Discrete Hamilton Jacobi Theory and Discrete Optimal Control
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1 49th IEEE Conference on Decision an Control December 15-17, 2010 Hilton Atlanta Hotel, Atlanta, GA, USA Discrete Hamilton Jacobi Theory an Discrete Optimal Control Tomoi Ohsawa, Anthony M. Bloch, an Melvin Leo Abstract We evelop a iscrete analogue of Hamilton Jacobi theory in the framewor of iscrete mechanics. The resulting iscrete Hamilton Jacobi equation is iscrete only in time. The corresponence between iscrete an continuous mechanics naturally gives rise to a iscrete analogue of Jacobi s solution to the Hamilton Jacobi equation. We prove iscrete analogues of Jacobi s solution to the Hamilton Jacobi equation an of the geometric Hamilton Jacobi theorem. These results are reaily applie to the iscrete optimal control setting, an some well-nown results in iscrete optimal control theory, such as the Bellman equation, follow immeiately. We also apply the theory to iscrete linear systems, an show that the iscrete Riccati equation follows as a special case. A. Discrete Mechanics I. INTRODUCTION Discrete mechanics is a reformulation of Lagrangian an mechanics with iscrete time, as oppose to a iscretization of the equations in the continuous-time theory. It not only provies a systematic view of structurepreserving integrators, but also has interesting theoretical aspects analogous to continuous-time Lagrangian an mechanics [see, e.g., 15; 17; 18]. The main feature of iscrete mechanics is its use of iscrete versions of variational principles. Namely, iscrete mechanics assumes that the ynamics is efine at iscrete times from the outset, formulates a iscrete variational principle for such ynamics, an then erives a iscrete analogue of the Euler Lagrange or Hamilton s equations from it. The avantage of this construction is that it naturally gives rise to iscrete analogues of the concepts an ieas in continuous time that have the same or similar properties, such as symplectic forms, the Legenre transformation, momentum maps, an Noether s theorem [15]. Whereas the main topic in iscrete mechanics is the evelopment of structurepreserving algorithms for Lagrangian an systems [see, e.g., 15], the theoretical aspects of it are interesting in their own right, an furthermore provie insight into the numerical aspects as well. Another notable feature of iscrete mechanics is that it is a generalization of nonsingular) iscrete optimal control problems. In fact, as state in Marsen an West [15], iscrete mechanics is inspire by iscrete formulations of This wor was partially supporte by NSF grants DMS , DMS , DMS , an DMS M. Leo an T. Ohsawa are with Department of Mathematics, University of California, San Diego, 9500 Gilman Drive, La Jolla, California , mleo@math.ucs.eu, tohsawa@ucs.eu A.M. Bloch is with Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, Michigan , abloch@umich.eu optimal control problems see, e.g., Joran an Pola [10] an Cazow [5]). B. Hamilton Jacobi Theory In classical mechanics [see, e.g., 3; 8; 13; 14], the Hamilton Jacobi equation is first introuce as a partial ifferential equation satisfie by the action integral. Specifically, let Q be a configuration space an T Q be its cotangent bunle, an suppose that ˆqs), ˆps)) T Q is a solution of Hamilton s equations q = H p, ṗ = H q, where H : T Q R is the of the system. Then calculate the action integral along the solution starting from s = 0 an ening at s = t with t > 0: t [ Sq, t) := ˆps) ˆqs) ] Hˆqs), ˆps)) s, 1) 0 where q := ˆqt) an we regar the resulting integral as a function of the enpoint q, t) Q R +, where R + is the set of positive real numbers. Then by taing a variation of the enpoint q, t), one obtains a partial ifferential equation satisfie by Sq, t): S t + H q, S ) = 0. 