufficient conditions for exponential stability of (1)-(3) have been obtained in [5] using aLyapunov function. In this section, we present an extension of this result. This extension willbe also useful for subsequent work on switched linear hyperbolic systems.Let Λ + = diag ( | λ | , . . . , | λ n | ) . Proposition 2.1:
Let us assume that there exist ν > , µ ∈ R and symmetric positive definitematrices Q − in R m × m and Q + in R ( n − m ) × ( n − m ) such that, defining for each x in [0 , , Q ( x ) = diag [ e µx Q − , e − µx Q + ] , Q ( x )Λ = Λ Q ( x ) , the following matrix inequalities hold − µ Q ( x )Λ + + F (cid:62) Q ( x ) + Q ( x ) F ≤ − ν Q ( x ) (5) I m m,n − m G + − G ++ (cid:62) Q (0)Λ I m m,n − m G + − G ++ ≤ G −− G − + n − m,m I n − m (cid:62) Q (1)Λ G −− G − + n − m,m I n − m . (6) Then there exists C such that (4) holds and the linear hyperbolic system (1)-(2) is GES.Proof: Let us consider the Lyapunov function, for all y ∈ L ((0 , R n ) , V ( y ) = (cid:90) y ( x ) (cid:62) Q ( x ) y ( x ) dx. November 5, 2018 DRAFT
Since Q ( x ) and Λ commute and are symmetric, then Q ( x )Λ is symmetric, ∂ x Q ( x )Λ = − µ Q ( x )Λ + ,and y (cid:62) Q ( x )Λ ∂ x y + ∂ x y Λ Q ( x ) y − µy (cid:62) Q ( x )Λ + y = ∂ x ( y (cid:62) Q ( x )Λ y ) . (7)Then, computing the time-derivative of V along the solutions of (1) yields the following: ˙ V ( y ) = (cid:90) ( y (cid:62) Q ( x ) ∂ t y + ∂ t y (cid:62) Q ( x ) y ) dx = − (cid:90) y (cid:62) Q ( x )Λ ∂ x y dx − (cid:90) ∂ x y (cid:62) Λ Q ( x ) y dx + (cid:90) y (cid:62) ( F (cid:62) Q ( x ) + Q ( x ) F ) y dx. Then, Equation (7) yields: ˙ V ( y ) = − [ y (cid:62) Q ( x )Λ y ] − (cid:90) µy (cid:62) Q ( x )Λ + y dx + (cid:90) y (cid:62) ( F (cid:62) Q ( x ) + Q ( x ) F ) y dx = y (cid:62) ( t, Q (0)Λ y ( t, − y (cid:62) ( t, Q (1)Λ y ( t, (cid:90) y (cid:62) ( − µ Q ( x )Λ + + F (cid:62) Q ( x ) + Q ( x ) F ) y dx = y − ( t, y + ( t, (cid:62) I m m,n − m G + − G ++ (cid:62) Q (0)Λ I m m,n − m G + − G ++ − G −− G − + n − m,m I n − m (cid:62) Q (1)Λ G −− G − + n − m,m I n − m y − ( t, y + ( t, + (cid:90) y (cid:62) (cid:0) − µ Q ( x )Λ + + F (cid:62) Q ( x ) + Q ( x ) F (cid:1) y dx where the last equality is obtained by the boundary conditions (2). Then, (5) and (6) imply that ˙ V ( y ( t, . )) ≤ − νV ( y ( t, . )) which yields, for all t ∈ R + , V ( y ( t, . )) ≤ V ( y ) e − νt . By remarkingthat there exist α > , β > (depending on the eigenvalues of Q − , Q + and on µ ) suchthat α (cid:107) y ( t, . ) (cid:107) L ((0 , R n ) ≤ (cid:112) V ( y ( t, . )) ≤ β (cid:107) y ( t, . ) (cid:107) L ((0 , R n ) , we obtain that, for all t ∈ R + , (cid:107) y ( t, . ) (cid:107) L ((0 , R n ) ≤ βα e − νt (cid:107) y (cid:107) L ((0 , R n ) .If all the diagonal elements of Λ are different, the assumption that Q ( x )Λ = Λ Q ( x ) is equiv-alent to Q being diagonal positive definite . The main contributions of the previous propositionwith respect to the result presented in [5] is double: first, we do not restrict the values of This equivalence follows from the computation of matrices Q ( x )Λ and Λ Q ( x ) , and from a comparison between each oftheir entries. November 5, 2018 DRAFT parameter µ to be positive, this allows us to consider non-contractive boundary conditions (itwill be the case for the numerical illustration considered in Example VI-B); second, we providean estimate of the exponential convergence rate (see Section V for computational aspects of thisestimate). When all the diagonal elements of the matrix Λ are positive, then Proposition 2.1 canbe interpreted in terms of two finite dimensional linear systems that share a common Lyapunovfunction: one in continuous-time associated to (1) and one in discrete-time associated to (2).Indeed we have the following result: Corollary 2.2:
Let us assume that m = 0 and there exists a diagonal positive definite matrix M in R n × n such that V : y ∈ R n (cid:55)→ y T M y is a common Lyapunov function for the continuous-time and discrete-time linear systems ˙ y ( t ) = (cid:0) Λ − F − µI (cid:1) y ( t ) , t ∈ R + , (8) y ( t + 1) = ( e µ G ) y ( t ) , t ∈ N . (9) Then, the linear hyperbolic system (1)-(2) is GES.Proof:
Remark first that Λ + = Λ since it is assumed that m = 0 . Let Q = M Λ − ,then Q is diagonal positive definite. By writing the Lyapunov equation of the continuous-timesystem (8), we obtain (Λ − F − µI ) (cid:62) M + M (Λ − F − µI ) < which can be rewritten as ( F − µ Λ) (cid:62) Q + Q ( F − µ Λ) = − µQ Λ + F (cid:62) Q + QF < . This implies existence of ν > suchthat (5) holds. Also the Lyapunov equation of the discrete-time system (9) gives e µ G (cid:62) M e µ G ≤ M. This can be rewritten as G (cid:62) Q Λ G ≤ e − µ Q Λ which is equivalent to (6), since m = 0 and Q ( x ) = e − µx Q . Thus the assumptions of Proposition 2.1 hold and this concludes the proof ofCorollary 2.2.Let us remark that increasing µ improves the stability of (8) and degrades that of (9) whiledecreasing µ will have the reverse effect. Another interpretation of the previous result is that thelinear hyperbolic system (1)-(2) is GES if there is a balance between the expansion (respectivelycontraction) rate of the continuous-time linear system ˙ y ( t ) = Λ − F y ( t ) and the contraction(respectively expansion) rate of discrete-time linear system y ( t + 1) = Gy ( t ) . November 5, 2018 DRAFT
