非线性动力学系统一般形式及其广义哈密顿体系下的几何积分方法
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摘要
几何积分方法无论在提高计算精度还是在保持系统的不变量性质等方面都比传统的积分算法有优势,同时,它还具有向后误差分析的性质,可用于研究数值方法的长期行为,以及进行数值方法的稳定性分析。本文主要研究了广义Hamilton系统及一般非线性动力学系统的几何积分方法。
首先,提出了求解一般动力学方程的李级数方法,并给出具体实施办法,它是泰勒展开方法的一个推广。另一方面将动力学微分方程用微分算子的形式表示之后,它的解算子可由它的无穷小生成元的预解式取Laplace逆变换得到,如此再次得到了李级数方法,对于自治系统它是一个李群方法。另外,提出了基于Laplace变换数值反演的非线性动力学方程的求解方法。
其次,基于李级数方法,提出了广义Hamilton系统及耗散广义Hamilton系统的李群积分法。广义Hamilton系统形式是动力学系统的一种恰当表述,它揭示了力学系统内蕴的某种对称性质,它的理论研究和实际应用在力学研究中具有十分重要的意义。本文在守恒系统解析解的理论基础上给出了构造广义Hamilton系统任意高阶显式保群积分格式的方法,同时讨论了算法的具体实施过程。对耗散广义Hamilton系统,就自治与非自治系统分别进行了讨论:对于自治系统,采用李级数方法并结合分裂合成的技巧直接进行求解;对于非自治系统,基于Magnus级数方法和Fer展开方法来构造其数值解。文中方法保持了原系统真解的典则性,因而也是稳定的。如果更关注系统的能量性质,如Hamilton函数性质,文中用离散梯度的方法给出了广义Hamilton系统及广义Hamilton控制系统的保持其Hamilton函数性质特征不变的数值解法。
同时,本文在伪Poisson流形上研究了广义Hamilton约束系统的求解问题。把广义Hamilton约束系统变形为无约束的广义Hamilton系统微分方程,提出了保持系统内在结构和约束不变性的李群积分方法,并就约束不变量的误差和稳定性等问题进行了理论分析和数值分析。另外通过引入拉格朗日乘子采用投影技术对广义Hamilton约束系统直接进行积分,进一步简化了积分过程。因为本文的讨论对完整与非完整约束不加区分,一样处理,所以也适用于非完整约束的情形。
然而,一般非线性动力学系统并不是都可以表示为(耗散)广义Hamilton系统的形式,即存在所谓广义Hamilton实现问题。为此,基于经典的Magnus和Fer展开式,在耗散广义Hamilton系统的保结构算法的基础上,主要从两个不同的角度,进一步深入地研究了一般非线性动力学系统的李群积分方法:一个是在算
Geometric integration has advantages over classical integration either in higher precision or in preservation of the invariant of the systems. And it has backward error analysis property that can be used to analyze the long time behavior and the stability of the numerical method. In this paper the geometric integration methods for solving generalized Hamiltonian systems and furthermore for solving general nonlinear dynamic system are investigated.Firstly, Lie series method to solve general nonlinear dynamic system is presented, and the implementation method is given in detail. It is an expansion of the Talor series method. In the other hand, if dynamic differential equation is expressed in the form of linear differential operator action, its solver can be obtained by inversing the Laplace transform of the resolvent of its infinitesimal generator (differential operator). In such a way the Lie series method is again obtained. The Lie series method is a Lie group method for an autonomous dynamic system. In addition, another numerical solution for solving nonlinear dynamic equations is presented based on numerical Laplace transform inversion.Secondly, based on Lie series method, Lie group methods for solving generalized Hamiltonian systems and generalized Hamiltonian systems with dissipation are presented. Generalized Hamiltonian formalism is a proper description of dynamic system; it reveals some intrinsic symmetric properties of the dynamic systems. So it is much more important to study generalized Hamiltonian systems theoretically and practically. In this paper, arbitrary high-order explicit integration methods for solving generalized Hamiltonian systems are developed based on the theory of the analytical solution of conservation systems, and the implementation of algorithms is discussed in detail. Then the generalized Hamiltonian systems with dissipation are differentiated between autonomous and non-autonomous systems for the convenience of the study. Autonomous systems are solved by using Lie series method and operator-splitting and composition method. For solving non-autonomous systems, the numerical methods are developed based on the Magnus series method and Fer expansion method. The methods in this paper preserve the canonical property of the exact solution of the original systems, so it is stable. Sometimes, the energy property of the systems, for
example, the Hamiltonian property, is more important. In this case, using discrete gradient, the numerical methods are given to solve generalized Hamiltonian systems and generalized Hamiltonian control systems, which preserve the Hamiltonian property.Furthermore, generalized Hamiltonian systems with constraints are studied on the pseudo-Poisson manifold. Generalized Hamiltonian systems with constraints are converted into unconstrained generalized Hamiltonian systems of ordinary differential equations. And some Lie group integration methods are given so as to preserve the intrinsic construction and constraint invariants. In the meantime, the stability and the error analysis of the constrain-invariants along solutions are investigated