2) q This is the Hamilton Jacobi H J) equation. Conversely, it is shown that if Sq, t) is a solution of the H J equation then Sq, t) is a generating function for the family of canonical transformations or symplectic flow) that escribe the ynamics efine by Hamilton s equations. This result is the theoretical basis for the powerful technique of exact integration calle separation of variables. The iea of H J theory is also useful in optimal control theory [see, e.g., 11]. Namely, the Hamilton Jacobi equation turns into the Hamilton Jacobi Bellman HJB) equation, which is a partial ifferential equation satisfie by the optimal cost function. It is also shown that the costate of the optimal solution is relate to the solution of the HJB equation. C. Discrete Hamilton Jacobi Theory The main objective of this paper is to present a iscrete analogue of H J theory within the framewor of iscrete mechanics [12]. There are some previous wors on iscrete-time analogues of the H J equation, such as Elnatanov an Schiff [6] an Lall an West [12]. Specifically, Elnatanov an Schiff [6] erive an equation for a generating function of /10/$ IEEE 5438
2 a coorinate transformation that trivializes the ynamics. This erivation is a iscrete analogue of the conventional erivation of the continuous-time H J equation [see, e.g., 13, Chapter VIII]. Lall an West [12] formulate a iscrete Lagrangian analogue of the H J equation as a separable optimization problem. D. Main Results Our wor was inspire by the result of Elnatanov an Schiff [6], an further extens the result by eveloping iscrete analogues of results in continuous-time) H J theory. Namely, we formulate a iscrete analogue of Jacobi s solution, which relates the iscrete action sum see Eq. 3) below) with a solution of the iscrete H J equation. Another important result in this paper is a iscrete analogue of the H J theorem, which relates the solution of the iscrete H J equation with the solution of the iscrete Hamilton s equations. We also show that the iscrete H J equation is a generalization of the iscrete Riccati equation an the Bellman equation iscrete HJB equation). See Fig. 1.) Specifically, we establish a lin with iscrete-time optimal control theory, an show that the Bellman equation of ynamic programming follows. This lin maes it possible to interpret iscrete analogues of Jacobi s solution an the H J theorem in the optimal control setting. Namely, we show that these results reuce to two well-nown results in optimal control theory that relate the Bellman equation with the optimal solution. We also show that the iscrete H J equation applie to linear iscrete systems reuces to the iscrete Riccati equation. This is again a iscrete analogue of the well-nown result that the H J equation applie to linear systems reuces to the Riccati equation [see, e.g., 11, p. 421]. Quaratic Discrete Linear Hamilton s Eq. Specific Choice of Matrix Discrete Hamilton s Eq. Linear Quaratic Regulator Control Discrete Optimal Control Linear System & Quaratic Cost Function Quaratic Discrete Riccati Eq. Specific Choice of Matrix Discrete Hamilton Jacobi Eq. Discrete Riccati Eq. for LQR Control Bellman Eq. Linear System & Quaratic Cost Function Fig. 1. Discrete evolution equations left) an corresponing iscrete H J-type equations right). Dashe lines are the lins establishe in the paper. E. Outline We first present a brief review of iscrete Lagrangian an mechanics in Section II. In Section III we escribe a iscrete analogue of Jacobi s solution to the iscrete H J equation, an also iscuss the left an right variants an more explicit forms of the iscrete H J equation. In Section IV we prove a iscrete version of the H J theorem. Section V establishes the lin with iscrete-time optimal control an interprets the results of the preceing sections in this setting. In Section VI we apply the theory to linear iscrete systems, an show that the iscrete Riccati equation follows from the iscrete H J equation. II. DISCRETE MECHANICS This section briefly reviews some ey results of iscrete mechanics following Marsen an West [15] an Lall an West [12]. A. Discrete Lagrangian Mechanics A iscrete Lagrangian flow {q } for = 0, 1,..., N on an n-imensional ifferentiable manifol Q can be escribe base on the following iscrete variational principle. Let S N be the following action sum of the iscrete Lagrangian L : Q Q R: S N {q } N =0) := N 1 =0 L q, q +1 ) tn 0 Lqt), qt)) t, 3) where