III. S
WITCHED LINEAR HYPERBOLIC SYSTEMS
We now consider the case of switched linear hyperbolic partial differential equation of theform (see [2]) ∂ t w ( t, x ) + L σ ( t ) ∂ x w ( t, x ) = A σ ( t ) w ( t, x ) , x ∈ [0 , , t ∈ R + (10)where w : R + × [0 , → R n , σ : R + → I , I is a finite set (of modes ), A i and L i are matricesin R n × n , for i ∈ I . The partial differential equation associated with each mode is hyperbolic,meaning that for all i ∈ I , there exists an invertible matrix S i in R n × n such that L i = S − i Λ i S i where Λ i is a diagonal matrix in R n × n satisfying Λ i = diag ( λ i, , . . . , λ i,n ) , with λ i,k < for k ∈ { , . . . , m i } and λ i,k > for k ∈ { m i + 1 , . . . , n } . The matrices S i can be written as S i = (cid:16) S −(cid:62) i S + (cid:62) i (cid:17) (cid:62) (11)where S − i and S + i are matrices in R m i × n and R ( n − m i ) × n . We define the matrices F i = S i A i S − i and Λ + i = diag ( | λ i, | , . . . , | λ i,n | ) for i ∈ I . The boundary conditions are given by B σ ( t ) w ( t,
0) + B σ ( t ) w ( t,
1) = 0 , t ≥ (12)where, for all i ∈ I , B i = G i S i and B i = G i S i , G i and G i being matrices in R n × n that satisfy G i = (cid:16) − G i −− mi,n − mi − G i + − I n − mi (cid:17) , G i = (cid:16) I mi − G i − + mi,n − mi − G i ++ (cid:17) . For i ∈ I , let us define the matrices in R n × n , G i = (cid:16) G i −− G i − + G i + − G i ++ (cid:17) . We shall consider an initialcondition given by w (0 , x ) = w ( x ) , x ∈ (0 , (13)where w ∈ L ((0 , R n ) .A switching signal is a piecewise constant function σ : R + → I , right-continuous, and witha finite number of discontinuities on every bounded interval of R + . This allows us to avoid the Zeno behavior , as described in [15]. The set of switching signals is denoted by S ( R + , I ) . Thediscontinuities of σ are called switching times . The number of discontinuities of σ on the interval ( τ, t ] is denoted by N σ ( τ, t ) . Following [12], for τ D > , N ∈ N , we denote by S τ D ,N ( R + , I ) the set of switching signals verifying, for all τ < t , N σ ( τ, t ) ≤ N + t − ττ D . The constant τ D iscalled the average dwell time and N the chatter bound .We first provide an existence and uniqueness result for the solutions of (10)-(13): November 5, 2018 DRAFT
Proposition 3.1:
For all σ ∈ S ( R + , I ) , w ∈ L ((0 , R n ) , there exists a unique (weak)solution w ∈ C ( R + ; L ((0 , R n )) to the initial value problem (10)-(13).Proof: We build iteratively the solution between successive switching times. Let ( t k ) k ∈ K denote the increasing switching times of σ , with t = 0 and K be a (finite or infinite) sub-set of N . Let us assume that we have been able to build a unique (weak) solution w ∈ C ([0 , t k ]; L ((0 , R n )) for some k ≥ . Then, let i k be the value of σ ( t ) for t ∈ [ t k , t k +1 ) .Let us introduce the following notation, for all k in K and for all x in [0 , , y k ( t, x ) = S i k w ( t, x ) , t ∈ [ t k , t k +1 ] . (14)Note that closed time intervals are used on both sides due to technical reasons in this proof.Then, (10) gives that, for all k in K , y k satisfies the following partial differential equation ∂ t y k ( t, x ) + Λ i k ∂ x y k ( t, x ) = F i k y k ( t, x ) , x ∈ [0 , , t ∈ [ t k , t k +1 ] . (15)Also, we use the notations y k = (cid:16) y − k y + k (cid:17) , where y − k : R + × [0 , → R m ik and y + k : R + × [0 , → R n − m ik . The boundary conditions (12) give, for all k in K , (cid:16) y − k ( t, y + k ( t, (cid:17) = G i k (cid:16) y − k ( t, y + k ( t, (cid:17) , t ∈ [ t k , t k +1 ] . (16)The initial condition ensuring the continuity of w at time t k is the following: y k ( t k , x ) = S i k w ( t k , x ) , x ∈ (0 , . (17)It follows from [5] that, for all k in K , there exists a unique (weak) solution y k ∈ C ([ t k , t k +1 ]; L ((0 , R n )) to the initial value problem (15)-(17). Then, we can extendthe (weak) solution to the initial value problem (10)-(13), from the initial time t k , up to theswitching time t k +1 ; (17) ensures that w ∈ C ([0 , t k +1 ]; L ((0 , R n )) , and the uniqueness of y k ensures that w is the unique solution. Finally, since there are only a finite number of switchingtimes on every bounded intervals of R + , the solution can be defined for all times, resulting ona unique solution w ∈ C ( R + ; L ((0 , R n )) .IV. S TABILITY OF SWITCHED LINEAR HYPERBOLIC SYSTEMS
Let
S ⊆ S ( R + , I ) . The switched linear hyperbolic system (10)-(12) is said to be globallyuniformly exponentially stable (GUES) with respect to the set of switching signals S if thereexist ν > and C > such that, for every w ∈ L ((0 , R n ) , for every σ ∈ S , the solution to November 5, 2018 DRAFT the initial value problem (10)-(13) satisfies (cid:107) w ( t, . ) (cid:107) L ((0 , R n ) ≤ Ce − νt (cid:107) w (cid:107) L ((0 , R n ) , ∀ t ∈ R + . In this section, we provide sufficient conditions for the stability of switched linear hyperbolicsystems.