numerically and analytically. Furthermore the integrations are simplified by introducing Lagrange multipliers and using projection techniques. Because the discussion does not differentiate holonomic constraints and non-holonomic constraints, the methods in this paper can be used to solve dynamic systems with non-holonomic constraints.However, general nonlinear dynamic system cannot always be expressed as a generalized Hamiltonian system (with dissipation). In other ward, there exist the generalized Hamiltonian realization problems. Based on the classical Magnus or Fer expansion and Lie group integration methods for generalized Hamiltonian systems with dissipation, Lie group integrations for solving general nonlinear dynamic systems are studied in a deep-going way in two different points of view: one is to discuss the question in the form of operator action, and the nonlinear dynamic equation can be expressed as model of linear map action, then new algorithm can be designed based on the theory of linear differential equation; the other is to expand the configuration space to the Minkowski space, where the original nonlinear dynamic system is converted into an augmented dynamic system of Lie type and it is convenient to design the algorithm and program. Among them, the integration methods based on Magnus expansion involve a large numbers of commutators. Taking advantage of the time-symmetry property of Magnus series and using the technique of Lie series expansion, a minimum of commutators are involved in the algorithms. Approximate schemes of 4-th, 6-th and 8-th order are constructed which involve only 1, 4 and 10 different commutators, and the time-symmetry properties of the schemes are proved. It should be point out that these methods are much more accurate and effective for solving autonomous dynamic systems. A second-order method can achieve computation precision up to 4-th order. And numerical examples show that large step size can be used in calculation. In the
meantime, the integration schemes of supper convergence based on Fer expansion are also attained. Then the Fer expansion methods are connected with Magnus expansion methods effectively and some techniques are given to simplify the construction of Fer expansion methods. Furthermore time-symmetric integrators of Fer type are constructed.Runge-Kutta/Munthe-Kaas (RKMK) methods are expanded Runge-Kutta methods for solving differential equation whose configuration space is on a Lie group. The differential equation evolving on a Lie group was converted into an equivalent differential equation evolving on the corresponding Lie algebra. The numerical solution of the equivalent differential equation can be obtained by using Runge-Kutta methods, then it can be pulled back to the original Lie group using exponential maps, and the numerical solution of original differential equation can be obtained. Exponential matrix is a mapping from the Lie algebra of matrix to the Lie group of matrix. Based on RKMK methods, a series of simple and effective integration methods of RKMK type are presented by combining Runge-Kutta methods with precise integration method in order to solve the nonlinear dynamic system and its augmented dynamic system in the Minkowski space. These methods are all Lie group methods.The numerical methods based on Magnus and Fer expansion involve a large number of exponential matrices too. The computation precision of the exponential matrices influences the accuracy of the schemes directly. Precise integration method can compute the exponential matrix quickly and effectively. In this paper, precise integration method for solving linear dynamic systems are expanded to solve general nonlinear dynamic systems. In the Minkowski space the nonlinear dynamic system is converted into an augmented dynamic system and it is very convenient to use the precise integration method. In this case, precise integration method is a Lie group method for autonomous dynamic systems. Additionally, precise integration method is ingenuously applied to solving nonlinear dynamic systems by means of expanding the dimension of the configuration space or combining the Runge-Kutta methods, which avoids the numerically non-stability of the matrix inversion and the non-existence of the inverse matrix.
