Nash equilibrium with wave dynamics and boundary control T-P Azevedo-Perdico ́ ulis SR–Coimbra Dep. de Engenharias, UTAD 5001-801 Vila Real, Portugal Email: tazevedo@utad.pt G. Jank Emeritus Professor at RWTH Department II of Mathematics 52056 Aachen, Germany Email: jank@math2.rwth-aachen.de P. Lopes dos Santos FEUP, 4200-465 Porto, Portugal Email:pjsantos@fe.up.pt Abstract —In this paper, the gas dynamics within the pipelines is written as a wave repetitive process, and modify it in a way that the dynamics is influenced by p decision makers, namely the boundary conditions. We obtain sufficient criteria for the existence of boundary equilibrium controls as well as controllability of the different agents and observability of the system when this is steered through initial and boundary data. From the point of view of some applications it seems to make sense to consider boundary data controls, e.g. in high pressure gas networks management. Keywords : Boundary control, Controllability, Differential games, Gas networks modelling, Nash equilibrium, Observ- ability, Repetitive processes, 2 D–systems. I. I NTRODUCTION We consider the discrete wave linear discrete model of repet- itive processes (RP), that has been introduced in [5] and modify it in a way that it can be steered independently by p decisions makers (or players) as in [2]. Wave model RP are a distinct class of 2 –D systems of both systems theoretic and applications interest and have been studied in [7], [17], [19]. The information propagation in this model occurs in two directions, showing an intrinsic non causal nature in the right upper quadrant, as pointed out in [5]. But, for some applications, the pass dynamics has a space characteristic and the causality is not physically required. Also, this type of models capture well the minimum neighbour characteristics of systems as well as the sometimes existing reverse dynamics. Furthermore, in a model of this type the final value of each pass is specified instead. Hence consider the autonomous wave model whose wave length is N : x k +1 ( ) = N ∑ i = − N | + i | ≤ N A i x k ( + i ) + p − 2 ∑ j =1 B j u j,k ( ) , (1) y k ( ) = Cx k ( ) , (2) k = 0 , 1 , . . . , T − 1 = 0 , 1 , . . . , L In model (1)–(2) the pass number is indexed by k and may be infinite. The steps per pass– k are indexed by ∈ [0 , L ] ∩ Z with L denoting the pass length that we define as L := [ − N, N ] ∩ Z Also, we consider that K := { k : x k ( ) = 0 , k = T + 1 , . . . and k = . . . , − 2 , − 1 } and L := [0 , L ] ∩ Z and, in order to obtain a more clear representation, we consider N := [ L 2 ] and relabel the steps along the pass as [ − N, . . . , N ] Therefore, we define the compact support of x k ( ) , y k ( ) and u j,k ( ) , j = 1 , 2 , . . . , p − 2 , as K × L , where p is the number of inputs. x k ( ) ∈ R n are the state vectors along pass– k. y k ( ) ∈ R m are the pass profile vectors along pass– k. u j,k ( ) ∈ R r j are the control vectors along pass– k. n, m and r j are the dimensions of the state, the pass profile and the control vectors, respectively. Matrices in (1)–(2) have the dimensions: A i ∈ R n × n , B j ∈ R n × r j , C ∈ R m × n , D j ∈ R m × r j , j = 1 , 2 , . . . , p − 2 In addition, we define the initial pass profile and the boundary conditions at every pass– k. Hence: x 0 ( ) = φ ( ) , = − N, . . . , N (3) y k ( − N ) = d k and y k ( N ) = g