A. Mode independent sign structure of characteristics
Assume first that the number of negative and positive characteristics of the linear partialdifferential equations associated with each mode is constant, that is for all i ∈ I , m i = m .We provide a first result giving sufficient conditions such that stability holds for all switchingsignals. The proof is based on a common Lyapunov function equivalent to the L norm. An alter-native proof can be obtained by checking some semigroup properties and by using [11] (wherethe equivalence is shown between the existence of a common Lyapunov function commensurablewith the squared norm and the global uniform exponential stability). Theorem 1:
Let us assume that, for all i ∈ I , m i = m and that there exist ν > , µ ∈ R anddiagonal positive definite matrices Q i in R n × n , i ∈ I such that the following matrix inequalitieshold, for all i ∈ I and for all x in [0 , , − µ Q i ( x )Λ + i + F (cid:62) i Q i ( x ) + Q i ( x ) F i ≤ − ν Q i ( x ) , (18) I m m,n − m G i + − G i ++ (cid:62) Q i (0)Λ i I m m,n − m G i + − G i ++ ≤ G i −− G i − + n − m,m I n − m (cid:62) Q i (1)Λ i G i −− G i − + n − m,m I n − m , (19) where Q i ( x ) = diag [ e µx Q − i , e − µx Q + i ] , Q i = (cid:16) Q − i Q + i (cid:17) , Q − i and Q + i are diagonal positivematrices in R m i × m i and R ( n − m i ) × ( n − m i ) , together with the following matrix equalities, for all i, j ∈ I , ( S + i ) (cid:62) Q + i S + i = ( S + j ) (cid:62) Q + j S + j , ( S − i ) (cid:62) Q − i S − i = ( S − j ) (cid:62) Q − j S − j . (20) Then, the switched linear hyperbolic system (10)-(12) is GUES with respect to the set of switchingsignals S ( R + , I ) .Proof: Given the diagonal matrices Q i satisfying the assumptions of Theorem 1, let M − =( S − i ) (cid:62) Q − i S − i and M + = ( S + i ) (cid:62) Q + i S + i , by (20), these matrices do not depend on the index i ∈ I . The proof is based on the use of a common Lyapunov function given by, for all w in L ((0 , R n ) , V ( w ) = (cid:90) w ( x ) (cid:62) M ( x ) w ( x ) dx, November 5, 2018 DRAFT0 where M ( x ) = e µx M − + e − µx M + . Let ( t k ) k ∈ K denote the increasing switching times of σ ,with t = 0 and K a (finite or infinite) subset of N . For k ∈ K , let i k be the value of σ ( t ) for t ∈ [ t k , t k +1 ) , and let y k be given by (14). It thus satisfies the boundary conditions (16). Let usremark that, due to (11) and (17), for t ∈ [ t k , t k +1 ) , V can be written as: V ( w ( t, . )) = (cid:90) y k ( t, x ) (cid:62) Q i k ( x ) y k ( t, x ) dx, where Q i k ( x ) = diag [ e µx Q − i k , e − µx Q + i k ] . Note that Q i k ( x ) commute with Λ since these matricesare diagonal. Using (18) and (19) and following the proof of Proposition 2.1, we obtain that,along the solutions of (10)-(12), it holds, for all k in K , ∀ t ∈ [ t k , t k +1 ) , V ( w ( t, . )) ≤ V ( w ( t k , . )) e − ν ( t − t k ) . (21)Moreover, V ( w ( t, . )) is continuous at the switching time t k +1 , thus ∀ k ∈ K, V ( w ( t k +1 , . )) ≤ V ( w ( t k , . )) e − ν ( t k +1 − t k ) . (22)Equations (21) and (22) allow us to prove that, for all t ∈ R + , V ( w ( t, . )) ≤ V ( w ) e − νt .By noting that there exist α > , β > such that α (cid:107) w ( t, . ) (cid:107) L ((0 , R n ) ≤ (cid:112) V ( w ( t, . )) ≤ β (cid:107) w ( t, . ) (cid:107) L ((0 , R n ) , we obtain that, for all t ∈ R + , (cid:107) w ( t, . ) (cid:107) L ((0 , R n ) ≤ βα e − νt (cid:107) w (cid:107) L ((0 , R n ) .This concludes the proof of Theorem 1.The numerical computation of the unknown variables, satisfying the sufficient conditions ofTheorem 1, is explained in Section V below.For systems that do not satisfy the assumptions of the previous theorem, but whose dynamicsin each mode satisfy independently the assumptions of Proposition 2.1 (i.e. the dynamics in eachmode is stable), it is possible to show that the system is stable provided that the switching isslow enough: Theorem 2:
Let us assume that, for all i ∈ I , m i = m and that there exist ν > , γ ≥ , µ i ∈ R , diagonal positive definite matrices Q i in R n × n , such that the following matrix inequalitieshold, for all x in [0 , , − µ i Q i ( x )Λ + i + F (cid:62) i Q i ( x ) + Q i ( x ) F i ≤ − ν Q i ( x ) , (23) I m m,n − m G i + − G i ++ (cid:62) Q i (0)Λ i I m m,n − m G i + − G i ++ ≤ G i −− G i − + n − m,m I n − m (cid:62) Q i (1)Λ i G i −− G i − + n − m,m I n − m , (24) November 5, 2018 DRAFT1 where Q i ( x ) = diag [ e µ i x Q − i , e − µ i x Q + i ] , Q i = (cid:16) Q − i Q + i (cid:17) , Q − i and Q + i are diagonal positivematrices in R m i × m i and R ( n − m i ) × ( n − m i ) , together with the following matrix inequalities, for all i, j ∈ I , ( S + i ) (cid:62) Q + i S + i ≤ γ ( S + j ) (cid:62) Q + j S + j , (25) ( S − i ) (cid:62) Q − i S − i ≤ γ ( S − j ) (cid:62) Q − j S − j . (26) Let ∆ µ = max( µ , . . . , µ n ) − min( µ , . . . , µ n ) , then, for all N ∈ N , for all τ D > ln( γ )2 ν + ∆ µ ν ,the switched linear hyperbolic system (10)-(12) is GUES with respect to the set of switchingsignals S τ D ,N ( R + , I ) .Proof: Let ( t k ) k ∈ K denote the increasing switching times of σ , with t = 0 and K is a(finite or infinite) subset of N . For k ∈ K , let i k be the value of σ ( t ) for t ∈ [ t k , t k +1 ) , and let y k be given by (14). It satisfies the boundary conditions (16). Given the diagonal matrices Q i satisfying the assumptions of Theorem 2, let M − i = ( S − i ) (cid:62) Q − i S − i and M + i = ( S + i ) (cid:62) Q + i S + i . Theproof is based on the use of multiple Lyapunov functions. More precisely, denoting M i k ( x ) = e µ ik x M − i k + e − µ ik x M + i k , let us define, for all w in C ([0 , ∞ ); L ((0 , R n )) , for all t in R + , V ( w ( t, . )) = (cid:90) w ( t, x ) (cid:62) M i k ( x ) w ( t, x ) dx, if t ∈ [ t k , t k +1 ) (27)which may be rewritten as V ( w ( t, . )) = (cid:82) y k ( t, x ) (cid:62) Q i k ( x ) y k ( t, x ) dx , if t ∈ [ t k , t k +1 ) .Note that Q i k ( x ) commute with Λ i k since these matrices are diagonal. Using (23) and (24),and following the proof of Proposition 1, we get that, along the solutions of (10)-(12), ∀ k ∈ K, ∀ t ∈ [ t k , t k +1 ) , V ( w ( t, . )) ≤ V ( w ( t k , . )) e − ν ( t − t k ) . (28)The function V may be not continuous at the switching times any more. Nevertheless, by (25)and (26), we have that, for all k in K , V ( w ( t k +1 , . )) = (cid:90) (cid:16) w ( t k +1 , x ) (cid:62) M − i k +1 w ( t k +1 , x ) e µ ik +1 x + w ( t k +1 , x ) (cid:62) M + i k +1 w ( t k +1 , x ) e − µ ik +1 x (cid:17) dx ≤ γ (cid:90) (cid:0) w ( t k +1 , x ) (cid:62) M − i k w ( t k +1 , x ) e µ ik +1 x + w ( t k +1 , x ) (cid:62) M + i k w ( t k +1 , x ) e − µ ik +1 x (cid:1) dx ≤ γe µ (cid:90) (cid:0) w ( t k +1 , x ) (cid:62) M − i k w ( t k +1 , x ) e µ ik x + w ( t k +1 , x ) (cid:62) M + i k w ( t k +1 , x ) e − µ ik x (cid:1) dx ≤ γe µ lim t → t − k +1 V ( w ( t, . )) November 5, 2018 DRAFT2 where the continuity of w is used in the last inequality (it follows from Proposition 3.1). Then,it follows from (28) that, for all k in K , V ( w ( t k +1 , . )) ≤ γe µ V ( w ( t k , . )) e − ν ( t k +1 − t k ) , and itallows us to prove recursively that, for all t ∈ R + , V ( w ( t, . )) ≤ (cid:0) γe µ (cid:1) N σ (0 ,t ) V ( w ) e − νt ≤ (cid:0) γe µ (cid:1) ( N + tτD ) V ( w ) e − νt . Let ¯ ν = ν − ∆ µ τ D − ln( γ )2 τ D , the assumption on the average dwell timegives that ¯ ν > and the previous inequality yields ∀ t ∈ R + , V ( w ( t, . )) ≤ (cid:0) γe µ (cid:1) N V ( w ) e − νt which allows us to conclude that the switched linear hyperbolic system is GUES with respectto the set of switching signals S τ D ,N ( R + , I ) . This concludes the proof of Theorem 2. Remark 4.1:
Setting γ = 1 and µ i = µ for all i ∈ I , we recover the assumptions ofTheorem 1. In that case we have ∆ µ = 0 : there is no positive lower bound imposed on theaverage dwell time, which is consistent with Theorem 1. ◦ Remark 4.2:
Note that the existence of γ ≥ such that (25) holds is equivalent to Ker ( S + i ) = Ker ( S + j ) , for all i, j ∈ I . Therefore the existence of γ ≥ such that (25) and (26) are satisfiedis equivalent to Ker ( S + i ) = Ker ( S + j ) and Ker ( S − i ) = Ker ( S − j ) , for all i, j ∈ I (and also, byrecalling L i = S − i Λ i S i , the subspace associated with all positive (resp. negative) eigenvalues of L i does not depend on i ). If this condition does not hold, stability can still be analyzed usingother stability results presented in the following section. ◦ B. Mode dependent sign structure of characteristics
We now relax the assumption on the number of negative and positive characteristics. As in theprevious section, we provide a first result giving sufficient conditions such that stability holdsfor all switching signals:
Theorem 3:
Let us assume that there exist ν > and diagonal positive definite matrices Q i in R n × n , i ∈ I such that, for all i ∈ I , F (cid:62) i Q i + Q i F i ≤ − νQ i , (29) G Ti Q i Λ + i G i ≤ Q i Λ + i , (30) and such that, for all i, j ∈ I , S (cid:62) i Q i S i = S (cid:62) j Q j S j . (31) Then, the switched linear hyperbolic system (10)-(12) is GUES with respect to the set of switchingsignals S ( R + , I ) . November 5, 2018 DRAFT3
Proof:
We use the same notations as in Theorem 1. We consider the candidate Lyapunovfunction, for all w in C ([0 , ∞ ); L ((0 , R n )) , for all t ∈ R + , V ( w ( t, . )) = (cid:90) y k ( t, x ) (cid:62) Q i k y k ( t, x ) dx, if t ∈ [ t k , t k +1 ) , where we used the change of variable (14). Using (29) and (30), and following the proof ofProposition 2.1, we obtain that, along the solutions of (10)-(12), it holds, for all k in K , andfor all t ∈ [ t k , t k +1 ) , V ( w ( t, . )) ≤ V ( w ( t k , . )) e − ν ( t − t k ) . Recalling (14), we have y k ( t k +1 , . ) = S i k w ( t k +1 , . ) and y k +1 ( t k +1 , . ) = S i k +1 w ( t k +1 , . ) , which gives y k +1 ( t k +1 , . ) = S i k +1 S − i k y k ( t k +1 , . ) .Hence, Equation (31) yields, for all k in K , V ( w ( t k +1 , . )) = (cid:90) y k +1 ( t k +1 , x ) (cid:62) Q i k +1 y k +1 ( t k +1 , x ) dx = (cid:90) y k ( t k +1 , x ) (cid:62) S −(cid:62) i k S (cid:62) i k +1 Q i k +1 S i k +1 S − i k y k ( t k +1 , x ) dx = (cid:90) y k ( t k +1 , x ) (cid:62) Q i k y k ( t k +1 , x ) dx = lim t → t − k +1 V ( w ( t, . )) . Thus, V is continuous at the switching time t k +1 . The end of the proof is similar to that ofTheorem 1.The assumptions of the previous theorem are quite strong. To assure the asymptotic stabilityfor switching signals with a sufficiently large dwell time, weaker assumptions are needed. Moreprecisely, considering the assumptions of Theorem 2, the last main result of this paper can bestated: Theorem 4:
Let us assume that there exist ν > , γ ≥ , µ i ∈ R , and diagonal positivedefinite matrices Q i in R n × n , i ∈ I such that the matrix inequalities (23), (24) hold (where thesame notation for Q i ( x ) is used) together with the following matrix inequalities, for all i, j ∈ I , S (cid:62) i Q i S i ≤ γS (cid:62) j Q j S j . (32) Let ¯∆ µ = 2 | µ i | if I is a singleton and ¯∆ µ = 2 max i (cid:54) = j ∈ I ( | µ i | + | µ j | ) else. Then, for all N ∈ N ,for all τ D > ln( γ )2 ν + ¯∆ µ ν , the switched linear hyperbolic system (10)-(12) is GUES with respectto the set of switching signals S τ D ,N ( R + , I ) .Proof: We use the same notations as in Theorem 2, and we consider the candidate Lyapunovfunction (27). Using (23) and (24), Equation (28) still holds along the solutions of (10)-(12).