首先,提出了求解一般动力学方程的李级数方法,并给出具体实施办法,它是泰勒展开方法的一个推广。另一方面将动力学微分方程用微分算子的形式表示之后,它的解算子可由它的无穷小生成元的预解式取Laplace逆变换得到,如此再次得到了李级数方法,对于自治系统它是一个李群方法。另外,提出了基于Laplace变换数值反演的非线性动力学方程的求解方法。
其次,基于李级数方法,提出了广义Hamilton系统及耗散广义Hamilton系统的李群积分法。广义Hamilton系统形式是动力学系统的一种恰当表述,它揭示了力学系统内蕴的某种对称性质,它的理论研究和实际应用在力学研究中具有十分重要的意义。本文在守恒系统解析解的理论基础上给出了构造广义Hamilton系统任意高阶显式保群积分格式的方法,同时讨论了算法的具体实施过程。对耗散广义Hamilton系统,就自治与非自治系统分别进行了讨论:对于自治系统,采用李级数方法并结合分裂合成的技巧直接进行求解;对于非自治系统,基于Magnus级数方法和Fer展开方法来构造其数值解。文中方法保持了原系统真解的典则性,因而也是稳定的。如果更关注系统的能量性质,如Hamilton函数性质,文中用离散梯度的方法给出了广义Hamilton系统及广义Hamilton控制系统的保持其Hamilton函数性质特征不变的数值解法。
同时,本文在伪Poisson流形上研究了广义Hamilton约束系统的求解问题。把广义Hamilton约束系统变形为无约束的广义Hamilton系统微分方程,提出了保持系统内在结构和约束不变性的李群积分方法,并就约束不变量的误差和稳定性等问题进行了理论分析和数值分析。另外通过引入拉格朗日乘子采用投影技术对广义Hamilton约束系统直接进行积分,进一步简化了积分过程。因为本文的讨论对完整与非完整约束不加区分,一样处理,所以也适用于非完整约束的情形。
然而,一般非线性动力学系统并不是都可以表示为(耗散)广义Hamilton系统的形式,即存在所谓广义Hamilton实现问题。为此,基于经典的Magnus和Fer展开式,在耗散广义Hamilton系统的保结构算法的基础上,主要从两个不同的角度,进一步深入地研究了一般非线性动力学系统的李群积分方法:一个是在算
Geometric integration has advantages over classical integration either in higher precision or in preservation of the invariant of the systems. And it has backward error analysis property that can be used to analyze the long time behavior and the stability of the numerical method. In this paper the geometric integration methods for solving generalized Hamiltonian systems and furthermore for solving general nonlinear dynamic system are investigated.Firstly, Lie series method to solve general nonlinear dynamic system is presented, and the implementation method is given in detail. It is an expansion of the Talor series method. In the other hand, if dynamic differential equation is expressed in the form of linear differential operator action, its solver can be obtained by inversing the Laplace transform of the resolvent of its infinitesimal generator (differential operator). In such a way the Lie series method is again obtained. The Lie series method is a Lie group method for an autonomous dynamic system. In addition, another numerical solution for solving nonlinear dynamic equations is presented based on numerical Laplace transform inversion.Secondly, based on Lie series method, Lie group methods for solving generalized Hamiltonian systems and generalized Hamiltonian systems with dissipation are presented. Generalized Hamiltonian formalism is a proper description of dynamic system; it reveals some intrinsic symmetric properties of the dynamic systems. So it is much more important to study generalized Hamiltonian systems theoretically and practically. In this paper, arbitrary high-order explicit integration methods for solving generalized Hamiltonian systems are developed based on the theory of the analytical solution of conservation systems, and the implementation of algorithms is discussed in detail. Then the generalized Hamiltonian systems with dissipation are differentiated between autonomous and non-autonomous systems for the convenience of the study. Autonomous systems are solved by using Lie series method and operator-splitting and composition method. For solving non-autonomous systems, the numerical methods are developed based on the Magnus series method and Fer expansion method. The methods in this paper preserve the canonical property of the exact solution of the original systems, so it is stable. Sometimes, the energy property of the systems, for
example, the Hamiltonian property, is more important. In this case, using discrete gradient, the numerical methods are given to solve generalized Hamiltonian systems and generalized Hamiltonian control systems, which preserve the Hamiltonian property.Furthermore, generalized Hamiltonian systems with constraints are studied on the pseudo-Poisson manifold. Generalized Hamiltonian systems with constraints are converted into unconstrained generalized Hamiltonian systems of ordinary differential equations. And some Lie group integration methods are given so as to preserve the intrinsic construction and constraint invariants. In the meantime, the stability and the error analysis of the constrain-invariants along solutions are investigated numerically and analytically. Furthermore the integrations are simplified by introducing Lagrange multipliers