k , (4) k = 0 , 1 , . . . , T − 1 where the compact support of φ and d k , g k is L and K , respectively. Define I := K × L The state x · ( ) ∈ 2 ,n ( I ) , the pass profile y · ( ) ∈ 2 ,m ( I ) and the controls are admissible if u j, · ( ) ∈ 2 ,r j ( I ) , where 2 ,ν ( I ) denotes the Hilbert space of ν -dimensional sequences defined on I with the standard scalar product. Furthermore, define X := 2 ,n ( I ) and U j := 2 ,r j ( I ) The used to represent 2 ,ν ( I ) does not have anything to do with the indexation of every pass, e.g. x k ( ) But we use the same notation, since it looks to us that its meaning becomes clear from the context. Also, we assume that there is a quadratic cost functional 978-1-4799-8739-9/15/$31.00 c © 2015 European Union associated to each player: J j ( u 1 , . . . , u p , Φ) = N − 1 ∑ = − N +1 x ∗ T ( ) K j x T ( ) + T − 1 ∑ k =0 N − 1 ∑ = − N +1 x ∗ k ( ) Q j x k ( )+ + p − 2 ∑ i =1 T − 1 ∑ k =0 N − 1 ∑ = − N +1 u ∗ i,k ( ) R ji u i,k ( ) , (5) where − ∗ is the hermitian transpose, K j , Q j ∈ R n × n , R ji ∈ R r j × r i ; j, i = 1 , . . . , p, k = 0 , . . . , T − 1 Next, we introduce the stacked vectors for every pass number k : X k := ⎛ ⎜ ⎝ x k ( − N + 1) x k ( N − 1) ⎞ ⎟ ⎠ and Φ : = ⎛ ⎜ ⎝ φ ( − N + 1) φ ( N − 1) ⎞ ⎟ ⎠ , where both belong to X (2 N − 1) Also u j,k : = ( u j,k ( − N + 1) · · · u j,k ( N − 1)) ∗ ∈ U (2 N − 1) j , j = 1 , . . . , p, and define the square state matrix, A , of dimension (2 N − 1) n as: ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ A 0 A 1 · · · A N 0 . . . 0 0 A − 1 A 0 · · · . . . A N . . . 0 0 . . . A − N A N 0 A − N A N − 1 . . . 0 0 0 0 A − N . . . A − 1 A 0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ Define R ( m, p ) := R (2 N − 1) m × (2 N − 1) p and matrices B j := I 2 N − 1 ⊗ B j ( t ) ∈ R ( n, r j ) , R ji := I 2 N − 1 ⊗ R ji ∈ R ( r j , r i ) , S j = B j R − 1 jj B T j ∈ R ( n, n ) and Q j := I 2 N − 1 ⊗ Q j ∈ R ( n, n ) , K j := I 2 N − 1 ⊗ K j ∈ R ( n, n ) , j = 1 , . . . , p − 2 , where ⊗ is the Kronecker product and I i is the i -dimension unit matrix. Define u p − 1 ,k := d k and u p,k := g k , k = 0 , . . . , T − 1 , and the respective input matrices B p − 1 := ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ A − 1 A − 2 A − N 0 0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ and B p := ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 0 0 A N A N − 1 A 0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ Hence: X k +1 = A X k + p ∑ j =1 B j u j,k (6) with initial data X 0 = Φ The last two of these controls are boundary controls. Using this compact notation, (5) is written as: J j ( u 1 , . . . , u p , Φ) = X ∗ T K X T + T − 1 ∑ k =0 X ∗ k Q X k (7) + p ∑ i =1 T − 1 ∑ k =0 U ∗ j,k R ji U ∗ i,k , where the two last players are boundary controls. In [13], [18], and related articles cited therein, controllability, reachability and observability criteria are obtained based on an implicit representation of the solutions. Also, general algebraic meth- ods, i.e., module theoretic or behavioural approaches could be used to obtain controllability results. Controllability properties by boundary data control in continuous time 2 –D systems are also obtained in [8], [10]. Nash equilibria have been considered in literature in a vast amount of articles with different underlying dynamics. See for example [1] and the literature therein for