L : T Q R is the Lagrangian of the corresponing continuous system. Consier iscrete variations q q + ɛδq for = 0, 1,..., N with δq 0 = δq N = 0. Then the iscrete variational principle δs N = 0 gives the iscrete Euler Lagrange equations: D 2 L q 1, q ) + D 1 L q, q +1 ) = 0, 4) where D i stans for the partial erivative with respect to the variables) in the i-th slot. This etermines the iscrete flow F L : Q Q Q Q: F L : q 1, q ) q, q +1 ), 5) B. Discrete Mechanics Lall an West [12] introuce iscrete mechanics in the following way: Introuce the right an left iscrete Legenre transforms FL ± : Q Q T Q by FL + : q, q +1 ) q +1, D 2 L q, q +1 )), FL : q, q +1 ) q, D 1 L q, q +1 )). With the iscrete Legenre transform 6a) 6b) p +1 = FL + q, q +1 ) = D 2 L q, q +1 ), 7) we can efine the following right iscrete : H + q, p +1 ) = p +1 q +1 L q, q +1 ). 8) Then the iscrete map F L : q, p ) q +1, p +1 ) is efine implicitly by the right iscrete Hamilton s equations q +1 = D 2 H + q, p +1 ), p = D 1 H + q, p +1 ). Similarly, with the iscrete Legenre transform 9a) 9b) p = FL q, q +1 ) = D 1 L q, q +1 ), 10) we can efine the following left iscrete : H p, q +1 ) = p q L q, q +1 ). 11) Then we have the left iscrete Hamilton s equations q = D 1 H p, q +1 ), p +1 = D 2 H p, q +1 ). 12a) 12b) 5439
3 III. DISCRETE HAMILTON JACOBI EQUATION A. Discrete Analogue of Jacobi s Solution This section presents a iscrete analogue of Jacobi s solution. This also gives an alternative erivation of the iscrete H J equation that is much simpler than that of Elnatanov an Schiff [6]. Theorem 1 Consier the action sums Eq. 3) written in terms of the right iscrete, Eq. 8): 1 Sq [ ) := pl+1 q l+1 H + q l, p l+1 ) ] 13) evaluate along a solution of the right iscrete Hamilton s equations 9); each S q ) is seen as a function of the en point coorinates q an the iscrete en time. Then these action sums satisfy the right iscrete H J equation q +1 ) Sq ) D q +1 ) q +1 + H + q, D q +1 ) ) = 0, 14) where DS q ) = S / q1,..., S / qn ). Proof: From Eq. 13), we have q +1 ) S q ) = p +1 q +1 H + q, p +1 ), 15) where p +1 is consiere to be a function of q an q +1, i.e., p +1 = p +1 q, q +1 ). Taing the erivative of both sies with respect to q +1 an using Eq. 12a), we have p +1 = D q +1 ). 16) Substituting this into Eq. 15) gives Eq. 14). Remar 2 Recall that, in the erivation of the continuous H J equation [see, e.g., 7, Section 23], we consier the variation of the action integral Eq. 1) with respect to the en point q, t) an fin This gives S t S = p q Hq, p) t. 17) = Hq, p), p = S q, 18) an hence the H J equation 2). Table I summarizes the corresponence between the ingreients in the continuous an iscrete theories see also Remar 2). B. The Right an Left Discrete H J Equations We can also write the action sum Eq. 3) in terms of the left iscrete, Eq. 11), as follows: 1 Sq [ ) = pl q l H p l, q l+1 ) ]. 19) Then we can procee as in the proof of Theorem 1 see Ohsawa et al. [16] for etails) to obtain the left iscrete H J equation: q +1 ) Sq ) + DSq ) q + H ) DS q ), q +1 = 0. 20) TABLE I CORRESPONDENCE BETWEEN INGREDIENTS IN CONTINUOUS AND DISCRETE THEORIES; R 0 IS THE SET OF NON-NEGATIVE REAL NUMBERS AND N 0 IS THE SET OF NON-NEGATIVE INTEGERS. Continuous Discrete q, t) Q R 0 q, ) Q N 0 Sq, t) := q = H/ p, q +1 = D 2H + q, p +1 ), ṗ = H/ q p = D 1H + q, p +1 ) t 0 p q H s S = S S q + q t t Sq ) := 1 [ pl+1 q l+1 H + q l, p l+1 ) ] S+1 q +1 ) S q ) p q Hq, p) t p +1 q +1 H + q, p +1 ) S t + H q, S ) = 0 q q +1 ) S q ) D q +1 ) q +1 ) + H + q, D q +1 ) = 0 As mentione above, Eqs. 13) an 19) are the same action sum Eq.3) expresse in ifferent ways. Therefore we may summarize the above argument as follows: Proposition 3 The action sums, Eq. 13) or equivalently Eq. 19), satisfy both the right an left iscrete H J equations 14) an 20). C. Explicit Forms of the Discrete H J Equations The expressions for the right an left iscrete H J equations in Eqs. 14) an 20) are implicit in the sense that they contain two spatial variables q an q +1. However, Theorem 1 suggests that q an q +1 may be consiere to be relate by the ynamics efine by either Eq. 9) or 12). More specifically, we may write q +1 in terms of q. This results in explicit forms of the iscrete H J equations, an we shall efine the iscrete H J equations by the resulting explicit forms. For the right iscrete H J equation 14), we first efine the map f + : Q Q as follows: Replace p +1 in Eq. 9a) by D q +1 ) as suggeste by Eq. 16): q +1 = D 2 H + q, D q +1 ) ). 