and using projection techniques. Because the discussion does not differentiate holonomic constraints and non-holonomic constraints, the methods in this paper can be used to solve dynamic systems with non-holonomic constraints.However, general nonlinear dynamic system cannot always be expressed as a generalized Hamiltonian system (with dissipation). In other ward, there exist the generalized Hamiltonian realization problems. Based on the classical Magnus or Fer expansion and Lie group integration methods for generalized Hamiltonian systems with dissipation, Lie group integrations for solving general nonlinear dynamic systems are studied in a deep-going way in two different points of view: one is to discuss the question in the form of operator action, and the nonlinear dynamic equation can be expressed as model of linear map action, then new algorithm can be designed based on the theory of linear differential equation; the other is to expand the configuration space to the Minkowski space, where the original nonlinear dynamic system is converted into an augmented dynamic system of Lie type and it is convenient to design the algorithm and program. Among them, the integration methods based on Magnus expansion involve a large numbers of commutators. Taking advantage of the time-symmetry property of Magnus series and using the technique of Lie series expansion, a minimum of commutators are involved in the algorithms. Approximate schemes of 4-th, 6-th and 8-th order are constructed which involve only 1, 4 and 10 different commutators, and the time-symmetry properties of the schemes are proved. It should be point out that these methods are much more accurate and effective for solving autonomous dynamic systems. A second-order method can achieve computation precision up to 4-th order. And numerical examples show that large step size can be used in calculation. In the
meantime, the integration schemes of supper convergence based on Fer expansion are also attained. Then the Fer expansion methods are connected with Magnus expansion methods effectively and some techniques are given to simplify the construction of Fer expansion methods. Furthermore time-symmetric integrators of Fer type are constructed.Runge-Kutta/Munthe-Kaas (RKMK) methods are expanded Runge-Kutta methods for solving differential equation whose configuration space is on a Lie group. The differential equation evolving on a Lie group was converted into an equivalent differential equation evolving on the corresponding Lie algebra. The numerical solution of the equivalent differential equation can be obtained by using Runge-Kutta methods, then it can be pulled back to the original Lie group using exponential maps, and the numerical solution of original differential equation can be obtained. Exponential matrix is a mapping from the Lie algebra of matrix to the Lie group of matrix. Based on RKMK methods, a series of simple and effective integration methods of RKMK type are presented by combining Runge-Kutta methods with precise integration method in order to solve the nonlinear dynamic system and its augmented dynamic system in the Minkowski space. These methods are all Lie group methods.The numerical methods based on Magnus and Fer expansion involve a large number of exponential matrices too. The computation precision of the exponential matrices influences the accuracy of the schemes directly. Precise integration method can compute the exponential matrix quickly and effectively. In this paper, precise integration method for solving linear dynamic systems are expanded to solve general nonlinear dynamic systems. In the Minkowski space the nonlinear dynamic system is converted into an augmented dynamic system and it is very convenient to use the precise integration method. In this case, precise integration method is a Lie group method for autonomous dynamic systems. Additionally, precise integration method is ingenuously applied to solving nonlinear dynamic systems by means of expanding the dimension of the configuration space or combining the Runge-Kutta methods, which avoids the numerically non-stability of the matrix inversion and the non-existence of the inverse matrix.