further reading. A game theoretic approach in a continuous 2 D -system also has been developed in [9]. In this article, we formulate the wave model RP as a differen- tial game where the boundary conditions are seen as boundary controls of a differential game. In Section II, we withdraw sufficient conditions for the existence of an equilibrium in the sense of Nash for the open-loop information structure and supply a representation for equilibrium controls. In Section III we study the controllability of this noncooperative differential game. For completeness reasons we also study observability of the Nash game. Issues as uniqueness of the equilibrium or stabilization of the solution are not treated in this work, since we focus on the finite time horizon case. This formulation is very suitable to some applications, as for instance gas pipelines since the wave model captures well the reverse dynamics that exists in the pipes and also allows for studying the sole influence of boundary data on the system dynamics. In Section IV we write the pipeline dynamics as a wave model. In Section V, after withdrawing some conclusions, we outline some guidelines for future work. II. O PEN -L OOP N ASH EQUILIBRIUM Consider a p -player game, Γ p , on a finite time horizon, T < ∞ , with an open-loop information structure. In game theory terminology, an ”open-loop” strategy means that the players have to choose their strategies (the controls) u 1 , u 2 , . . . , u p prior to the game start and that their only information on the state of the game is at the initial pass, x 0 ( ) = φ ( ) , ∈ L . We recall from [3] the definition of a Nash equilibrium: Definition II.1 (Open-loop Nash equilibrium) (ˆ u 1 , ˆ u 2 , . . . , ˆ u p ) is called a ( p -player) open-loop Nash equilibrium strategy on the system (1) – (4) if J 1 (ˆ u 1 , ˆ u 2 , . . . , ˆ u p , Φ) ≤ J 1 ( u 1 , ˆ u 2 , . . . , ˆ u p , Φ) , J 2 (ˆ u 1 , ˆ u 2 , . . . , ˆ u p , Φ) ≤ J 2 (ˆ u 1 , u 2 , . . . , ˆ u p , Φ) , J p (ˆ u 1 , ˆ u 2 , . . . , ˆ u p , Φ) ≤ J p (ˆ u 1 , ˆ u 2 , . . . , u p , Φ) (8) for all initial states Φ ∈ X (2 N − 1) and all admissible strategies u 1 , . . . , u p ∈ U (2 N − 1) 1 × · · · × U (2 N − 1) p Nash strategies safeguard against every player deviating from the equilibrium strategy, and becomes reasonable when coop- eration between decision makers cannot be guaranteed. Necessary and sufficient conditions for the existence of Nash equilibria under the dynamics of a wave model RP were established in [2], using a value function approach that allows for obtaining sufficient conditions for the existence of Nash equilibria under very relaxed assumptions on the definiteness of the coefficient matrices. This is very interesting from the applications point of view, since the usual convexity require- ments are usually very restrictive. Also, a representation for the equilibrium controls and of the respective cost function of every player are withdrawn. In [2] it was shown that if the symmetric standard discrete-time matrix Riccati equation admits a solution then necessary and sufficient conditions for the existence of a Nash equilibrium are obtained in terms of the solvability of a boundary value problem. Furthermore, the uniqueness of the solution of this boundary value prob- lem implies the uniqueness of the Nash equilibrium. Also, the existence of the solution of the discrete-time