21) Assuming this equation is solvable for q +1, we efine f + : Q Q by the resulting q +1, i.e., f + is implicitly efine by f + q ) = D 2 H + q, D f + q )) ). 22) We may now ientify q +1 with f + q ) in the implicit form of the right H J equation 14): f + q)) S q) D f + q)) f + q) + H + q, DS +1 f + q))) = 0, 23) 5440
4 where we suppresse the subscript of q since it is now clear that q is an inepenent variable as oppose to a function of the iscrete time. We efine Eq. 23) to be the right iscrete H J equation. Notice that these are ifferentialifference equations efine on Q N, with the spatial variable q an the iscrete time. For the left iscrete H J equation 20), we efine the map : Q Q as follows: f f q ) := π Q F L S q ) ), 24) where π Q : T Q Q is the cotangent bunle projection; equivalently, f is efine so that the iagram below commutes. S T Q Q F L f T Q Q π Q S q ) FL S q ) ) q f q ) 25) Notice also that, since the map F L : q, p ) q +1, p +1 ) is efine by Eq. 12), f is efine implicitly by q = D 1 H DS q ), f q ) ). 26) In other wors, replace p in Eq. 12a) by DS q ), an efine f q ) as the q +1 in the resulting equation. We may now ientify q +1 with f q ) in Eq. 20): f q)) S q) + DSq) q + H DS q), f q)) = 0, 27) where we again suppresse the subscript of q. We efine Eqs. 23) an 27) to be the right an left iscrete H J equations, respectively. Notice that these are ifferentialifference equations efine on Q N, with the spatial variable q an the iscrete time. D. Discrete H J Equation an Generating Functions Assuming the uniqueness of the solution of the iscrete H J equation, Theorem 1 tells us that, the solution S n q) is the action integral written in terms of the en time an en point, which is the generating function of the the ynamics q 0, p 0 ) q n, p n ). If we construct FS n from S n using the corresponence 1 L F L, then F S n gives the map q 0, p 0 ) q n, p n ), i.e., we have F S n = F L F L, 28) }{{} n In other wors, a solution of the iscrete H J equation generates the n-step ynamics of the corresponing iscrete Lagrangian/ system see also Fig. 2). Remar 4 Note that Eq. 28) hols exactly. This property is not guarantee for those solutions obtaine by irect iscretizations of the H J equation. 1 Recall that a iscrete Lagrangian is nothing but a generating function. See [15] an [12]. F S n q 0, p 0 ) q 1, p 1 ) q 2, p 2 ) q n 1, p n 1 ) q n, p n ) F L FL FL Fig. 2. The generating function S n generates the flow efine by the n-fol composition F L F L. IV. DISCRETE HAMILTON JACOBI THEOREM The following gives a iscrete analogue of the geometric H J theorem Theorem 5.2.4) by Abraham an Marsen [1]: Theorem 5 Discrete Hamilton Jacobi) Suppose that S satisfies the right iscrete H J equation 23), an let {c } N =0 Q be a set of points such that c +1 = f + c ) for = 0, 1,..., N 1. 29) Then the set of points {c, p )} N =0 T Q with p := DS c ) 30) is a solution of the right iscrete Hamilton s equations 9). Similarly, suppose that S satisfies the left iscrete H J equation 27), an let {c } N =0 Q be a set of points that satisfy c +1 = f c ) for = 0, 1,..., N 1. 31) Furthermore, assume that the Jacobian Df is invertible at each point c. Then the set of points {c, p )} N =0 T Q with p := DSc ) 32) is a solution of the left iscrete Hamilton s equations 12). Proof: See Ohsawa et al. [16]. V. RELATION TO THE DISCRETE-TIME HJB EQUATION In this section we apply the above results to the optimal control setting. We will show that the right) iscrete H J equation 23) gives the Bellman equation iscrete-time HJB equation) as a special case. A. Discrete Optimal Control Problem Let {q } N =0 be the state variables in a vector space V = R n with q 0 an q N fixe an u := {u } N =0 be controls in the set U R m. With a given function C : V U R, efine the cost functional J := N 1 =0 C q, u ). 33) Then a typical iscrete optimal control problem is formulate as follows [see, e.g., 4; 5; 9; 10]: Problem 6 Minimize the cost functional, i.e., min u N 1 J = min u =0 C q, u ) 34) 5441