引文
[1] A. Iserles, Multistep methods on manifolds. Technical Report 1997/NA13, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, England, 1997.
[2] R. I. McLachlan, G. R. W. Quispel and N. Robidoux Geometric integration using discrete gradients. Phil. Trans. R. Soc. Lond. A,357:1021-1045(1999).
[3] G. R. W. Quispel and G. S. Turner, Discrete gradient methods for solving ODEs numerically while preserving a first integral. J. Phys. A,29:L341-L349(1996).
[4] P. E. Crouch and R. Grossman, Numerical integration of ordinary differential equations on manifolds, J. Nonlinear Sci., 3: 1-33(1993)
[5] R. I. McLachlan On the numerical integration of ordinary differential equations by symmetric composition methods SIAMJ. Sci. Comp., 16:151-168(1995).
[6] A. Iserles, H.Z. Munthe-Kaas, S. P. Nφprsett, and A. Zanna, Lie group methods, Acta Numerica, 9:215-365(2000).
[7] F. Cacas, J. A. Oteo, and J. Ros, Lie algebraic approach to Fer's expansion for classical Hamiltonian systems, J. Phys. A: Math. Gen. 24(17): 4037-4041(1991).
[8] S. Klarsfeld, and J. A. Oteo, Exponential infinite-product representations of the time-displacement operator. J. Phys. A: Math. Gen. 22:2687-2694(1989).
[9] L. Dieci, R. D. Russell, and E. S. Van Vleck, Unitary integrators and applications to continuous othonormalization techniques. SIAM J. Numer. Anal. 31:261-281(1994).
[10] A. Iserles and S. P. Ncprsett, On the solution of linear differential equations in Lie groups. Phil. Trans. R. Soc, Lond. A 357:983-1019(1999).
[11] S. Faltinsen, Backward error analysis for Lie-group methods, BIT, 40(4): 652-670(2000)
[12] A. Iserles, Solving linear ordinary differential equations by exponentials of iterated commutators, Numer. Math., 45:183-199(1984)
[13] H. Munthe-Kaas, Runge-Kutta methods on Lie group, BIT 38(1): 92-111(1998).
[14] H. Munthe-Kaas, High order Runge-Kutta methods on manifolds, Applied Numerical Mathematics, 29(1): 115-127,1999.
[15] W. Magnus, On the exponential solution of differential equations for a linear operator, Commun. Pure Appl. Math. 7:649-673(1954).
[16] F. Fer, Resolution de l'equation matricielle U = pU par produit infini
d'exponentielles matricielles, Bull. Classes Sci. Acad. Roy. Bel., 44:818-829 (1958).
[17] S. Blanes, F. Cacas and J. Ros, Improved high order integrators based on the Magnus expansion, BIT, 40(3): 434-450(2000).
[18] H. Yoshida. Construction of higher order symplectic integrators. Physics Letters A, 150:262-268(1990).
[19] 冯康.Hamilton形式体系和辛几何的差分格式——纪念华罗庚教授.力学未来15年国际学术讨论会论文集,科学出版社,1989.