open-loop Riccati difference equations is a sufficient condition for the existence of Nash equilibrium that avoids the calculation of the solution of the boundary value problem and conveys a linear representation of the equilibrium controls, which yields a very efficient numerical method. This result is synthesised in Theorem II.1. The problem of finding an OL Nash equilibrium has been considered in several articles (see for instance [3], [11]). In this section, we collect the most important results. Since, we are concerned about boundary control, we consider in the cost function R ji = 0 , j = 1 , . . . , p − 2 and (7) becomes: J j ( u 1 , u 2 , Φ) = X ∗ T K X T + T − 1 ∑ k =0 X ∗ k Q X k + 2 ∑ s =1 T − 1 ∑ k =0 u ∗ j,k R js u ∗ s,k (9) In what follows we consider two boundary controls, however this can be generalised to any number p . In what follows, we consider p = 2 Definition II.2 (Best reply) 1) An admissible control ˆ u j , j = 1 , 2 , is called the best reply of player- j , to any set of admissible controls u ̄ j = { u i | i ∈ { 1 , 2 }\{ j }} on system (6) if J j (ˆ u j , u ̄ j , Φ) ≤ J j ( u j , u ̄ j , Φ) and J j , j = 1 , 2 , is given in (9) 2) ˆ u 1 , ˆ u 2 is an open-loop Nash equilibrium in a 2-player game for system (6) if both players simultaneously achieve their best replies. Theorem II.1 (Sufficient condition for Nash equilibrium) Let the solutions E j ( k ) , j = 1 , 2 , of the symmetric standard discrete time matrix Riccati equations: 0 = A ∗ E j ( k + 1) A − E j ( k ) + Q j − (10) − A ∗ E j ( k + 1) B j × ( R jj + B ∗ j E j ( k + 1) B j ) − 1 B ∗ j E j ( k + 1) A E j ( T ) = K j , j = 1 , 2 , (11) exist for t = 0 , . . . , T, and where, necessarily, S j ( k ) = R jj + B ∗ j E j ( k + 1) B j is invertible (i.e., S j ( k ) > 0 ). Whence, considering Ω := I + 2 ∑ s =1 B s R − 1 ss B ∗ s K s ( k + 1) , if the discrete time open-loop Nash Riccati difference equations K j ( k ) = Q j + A ∗ K j ( k + 1)Ω − 1 A , (12) K j ( T ) = K j , j = 1 , 2 , t = 0 , . . . , T − 1 , (13) admit a solution then there exists a unique open-loop Nash equilibrium defined in quasi feedback form by ˆ u j,k = − R − 1 jj B ∗ j ( K j ( k + 1) X k +1 + D j ( k + 1)) , (14) k = 0 , . . . , T − 1 , with D j ( k ) , G j ( k ) defined as: D j ( k ) = A ∗ D j ( k + 1) + G j ( k ) , D j ( T ) = 0 , (15) G j ( k ) = − A ∗ K j ( k + 1)Ω − 1 p ∑ s =1 B s R − 1 ss B ∗ s D s ( k + 1) (16) Proof: See [2]. The proof reveals that solvability of the discrete time open- loop Nash Riccati difference equation is sufficient for the solvability of the boundary value problem and, moreover, under the given conditions, for the finite time horizon, a unique Nash equilibrium is obtained. Thus, we obtained in this way a representation for the optimal boundary controls. III. C ONTROLLABILITY AND OBSERVABILITY Controllability of differential games has been discussed in [14]. In this article, we are interested in conditions for individual boundary control of wave model RP. Definition III.1 (Individual controllability) Let Γ 2 be a 2- player game. Suppose that strategies are chosen such that (ˆ u 1 , ˆ u 2 ) is an equilibrium for Γ 2 Then, we say that the game is controllable at this equilibrium point, from the point of view of the j th player, if the control system X k +1 = f ( k, X k , u j,k , ˆ u ̄ j ,k ) = A X k + B j u j,k + B ̄ j ˆ u ̄ j ,k is controllable in the