5 subject to the constraint q +1 = fq, u ). 35) B. Necessary Conition for Optimality an the Discrete- Time HJB Equation We woul lie to formulate the necessary conition for optimality. First introuce the augmente cost functional: 1 Ŝq, p, u ) := {C q l, u l ) + p l+1 [q l+1 fq l, u l )]} 1 [ ] = p l+1 q l+1 Ĥ+ q l, p l+1, u l ), where we efine the Ĥ + q l, p l+1, u l ) := p l+1 fq l, u l ) C q l, u l ) 36) an the shorthan notation q := {q l }, p := {p l } l=1, an u := {u l } 1. Then the optimality conition Eq. 34) is restate as min q,p,u Ĵ q, p, u ) or equivalently min Ŝ q,p,u q, p, u ). 37) In particular, extremality with respect to the control u implies D 3 Ĥ + q l, p l+1, u l ) = 0, l = 0, 1,..., 1. 38) Now we assume that Ĥ + is sufficiently regular so that the optimal control u := {u l } 1 is etermine by D 3 Ĥ + q l, p l+1, u l ) = 0, l = 0, 1,..., 1. 39) Therefore u l is a function of q l an p l+1, i.e., u l = u l q l, p l+1 ). Then we can eliminate u in the minimization problem Eq. 37): min S q, p ) = min q,p q,p 1 [ pl+1 q l+1 H + q l, p l+1 ) ], 40) where we efine H + q l, p l+1 ) := Ĥ+ q l, p l+1, u l ) an S q, p ) := Ŝ q, p, u ). So now the problem is reuce to minimizing an action sum that has exactly the same form as the one in Eq. 13) formulate in the framewor of iscrete mechanics. The corresponing right iscrete Hamilton s equations are q +1 = fq, u ), p = p +1 D 1 fq, u ) D 1 C q, u ). 41) Therefore Eq. 22) gives the implicit efinition of f + as follows: f + q ) = f q, u q, D f + q )) )). 42) Hence the right) iscrete H J equation 23) applie to this case gives fq, u )) S q ) C q, u ) = 0, 43) or equivalently min u [ S +1 fq, u )) C q, u ) ] S q ) = 0, 44) which is the Bellman equation [see, e.g., 4]. C. Relation between the Discrete H J an HJB Equations an its Consequences Summarizing the observation mae above, we have Proposition 7 The right iscrete H J equation 23) applie to the formulation of the iscrete optimal control problem 6 gives the Bellman equation 44). Reinterpreting Theorems 1 an 5 in terms of this observation leas to the following well-nown facts: Proposition 8 The optimal cost function satisfies the Bellman equation 44). Proposition 9 Let S q ) be a solution to the Bellman equation 44). Then the costate p in the iscrete maximum principle is given as follows: p = DS c ), 45) where c +1 = fc, u ) with the optimal control u. VI. APPLICATION TO DISCRETE LINEAR HAMILTONIAN SYSTEMS A. Discrete Linear Systems an the Matrix Riccati Equation Example 10 Discrete linear systems) Consier a iscrete system on T R n = R n R n the configuration space is Q = R n ) efine by the quaratic left iscrete H p, q +1 ) = 1 2 pt M 1 p + p T Lq qt +1Kq +1, 46) where M, K, an L are real n n matrices; we assume that M an L are invertible an also that M an K are symmetric. The left iscrete Hamilton s equations 12) become ) ) ) q+1 L 1 L = 1 M 1 q KL 1 KL 1 M 1 L T. 47) p +1 Now let us solve the left iscrete H J equation 27) for this system. It is possible to show see Ohsawa et al. [16]) that, for this particular case, the solution S taes the form p S q) = 1 2 qt A q + b T q + c 48) where A are symmetric n n matrices, b are elements in R n, an c are in R. We substitute the above expression into the iscrete H J equation to fin the equations for A, b, an c. Notice first that the map f is given by the first half of Eq. 47) with p replace by DS q): f q) = L 1 I + M 1 A )q L 1 M 1 b. 49) Then substituting Eq. 48) into the left-han sie of the left iscrete H J equation 27) yiels the following recurrence 5442