[20] Feng Kang. On Difference Schemes and Symplectic Geometry. Proc. 1984 Beijing Int. Symp. on Differential Geometry and Differential Equation--COMPUTATION OF PARTIAL DIFFERENTIAL EQUATIONS, Ed. Feng Kang, Beijing: Science Press, 42-58, (1985).
[21] 钟万勰,欧阳华江,邓子辰.计算结构力学与最优控制.大连理工大学出版社,1993.
[22] 钱学森,钱令希,谈庆明.力学——迎接21世纪挑战.力学与实践.17(4):1-5(1995).
[23] B. M. J. Maschke, R. Ortega, A. J. van der Schaft, Energy-based Lyapunov functions for forced Hamiltonian systems with dissipation. Proc. of CDC98, 3599-3604.
[24] R. Ortega, A. Stankovic, R Stefanov, A passivation approach to power systems stabilization. IFAC Symp. Nonlinear Control Systems Design, Enschede, NL, July 1-3,1998.
[25] 程代展,席在荣,卢强, 梅生伟.广义Hamilton控制系统的几何结构及其应用.中国科学(E缉),30(4):341-355(2000).
[26] 王玉振,程代展,李春文.广义Hamilton实现及其在电力系统中的应用,自动化学报,28(5):745-753(2002).
[27] S. Blanes and P. C. Moan, Splitting methods for non-autonomous Hamiltonian equations, J. Comput. Phys., 170 (1): 205-230(2001).
[28] S. Blanes, F. Cacas and J. Ros, High order optimized geometric integrators for linear differential equations, BIT, 42(2): 262-284(2002).
[29] A. Zanna, Collocation and relaxed collocation for the Fer and the Magnus expansions, SIAM J. Numer Anal., 36:1145-1182(1999).
[30] S. Klarsfeld and J. A. Oteo, Recursive generation of higher-order terms in the Magnus expansion. Phys. Rev. A, 39:3270-3273(1989).
[31] A. Zanna, The Fer expansion and time-symmetry: A Strang-type approach, Appl. Numer. Math., 39:435-459(2001).
[32] J. M. Sanz-Serna, Symplectic integrators for Hamiltonian problems: An overview, Acta. Numerica, 1: 243-286 (1992).
[33] F. Cacas, Fer's factorization as a symplectic integrator, Numer. Math, 74(3): 283-303(1996).
[34] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983.
[35] Liu, C.-S., Cone of non-linear dynamical system and group preserving schemes, InternationalJournal of Non-linear Mechanics, 36,1047-1068(2001).
[36] Hong, H.-K., Liu, C.-S., Some physical models with Minkowski spacetime structure and Lorentz group symmetry, International Journal of Non-linear Mechanics, 36,1075-1084(2001).
[2] R. I. McLachlan, G. R. W. Quispel and N. Robidoux Geometric integration using discrete gradients. Phil. Trans. R. Soc. Lond. A,357:1021-1045(1999).
[3] G. R. W. Quispel and G. S. Turner, Discrete gradient methods for solving ODEs numerically while preserving a first integral. J. Phys. A,29:L341-L349(1996).
[4] P. E. Crouch and R. Grossman, Numerical integration of ordinary differential equations on manifolds, J. Nonlinear Sci., 3: 1-33(1993)
[5] R. I. McLachlan On the numerical integration of ordinary differential equations by symmetric composition methods SIAMJ. Sci. Comp., 16:151-168(1995).
[6] A. Iserles, H.Z. Munthe-Kaas, S. P. Nφprsett, and A. Zanna, Lie group methods, Acta Numerica, 9:215-365(2000).
[7] F. Cacas, J. A. Oteo, and J. Ros, Lie algebraic approach to Fer's expansion for classical Hamiltonian systems, J. Phys. A: Math. Gen. 24(17): 4037-4041(1991).
[8] S. Klarsfeld, and J. A. Oteo, Exponential infinite-product representations of the time-displacement operator. J. Phys. A: Math. Gen. 22:2687-2694(1989).