admissible set of u j,k The next Lemma characterises the individual controllability of the open-loop Nash equilibrium. Lemma III.1. Let Γ 2 be a linear open-loop quadratic differ- ential game. Suppose (ˆ u 1 , ˆ u 2 ) to be a Nash equilibrium for Γ 2 , based on the solutions K j ( k ) , j = 1 , 2 , of the correspondent discrete time open-loop Nash Riccati difference equations, then Γ 2 is individually controllable for the j th player if and only if any triple ( k 0 , Φ , Φ) ∈ K × X (2 N − 1) × X (2 N − 1) of the following linear control system ( X k +1 ˆ X k +1 ) = ( Ω − 1 ̄ j A 0 0 Ω − 1 A ) ( X k ˆ X k ) (17) + ( Ω − 1 ̄ j B j 0 ) u j,k , with Ω ̄ j := I + 2 ∑ s = 1 s = j B s R − 1 ss B ∗ s K s ( k + 1) Then it can be controlled to a pass X f × X (2 N − 1) for all X f ∈ X (2 N − 1) Remark III.1. The invertibility of Ω and of Ω ̄ j is required for the representation of the equilibrium controls in Theorem II.1. Proof: See [14]. In the sequel, we assume u j,k ( ) , ∈ L to be given and fixed, which is equivalent to having u j,k ( ) = 0 , ∈ L , j = 1 , . . . , p − 2 , that is, in equation (6) we consider only the boundary controls. We deal with pass boundary controllability whenever we are to steer the system with pass boundary data, { d k } k ∈ K and { g k } k ∈ K Definition III.2 (Individual pass controllability) System (1) is (completely) pass boundary controllable for player j in k 0 , k 0 + 1 , . . . , k 1 with k 0 , k 1 ∈ K if for any initial conditions φ ( − N + 1) , . . . , φ (0) , . . . , φ ( N − 1) in (3) and any vector pass x f ( ) , ∈ L , if there exists sequences of boundary data d k , g k , k = k 0 , . . . , k 1 such that x k 1 ( ) = x f ( ) , ∈ L Theorem III.1. The wave model (1) is completely pass con- trollable on 0 , 1 , . . . , T, if and only if the grammian matrix G T = T − 1 ∑ s =0 M ( s ) M ( s ) ∗ (18) is positive definite. and where M ( s ) := ( Ω − 1 ̄ j A 0 0 Ω − 1 A ) s ( Ω − 1 ̄ j B j 0 ) Notice that considering the discrete time interval 0 , 1 , . . . , T instead of k 0 , k 0 + 1 , . . . , k 0 + T = k 1 is, in the context of autonomous systems, no restriction. Proof: Consider the linear control system (17) written in terms of the initial pass and recall that the boundary conditions are written as controls in (6). Hence: ( X k ˆ X k ) = ( Ω − 1 ̄ j A 0 0 Ω − 1 A ) k ( Φ Φ ) + k ∑ s =1 ( Ω − 1 ̄ j A 0 0 Ω − 1 A ) k − 1 ( Ω − 1 ̄ j B j 0 ) u j,s − 1 Then the grammian G T is defined in terms of the transition ma- trix ( Ω − 1 ̄ j A 0 0 Ω − 1 A ) and the output matrix ( Ω − 1 ̄ j B j 0 ) The proof is the same as in classical systems and therefore omitted here. See for instance [2], [12]. There is another possibility of boundary control of system (1), which is by steering it with the data on the initial pass, that is, by an appropriate choice of Φ while keeping the boundary sequences { d k } k ∈ K and { g k } k ∈ K are fixed. Definition III.3. System (1) – (4) is completely pass control- lable by initial pass control if for any boundary condi- tions d 0 , d 1 , . . . , d T and g 0 , g 1 , . . . , g T in (4) and any vector pass x f ( ) , ∈ L , there exists a sequence of initial data φ ( − N + 1) , . . . , φ (0) , . . . , φ ( N − 1) , subsumed in Φ , such that X T = X f There is a simple criterion for this type of controllability. Theorem III.2. System (1) – (4) is completely pass controllable by initial pass control