November 5, 2018 DRAFT4
Moreover, for all k in K , V ( w ( t k +1 , . )) ≤ e | µ ik +1 | (cid:90) y k +1 ( t k +1 , x ) (cid:62) Q i k +1 y k +1 ( t k +1 , x ) dx ≤ γe | µ ik +1 | (cid:90) y k +1 ( t k +1 , x ) (cid:62) Q i k y k +1 ( t k +1 , x ) dx ≤ γe | µ ik +1 | +2 | µ ik | (cid:90) y k ( t k +1 , x ) (cid:62) Q i k ( x ) y k ( t k +1 , x ) dx ≤ γe µ lim t → t − k +1 V ( w ( t, . )) . The end of the proof follows the same lines as that of Theorem 2.Let us note that Theorem 3 can be deduced from Theorem 4 by selecting γ = 1 and µ i = 0 for all i ∈ I . V. C OMPUTATIONAL ASPECTS
The sufficient stability conditions of the results presented in this paper may be solved usingclassical numerical tools. More precisely, let us remark that the matrix inequalities (18) and (19)in the statement of Theorem 1 are linear in Q i but nonlinear in µ , some numerical methods maybe used: • by particularizing to the case µ = 0 ; • or when the source terms in (10)-(12) are diagonal; • or when the matrices Λ i are either all positive definite or all negative definite.Let us consider these three cases in the next three sections. A. Particularizing to the case µ = 0 By letting µ = 0 in the conditions (18) and (19), the matrix inequalities in Theorem 1 do notdepend on the x -variable. Therefore the following matrix inequalities F (cid:62) i Q i + Q i F i ≤ − νI n , Q i ≤ I n . (33) I m m,n − m G i + − G i ++ (cid:62) Q i Λ i I m m,n − m G i + − G i ++ ≤ G i −− G i − + n − m,m I n − m (cid:62) Q i Λ i G i −− G i − + n − m,m I n − m (34)imply (18) and (19). November 5, 2018 DRAFT5
The previous conditions (33) and (34) are linear in the unknown variables ν and Q i and cantherefore be solved using semi-definite programming (see e.g., [16] with [20]). It is thus obtainedthe following result as a corollary of Theorem 1: Corollary 5.1:
Let us assume that, for all i ∈ I , m i = m , and there exist ν > , and diagonalpositive definite matrices Q i in R n × n satisfying the matrix inequalities (20), (33) and (34). Then,the switched linear hyperbolic system (10)-(12) is GUES with respect to the set of switchingsignals S ( R + , I ) . Moreover the proof of Theorem 1 implies that the function given by, for all w in L ((0 , R n ) , V ( w ) = (cid:90) w ( x ) (cid:62) M w ( x ) dx, where M = ( S − i ) (cid:62) Q − i S − i + ( S + i ) (cid:62) Q + i S + i is a Lyapunov function of (10)-(12), and, since theestimation (cid:107) w ( t, . ) (cid:107) L ((0 , R n ) ≤ Ce − νt (cid:107) w (cid:107) L ((0 , R n ) holds along the solutions w of (10)-(12), for a suitable value C > (which does not dependon the solution), it implies that the value of ν is an estimation of the speed of the exponentialstability. Semi-definite programming (as in [20]) allows us to optimize this estimation and tocompute the largest positive value ν such that the linear matrix inequalities (33), (34), and (20)have a solution in the variables Q i and ν .Analogous corollaries may be written by letting µ i = 0 , for all i ∈ I , in Theorems 2 and 4,and by considering directly Theorem 3 (for which µ = 0 ).To conclude this section, let us emphasize that this approach is allowed since µ = 0 is possiblein our main results. This is not possible using the approach of [5], where µ should be a strictlypositive value. B. With a diagonal source term
If, for each i in I , the source term F i in (10)-(12) is diagonal, then (18) is equivalent to − µ Λ + i + F i ≤ − νI n . (35)Therefore the following result is a corollary of Theorem 1: Corollary 5.2:
Let us assume that for all i ∈ I , m i = m , the matrices F i are diagonal andthat there exist ν > , µ ∈ R , and diagonal positive definite matrices Q i in R n × n satisfying the November 5, 2018 DRAFT6 matrix inequalities, (19), (20) and (35). Then, the switched linear hyperbolic system (10)-(12)is GUES with respect to the set of switching signals S ( R + , I ) . The sufficient conditions (19), (20) and (35) of Corollary 5.2 are nonlinear in the unknownvariables ν , µ and Q i , due to the term Q i (1) in (19) which depends nonlinearly on Q i and µ .However, µ being a scalar variable, one may combine a line search algorithm with semi-definiteprogramming to solve (19), (20) and (35). Analogously, when the source terms are diagonal in(10)-(12), line search algorithms could be used to numerically check the sufficient conditions ofTheorems 2 and 4. C. When Λ i are all positive definite or all negative positive Let us assume in this section, that all velocities in (10)-(12) have the same sign, that is thateither, for all i in I , Λ i is positive definite or, for all i in I , Λ i is negative definite. To ease thepresentation of this section, it is assumed, that the first case occurs: for all i in I , Λ + i = Λ i and m i = 0 . Then the following matrix inequalities − µQ i Λ i + F (cid:62) i Q i + Q i F i ≤ − νI n , Q i ≤ I n , (36) G (cid:62) i Q i Λ i G i ≤ e − µ Q i Λ i (37)imply (18) and (19). This gives us the following corollary of Theorem 1: Corollary 5.3:
Let us assume that, for all i ∈ I , m i = 0 , and that there exist ν > , µ ∈ R ,and diagonal positive definite matrices Q i in R n × n satisfying the matrix inequalities (20), (36)and (37). Then, the switched linear hyperbolic system (10)-(12) is GUES with respect to the setof switching signals S ( R + , I ) . The conditions (20), (36) and (37) of Corollary 5.3 are again nonlinear in the unknownvariables ν , µ and Q i , due to the product of µ and Q i in (36) and the product of e − µ and Q i (37). However, since µ is a scalar variable, one may combine a line search algorithm withsemi-definite programming to solve (20), (36) and (37). Analogous techniques could be used aswell in Theorem 2. For Theorem 4, similar simplifications can be done if, for all i ∈ I , Λ i isnegative definite or positive definite. November 5, 2018 DRAFT7
VI. E
XAMPLES
A. Mode independent sign structure of characteristics
Consider the wave equation: ∂ t u ( t, x ) − ∂ x u ( t, x ) = 0 , where x ∈ [0 , , t ∈ R + , and u : R + × [0 , → R . The solutions of the previous equations can be written as u ( t, x ) = w ( t, x ) + w ( t, x ) with w = ( w w ) verifying ∂ t w ( t, x ) + L∂ x w ( t, x ) = 0 , x ∈ [0 , , t ∈ R + (38)where L = diag ( − , . We consider for this hyperbolic system the following switching bound-ary conditions: w ( t,
1) = − . w ( t, if i ( t ) = 1 − . w ( t, if i ( t ) = 2 , w ( t,
0) = . w ( t, if i ( t ) = 11 . w ( t, if i ( t ) = 2 . (39)This is a switched linear hyperbolic system of the form (10)-(12) with L = L = L , A = A = 0 , S = S = I , G = ( − . . ) and G = ( − . . ) . With the notations defined in theprevious sections, we also have Λ +1 = Λ +2 = I and F = F = 0 . We are in the particularcase described in Section V-B. Though, we were not able to apply Corollary 5.2 as we couldnot find ν > , µ ∈ R , and diagonal positive definite matrices Q and Q such that the set ofmatrix inequalities (35), (19), and (20) hold. Actually, this could be explained by the fact thatit is possible to find a switching signal that destabilizes the system as shown on the left part ofFigure 1 (where a periodic switching signal is used with a period equals to ).We can prove the exponential stability for a set of switching signals with an assumptionon the average dwell time using Theorem 2. Let us remark that since F = F = 0 , (23) isequivalent to µ i ≥ ν for i ∈ { , } . One can verify that Equations (24), (25) and (26) hold aswell for the choices Q = ( .
75 00 2 ) , Q = ( . ) and γ = 2 . Then, Theorem 2 guarantees thestability of the switched linear hyperbolic system for switching signals with average dwell timegreater than ln( γ )2 ν = 2 . . The right part of Figure 1 shows the stable behavior of the switchedlinear hyperbolic system for a periodic switching signal with a period equals to . . To illustrateCorollary 5.2, we add a diagonal damping term to (38): ∂ t w ( t, x ) + L∂ x w ( t, x ) = Aw ( t, x ) , x ∈ [0 , , t ∈ R + (40)where A = diag ( − . , − . . The boundary conditions are given by (39). Now, A = A = F = F = A and the other matrices of the system remain unchanged. In the present case November 5, 2018 DRAFT8
Fig. 1. Time evolution of u = w + w , solution of (38)-(39) for periodic switching signals of period (left) and . (right). (35) is equivalent to ν ≤ µ + 0 . . One can verify that Corollary 5.2 applies with µ = − . , ν = 0 . and Q = Q = ( . ) . Then, Theorem 1 guarantees the stability of the switched linearhyperbolic system for all switching signals. Figure 2 shows the stable behavior of the switchedlinear hyperbolic system for a periodic switching signal of period . B. Mode dependent sign structure of characteristics
To illustrate the results of Section IV-B, we consider the following switched linear hyperbolicsystem ∂ t w ( t, x ) + L i ( t ) ∂ x w ( t, x ) = F w ( t, x ) , x ∈ [0 , , t ∈ R + (41)where w : R + × [0 , → R , i ( t ) ∈ I = { , } , L = 1 and L = − and F ∈ R . The boundaryconditions are given by w ( t,
0) = Gw ( t, if i ( t ) = 1 w ( t,
1) = Gw ( t, if i ( t ) = 2 (42)and G > . This is a switched linear hyperbolic system of the form (10)-(12) with A = A = F , S = S = 1 , and G = G = G . With the notation defined in the previous sections, we alsohave Λ +1 = Λ +2 = 1 and F = F = F . November 5, 2018 DRAFT9
Fig. 2. Time evolution of u = w + w , solution of (39)-(40) for a periodic switching signal of period . We assume that
F < − ln( G ) ; if this does not hold, then it can be shown that the linearhyperbolic systems in each mode are both not asymptotically stable. If F < and G ≤ , thenTheorem 3 applies with Q = Q = 1 and ν = − F . Hence, in that case Theorem 3 guaranteesthe stability of the switched linear hyperbolic system for all switching signals. If G > (resp. F > ) then the condition (30) (resp. (29)) of Theorem 3 does not hold and thus Theorem 3does not apply.If G > , let F < µ < − ln( G ) , then Theorem 4 holds with µ = µ = µ , ν = µ − F , γ = 1 and Q = Q = 1 . Then, Theorem 4 guarantees the stability of the switched linearhyperbolic system for switching signals with average dwell time τ D greater than ¯∆ µ ν = − µµ − F for any µ ∈ ( F, − ln( G )) . The minimal value of − µµ − F in this interval is − G )ln( G )+ F ; therefore thestability of the switched linear hyperbolic system is guaranteed for switching signals with τ D greater than − G )ln( G )+ F . For G = 2 and F = − , in that case the minimal required τ D is . .Figure 3 shows unstable and stable behaviors for these values of G and F and for periods equalto . and . .If F > , let G and µ such that F < µ < − ln( G ) , then Theorem 4 holds with µ = µ = µ , ν = µ − F , γ = 1 and Q = Q = 1 . Then, Theorem 4 guarantees the stability of the November 5, 2018 DRAFT0 switched linear hyperbolic system for switching signals with τ D greater than ¯∆ µ ν = µµ − F for any µ ∈ ( F, − ln( G )) . The minimal value of µµ − F in this interval is G )ln( G )+ F ; therefore stability of theswitched linear hyperbolic system is guaranteed for switching signals with τ D > G )ln( G )+ F . For G = 0 . and F = − . , the minimal required τ D is . . Figure 4 shows unstable and stablebehaviors for these values of G and F and for periods equal . and . . Fig. 3. Time evolution of w , solution of (41)-(42) with F = − and G = 2 , for periodic switching signals of period . (left)and . (right). VII. C