6 relations for A, b, an c : A +1 = L T I + A M 1 ) 1 A L K, b +1 = L T I + A M 1 ) 1 b, c +1 = c 1 2 bt M + A ) 1 b, where we assume that I + A M 1 is invertible. 50a) 50b) 50c) Remar 11 For the A +1 efine by Eq. 50a) to be symmetric, it is sufficient that A is invertible; for if it is, then Eq. 50a) becomes A +1 = L T A 1 + M 1 ) 1 L K, where A, M, an K are symmetric. Remar 12 We can rewrite Eq. 50a) as follows: A +1 = [ KL 1 + KL 1 M 1 L T )A ] L 1 L 1 M 1 A ) 1. 51) Notice the exact corresponence between the coefficients in the above equation an the matrix entries in the iscrete linear equations 47). In fact, this is the iscrete Riccati equation that correspons to the iteration efine by Eq. 47). See Ammar an Martin [2] for etails on this corresponence. To summarize the above observation, we have Proposition 13 The iscrete H J equation 27) applie to the iscrete linear system 47) yiels the iscrete Riccati equation 51). In other wors, the iscrete H J equation is a nonlinear generalization of the iscrete Riccati equation. VII. CONCLUSION AND FUTURE WORK We evelope a iscrete-time analogue of the H J theory starting from the iscrete variational Hamilton equations formulate by Lall an West [12]. We showe that it possesses theoretical significance in iscrete mechanics that is equivalent to that of the continuous-time) H J equation in mechanics. Furthermore, we showe that the iscrete H J equation specializes to the Bellman equation if applie to iscrete optimal control problems, an also that it reuces to the iscrete Riccati equation with a quaratic. This again gives iscrete analogues of the corresponing nown results in the continuous-time theory. Application to iscrete optimal control also reveale that Theorems 1 an 5 specialize to two well-nown results in iscrete optimal control theory. We are intereste in the following topics for future wor: i) Application to integrable iscrete systems; ii) Development of numerical methos base on the iscrete H J equation; iii) Extension to iscrete nonholonomic an Dirac mechanics; iv) Relation to the power metho an iterations on the Grassmannian manifol. ACKNOWLEDGMENTS We woul lie to than Jerrol Marsen, Harris McClamroch, Matthew West, Dmitry Zenov, an Jingjing Zhang for helpful iscussions an comments. REFERENCES [1] R. Abraham an J. E. Marsen. Founations of Mechanics. Aison Wesley, 2n eition, [2] G. Ammar an C. Martin. The geometry of matrix eigenvalue methos. Acta Applicanae Mathematicae, 53): , [3] V. I. Arnol. Mathematical Methos of Classical Mechanics. Springer, [4] R. Bellman. Introuction to the Mathematical Theory of Control Processes, volume 2. Acaemic Press, [5] J. A. Cazow. Discrete calculus of variations. International Journal of Control, 113): , [6] N. A. Elnatanov an J. Schiff. The Hamilton Jacobi ifference equation. Functional Differential Equations, ), [7] I. M. Gelfan an S. V. Fomin. Calculus of Variations. Dover, [8] H. Golstein, C. P. Poole, an J. L. Safo. Classical Mechanics. Aison Wesley, 3r eition, [9] V. Guibout an A. M. Bloch. A iscrete maximum principle for solving optimal control problems. In 43r IEEE Conference on Decision an Control, volume 2, pages Vol.2, [10] B. W. Joran an E. Pola. Theory of a class of iscrete optimal control systems. Journal of Electronics an Control, 17: , [11] V. Jurjevic. Geometric control theory. Cambrige University Press, Cambrige, [12] S. Lall an M. West. Discrete variational mechanics. Journal of Physics A: Mathematical an General, 3919): , [13] C. Lanczos. The Variational Principles of Mechanics. Dover, 4th eition, [14] J. E. Marsen an T. S. Ratiu. Introuction to Mechanics an Symmetry. Springer, [15] J. E. Marsen an M. West. Discrete mechanics an variational integrators. Acta Numerica, pages , [16] T. Ohsawa, A. M. Bloch, an M. Leo. Discrete Hamilton Jacobi theory. Preprint arxiv: ). [17] Y. B. Suris. The problem of integrable iscretization: approach. Birhäuser, Basel, [18] Y. B. Suris. Discrete Lagrangian moels. In Discrete Integrable Systems, volume 644 of Lecture Notes in Physics, pages Springer,
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SIAM J. CONTROL OPTIM. Vol. 49, No. 4, pp. 1829 1856 c 211 Society for Inustrial an Applie Mathematics DISCRETE HAMILTON JACOBI THEORY TOMOKI OHSAWA, ANTHONY M. BLOCH, AND MELVIN LEOK Abstract. We evelop
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