if ( Ω − 1 ̄ j A 0 0 Ω − 1 A ) ∈ R (2 n, 2 n ) has full rank. Proof: See [2]. Definition III.4 (Boundary observability) System (1) – (2) is pass-boundary observable in { 0 , 1 , . . . , T } , if for all t 1 ∈ N , 0 < t 1 ≤ T and boundary data Φ for any two trajectories X k , ̃ X k , 0 < k ≤ t 1 , corresponding to the same input u j,k , j = 1 , 2 , 0 < k ≤ t 1 , from C X k = C ̃ X k , 0 < k ≤ t 1 , it follows necessarily that X k = ̃ X k , 0 < k ≤ t 1 Remark III.2. If we set ˆ X k = X k − ̃ X k , k = 1 , . . . , T, i.e., ˆ X k is the solution of the homogeneous equation, then pass boundary observable is equivalent to the condition: C ˆ X k = 0 = ⇒ ˆ X k = 0 , 0 < k ≤ t 1 , considering Φ = 0 Theorem III.3 (pass-boundary observable) System (1) and (2) is pass-boundary observable in { 0 , 1 , . . . , T } , if rank ( CA k − 1 B s ) = n. Proof: Using the compact notation, we define C = diag { C, . . . , C } Hence: ˆ Y k = Y k − ̃ Y k = C X k − C ̃ X k = C 2 ∑ s =1 k − 1 ∑ i =0 A k − 1 B s ( u s,i − ̃ u s,i ) Considering ˆ Y k = 0 , k = 1 , 2 , . . . , t 1 , we obtain: ˆ Y 1 = C 2 ∑ s =1 B s ( u s, 0 − ̃ u s, 0 ) = 0 = ⇒ u s, 0 = ̃ u s, 0 ˆ Y 2 = C 2 ∑ s =1 B s ( u s, 0 − ̃ u s, 0 ) + AB s ( u s, 1 − ̃ u s, 1 ) = 0 = ⇒ u s, 1 = ̃ u s, 1 Then we have that the boundary controls are uniquely defined by a measured output. IV. T HE GAS NETWORK AS A WAVE MODEL RP Following [15], the gas dynamics within the pipelines is represented by a set of partial differential equations. Different models exist, that vary according to the assumptions made and working conditions. The derivation of some of these models can be seen in [16]. Here, we use the hyperbolic model: ⎧ ⎪ ⎨ ⎪ ⎩ ∂q ( t, x ) ∂t = − S ∂p ( t, x ) ∂x − λc 2 2 dS q 2 ( t, x ) p ( t, x ) , ∂p ( t, x ) ∂t = − c 2 S ∂q ( t, x ) ∂x , (19) where x is space, t is time, p is pressure, q is mass flow, S is the cross-sectional area, d is the pipe diameter, c is the isothermal speed of sound, and λ is a friction factor. Model (19) is linearised around the operation levels ( ̄ p ( x ) , ̄ q ) , where we assume a constant flow rate ̄ q. Thence we set p = ̄ p ( x ) + Δ p ( t, x ) and q = ̄ q + Δ q ( t, x ) , with Δ p and Δ q as the deviations from the reference values of pressure and flow, ̄ p ( x ) and ̄ q . Hence: q 2 p = ( ̄ q + Δ q ) 2 ̄ p + Δ p ∼ = ̄ q 2 ̄ p ( x ) + 2 ̄ q ̄ p ( x ) Δ q − ̄ q 2 ̄ p ( x ) 2 Δ p. (20) Substituting (20) into (19), we obtain: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ∂ Δ q ( t, x ) ∂t = − S ∂ Δ p ( t, x ) ∂x − S ∂ ̄ p ( x ) ∂x + − λc 2 2 dS ( ̄ q 2 ̄ p ( x ) + 2 ̄ q ̄ p ( x ) Δ q ( t, x ) ) + λc 2 2 dS ̄ q 2 ̄ p ( x ) 2 Δ p ( t, x ) ∂ Δ p ( t, x ) ∂t = − c 2 S ∂ Δ q ( t, x ) ∂x (21) Divide the pipeline into several segments L = [ x − 1 , x ] , = 1 , . . . , n with x 0 = 0 and x n = L, where L is the pipe length, and consider every segment to have equal mass flow. Thus, we assume the linearisation according to a constant flow rate. Therefore, from the first equation of (19), a representation for the operational level of pressure is: ̄ p ( x ) = √ ̄ p 2 ( x 0 ) − λc 2 2 dS 2 ̄ q 2 ( x − x 0 ) , where ̄ p ( x 0 ) is the initial nodal pressure at a vertex of the pipeline and is the average value along time 1 T [ T 0 p ( t, x ) dt ∣ ∣ ∣ ∣ = x 0 A combined discretisation scheme is used, where we discretise according to time using a forward formula and according to space