[9] L. Dieci, R. D. Russell, and E. S. Van Vleck, Unitary integrators and applications to continuous othonormalization techniques. SIAM J. Numer. Anal. 31:261-281(1994).
[10] A. Iserles and S. P. Ncprsett, On the solution of linear differential equations in Lie groups. Phil. Trans. R. Soc, Lond. A 357:983-1019(1999).
[11] S. Faltinsen, Backward error analysis for Lie-group methods, BIT, 40(4): 652-670(2000)
[12] A. Iserles, Solving linear ordinary differential equations by exponentials of iterated commutators, Numer. Math., 45:183-199(1984)
[13] H. Munthe-Kaas, Runge-Kutta methods on Lie group, BIT 38(1): 92-111(1998).
[14] H. Munthe-Kaas, High order Runge-Kutta methods on manifolds, Applied Numerical Mathematics, 29(1): 115-127,1999.
[15] W. Magnus, On the exponential solution of differential equations for a linear operator, Commun. Pure Appl. Math. 7:649-673(1954).
[16] F. Fer, Resolution de l'equation matricielle U = pU par produit infini
d'exponentielles matricielles, Bull. Classes Sci. Acad. Roy. Bel., 44:818-829 (1958).
[17] S. Blanes, F. Cacas and J. Ros, Improved high order integrators based on the Magnus expansion, BIT, 40(3): 434-450(2000).
[18] H. Yoshida. Construction of higher order symplectic integrators. Physics Letters A, 150:262-268(1990).
[19] 冯康.Hamilton形式体系和辛几何的差分格式——纪念华罗庚教授.力学未来15年国际学术讨论会论文集,科学出版社,1989.
[20] Feng Kang. On Difference Schemes and Symplectic Geometry. Proc. 1984 Beijing Int. Symp. on Differential Geometry and Differential Equation--COMPUTATION OF PARTIAL DIFFERENTIAL EQUATIONS, Ed. Feng Kang, Beijing: Science Press, 42-58, (1985).
[21] 钟万勰,欧阳华江,邓子辰.计算结构力学与最优控制.大连理工大学出版社,1993.
[22] 钱学森,钱令希,谈庆明.力学——迎接21世纪挑战.力学与实践.17(4):1-5(1995).
[23] B. M. J. Maschke, R. Ortega, A. J. van der Schaft, Energy-based Lyapunov functions for forced Hamiltonian systems with dissipation. Proc. of CDC98, 3599-3604.
[24] R. Ortega, A. Stankovic, R Stefanov, A passivation approach to power systems stabilization. IFAC Symp. Nonlinear Control Systems Design, Enschede, NL, July 1-3,1998.
[25] 程代展,席在荣,卢强, 梅生伟.广义Hamilton控制系统的几何结构及其应用.中国科学(E缉),30(4):341-355(2000).
[26] 王玉振,程代展,李春文.广义Hamilton实现及其在电力系统中的应用,自动化学报,28(5):745-753(2002).
[27] S. Blanes and P. C. Moan, Splitting methods for non-autonomous Hamiltonian equations, J. Comput. Phys., 170 (1): 205-230(2001).
[28] S. Blanes, F. Cacas and J. Ros, High order optimized geometric integrators for linear differential equations, BIT, 42(2): 262-284(2002).
[29] A. Zanna, Collocation and relaxed collocation for the Fer and the Magnus expansions, SIAM J. Numer Anal., 36:1145-1182(1999).
[30] S. Klarsfeld and J. A. Oteo, Recursive generation of higher-order terms in the Magnus expansion. Phys. Rev. A, 39:3270-3273(1989).
[31] A. Zanna, The Fer expansion and time-symmetry: A Strang-type approach, Appl. Numer. Math., 39:435-459(2001).
[32] J. M. Sanz-Serna, Symplectic integrators for Hamiltonian problems: An overview, Acta. Numerica, 1: 243-286 (1992).
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