ONCLUSION
In this paper, some sufficient conditions have been derived for the exponential stability ofhyperbolic PDE with switching signals defining the dynamics and the boundary conditions.This stability analysis has been done with Lyapunov functions and exploiting the dwell timeassumption, if it holds, of the switching signals. The sufficient stability conditions are writtenin terms of matrix inequalities which lead to numerically tractable problems.This work lets many questions open and may have natural applications on physical applica-tions. In particular, exploiting the sufficient conditions for the derivation of switching stabilizingboundary controls (as for the physical application considered in [6]) seems to be a natural
November 5, 2018 DRAFT1
Fig. 4. Time evolution of w , solution of (41)-(42) with F = 0 . and G = 0 . , for periodic switching signals of period . (left) and . (right). extension. The generalization of the results to linear hyperbolic with space-varying entries mayalso be studied. R EFERENCES [1] I. Aksikas, A. Fuxman, J.F. Forbes, and J.J. Winkin. LQ control design of a class of hyperbolic PDE systems: Applicationto fixed-bed reactor.
Automatica , 45(6):1542–1548, 2009.[2] S. Amin, F.M. Hante, and A.M. Bayen. Exponential stability of switched linear hyperbolic initial-boundary value problems.
IEEE Transactions on Automatic Control , 57(2):291–301, 2012.[3] J.-M. Coron, G. Bastin, and B. d’Andr´ea Novel. Dissipative boundary conditions for one-dimensional nonlinear hyperbolicsystems.
SIAM Journal on Control and Optimization , 47(3):1460–1498, 2008.[4] J.-M. Coron, B. d’Andr´ea Novel, and G. Bastin. A strict Lyapunov function for boundary control of hyperbolic systemsof conservation laws.
IEEE Transactions on Automatic Control , 52(1):2–11, 2007.[5] A. Diagne, G. Bastin, and J.-M. Coron. Lyapunov exponential stability of linear hyperbolic systems of balance laws.
Automatica , 48(1):109–114, 2012.[6] V. Dos Santos and C. Prieur. Boundary control of open channels with numerical and experimental validations.
IEEETransactions on Control Systems Technology , 16(6):1252–1264, 2008.[7] N.H. El-Farra and P.D. Christofides. Coordinating feedback and switching for control of spatially distributed processes.
Computers & chemical engineering , 28(1-2):111–128, 2004.[8] M. Gugat and M. Tucsnak. An example for the switching delay feedback stabilization of an infinite dimensional system:The boundary stabilization of a string.
Systems & Control Letters , 60(4):226–233, 2011.
November 5, 2018 DRAFT2 [9] F. Hante, M. Sigalotti, and M. Tucsnak. On conditions for asymptotic stability of dissipative infinite-dimensional systemswith intermittent damping.
Journal of Differential Equations , to appear.[10] F.M. Hante, G. Leugering, and T.I. Seidman. Modeling and analysis of modal switching in networked transport systems.
Applied Mathematics & Optimization , 59(2):275–292, 2009.[11] F.M. Hante and M. Sigalotti. Converse Lyapunov theorems for switched systems in Banach and Hilbert spaces.
SIAMJournal on Control and Optimization , 49(2):752–770, 2011.[12] J.P. Hespanha and A.S. Morse. Stability of switched systems with average dwell-time. In , volume 3, pages 2655–2660, Phoenix, AZ, USA, 1999.[13] M. Krstic and A. Smyshlyaev. Backstepping boundary control for first-order hyperbolic PDEs and application to systemswith actuator and sensor delays.
Systems & Control Letters , 57:750–758, 2008.[14] T.-T. Li.
Global classical solutions for quasilinear hyperbolic systems , volume 32 of
RAM: Research in AppliedMathematics . Masson, Paris, 1994.[15] D. Liberzon.
Switching in systems and control . Springer, 2003.[16] J. L¨ofberg. Yalmip : A toolbox for modeling and optimization in MATLAB. In
Proc. of the CACSD Conference , Taipei,Taiwan, 2004. http://users.isy.liu.se/johanl/yalmip/.[17] Z.-H. Luo, B.-Z. Guo, and O. Morgul.
Stability and stabilization of infinite dimensional systems and applications .Communications and Control Engineering. Springer-Verlag, New York, 1999.[18] C. Prieur and J. de Halleux. Stabilization of a 1-D tank containing a fluid modeled by the shallow water equations.
Systems& Control Letters , 52(3-4):167–178, 2004.[19] C. Prieur and F. Mazenc. ISS-Lyapunov functions for time-varying hyperbolic systems of balance laws.
Mathematics ofControl, Signals, and Systems , 24(1):111–134, 2012.[20] J.F. Sturm. Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones.
Optimization Methods andSoftware , 11-12:625–653, 1999. http://sedumi.ie.lehigh.edu/.[21] E. Zuazua. Switching control.
European Mathematical Society , 13:85–117, 2011., 13:85–117, 2011.