using a centred 2-point formula; consider h 1 as the time step and h 2 as the space step. Thence, consider Δ q ( t, x ) and Δ p ( t, x ) at points ( k h 1 , h 2 ) , k = 0 , 1 , . . . , T, T is gas network operation period and = 0 , 1 , . . . , L. We define: β := Sh 1 2 h 2 , ξ ( ) := λc 2 dS h 1 ̄ q ̄ p ( ) , γ ( ) := ξ ( ) 2 ̄ p ( ) ̄ q, α ( ) := 1 − ξ ( ) , ρ := c 2 h 1 2 Sh 2 , F ( ) := − γ ( ) ̄ p ( ) − β ( ̄ p ( + 1) − ̄ p ( − 1)) Using the above definitions and changing the notation a bit more, that is f ( k, ) = f k ( ) , we have: ⎧ ⎨ ⎩ Δ q k +1 ( ) = α ( )Δ q k ( ) + β Δ p k ( − 1) + γ ( )Δ p k ( ) − β Δ p k ( + 1) + F ( ) Δ p k +1 ( ) = Δ p k ( ) + ρ Δ q k ( + 1) − ρ Δ q k ( − 1) (22) In matrix form and defining x 1 := Δ q and x 2 := Δ p and x = ( x 1 x 2 ) In this new notation, we write: x k +1 ( ) = ( 0 β α ( ) 0 ) ︸ ︷︷ ︸ := A − 1 x k ( − 1) + ( α ( ) γ ( ) 0 1 ) ︸ ︷︷ ︸ := A 0 x k ( ) + ( 0 − β − α ( ) 0 ) ︸ ︷︷ ︸ := A 1 x k ( + 1) + ( F ( ) 0 ) that is x k +1 ( ) = A − 1 x k ( − 1) + A 0 x k ( ) + A 1 x k ( + 1) + ( F ( ) 0 ) If we opt for using discretisation with more points, we obtain more full matrices. To this model, we also need to add initial and boundary conditions of type (3)–(4). Whence the boundary conditions can be interpreted as the most convenient regime of operation of the controllable units (or players), i.e., gas pressure and mass flow need to be kept at some desirable levels through time. The initial conditions can be seen as a starting regime of operation, namely two possibilities to initialise the flow/pressure vector are: (i) using the optimum solution found at the previous period of opera- tion, which is reasonable since the network behaviour is not expected to change drastically from one period of operation Fig. 1. to the next one; or (ii) a starting value could be found in pre-computation, for example using steady-state optimisation. These initialisation values are also the operation levels consid- ered in the linearisation of the PDE system (19). d k , g k can be understood as the pumping regimes at the pipe inlet and contracted delivery levels at the outlet, respectively, which means that the boundary data can be seen as boundary controls. Besides taking into account the operation of the network, these boundary levels can be calculated having in view some economic goals. V. C ONCLUSIONS AND FUTURE WORK We have formulated a wave RP as an open-loop Nash game where the strategies are the boundary settings. We state suffi- cient conditions for the existence of the equilibrium strategies suitable for numerical calculations and study structural prop- erties of the equilibrium strategies. In future work, we intend to investigate the same problem but considering an infinite time horizon, where questions such as individual stabilisation of the solution by the different players become relevant as well as uniqueness of the equilibrium strategies. Also, we would like to consider other type of information structures and equilibria for the same problem. Also, these results may have applications in the modeling of physical systems or as an analysis setting for addressing systems theoretic questions. Iterative learning control laws is one possible application area that uses the wave model [4]. VI. A CKNOWLEDGMENTS The authors would like to thank the reviewers for their insightful comments. R EFERENCES [1] H. Abou Kandil, G. Freiling, V. Ionescu and G. Jank, Matrix Riccati Equations in Control and Systems Theory , Birkh ̈ auser Verlag, Basel, 2003. [2] Azevedo-Perdico ́ ulis, T. P. and G. Jank, OL-Nash Games on Affine Repetitive Processes in Proceedings of MTNS 2008: 18th International Symposium on the Mathematical Theory of Networks and Systems , Virginia Tech, Blacksburg, Virginia, USA, 2008. [3] T. Bas ̧ar and G. J. Olsder, Dynamic Noncooperative Game Theory , New York: Academic, 1995. [4] B. Cichy, K. Galkowski and E. Rogers, 2D systems based robust iterative learning control using noncausal finite-time interval data in Systems and Control Letters , Volume 64, pp. 36–42, 2014. [5] B. Cichy, K. Galkowski, E. Rogers and D. H. Owens, Control of a class of wave discrete linear repetitive processes in Proceedings of the 13th Mediterranean Conference on Control and Automation , Limassol, Cyprus, June 27-29, 2005. [6] S. Dymkou, G. Jank and T.P. Azevedo-Perdico ́ ulis, Graph and 2D Systems Approach in Gas Transport Network Modeling in International Journal of Tomography & Statistics , Volume 6, pp. 21–26, 2007. [7] K. Galkowski, B. Cichy, E. Rogers, J. Lam, Stabilisation of a class of a uncertain “wave” discrete linear repetitive processes, in Proceedings of the 45th IEEE Conference on Decision and Control , pp. 1435–1440, December 2006. [8] E. Gyurkovics and G. Jank, Instant Controllability for Goursat Problems in Pure Mathematics and Applications , Volume 12, pp. 51–65, 2001. [9] G. Jank, Disturbance Attenuation in Hyperbolic 2D-systems in Multi- dimensional Systems and Signal Processing , Volume 17, pp. 257–270, 2006. [10] G. Jank, Controllability, Observability and Optimal Control of Continuous-Time 2-D Systems in Int. J. Appl.Math. Comput. Sci. , Volume 12, pp. 181–195, 2002 [11] G. Jank and H.Abou Kandil, Existence and Uniqueness of Open- Loop Nash Equilibria in Linear-Quadratic Discrete Time Games , IEEE Transactions on Automatic Control, Volume 48, pp. 267–271, 2003. [12] H.W. Knobloch and H. Kwakernaak. Lineare Kontrolltheorie . Springer- Verlag, Berlin, 1985. [13] J. Klamka, Controllability of 2-D Systems: a Survey in Applied Math- ematical and Computational Sciences , Volume 7, pp. 835–854, 1997. [14] G. Kun, Stabilizability, Controllability and Optimal Strategies of Linear and Nonlinear Dynamical Games , PhD Thesis, RWTH, Germany, 2000. [15] J. Niepłocha, Discrete-time Quadratic Optimal Control of Gas Network Systems in Foundations of Control Engineering , Volume 13 (4), pp. 175–186, 1988. [16] A. Osiadacz, S imulation and Analysis of Gas Networks. E. & F.N. Spon, London, 1987. [17] B. Palucki, K. Galkowski, A. Kummert and B. Cichy, Wave repetitive process approach to a class of ladder circuits, in Proceedings of 2012 IEEE International Symposium on Circuits and Systems (ISCAS) , pp. 950–953, 20–23 May 2012. [18] E. Rogers, K. Galkowski and D.H. Owens, Control Systems Theory and Applications for Linear Repetitive Processes , Lecture Notes in Control and Information Sciences, Volume 349, Springer-Verlag, Berlin, 2007. [19] T. Schwerdtfeger, K. Galkowski and A. Kummert, Stabilisation of a class of active ladder circuits based on a wave repetitive process approach by means of multidimensional wave digital filters in Proceed- ings of the 8th International Workshop on Multidimensional Systems (nDS2013), Erlangen, Germany, September 9–11, 2013.