Equivalent Hamiltonian Equations Modelling and Energy Function Construction for MMC-HVDC in Hybrid AC/DC Power Systems

This paper proposes an equivalent Hamiltonian equations model for the modular multilevel converter-based high-voltage direct-current(MMC-HVDC)transmission system,and constructs an energy function for multi-machine power systems with MMC-HVDC transmission lines.The equivalent Hamiltonian equations model is verified to be able to track the power output dynamics of the full model of an MMC-HVDC transmission system.Both theoretical and numerical studies have been undertaken to validate that the energy function proposed for hybrid AC/DC systems satisfies the conditions of an energy function.The work of this paper bridges the gap between the well-developed direct methods of transient stability analysis and power systems with MMC-HVDC transmission lines.

Birkhoffian symplectic algorithms derived from Hamiltonian symplectic algorithms

In this paper, we focus on the construction of structure preserving algorithms for Birkhoffian systems, based on existing symplectic schemes for the Hamiltonian equations. The key of the method is to seek an invertible transformation which drives the Birkhoffian equations reduce to the Hamiltonian equations. When there exists such a transformation, applying the corresponding inverse map to symplectic discretization of the Hamiltonian equations, then resulting difference schemes are verified to be Birkhoftian symplectic for the original Birkhoffian equations. To illustrate the operation process of the method, we construct several desirable algorithms for the linear damped oscillator and the single pendulum with linear dissipation respectively. All of them exhibit excellent numerical behavior, especially in preserving conserved quantities.

HAMILTONIAN FORMULATION OF NONLINEAR WATER WAVES IN A TWO-FLUID SYSTEM

In this paper, it is dealt with that the Hamiltonian formulation of nonlinear water waves in a two_fluid system,which consists of two layers of constant_density incompressible inviscid fluid with a horizontal bottom,an interface and a free surface. The velocity potentials are expanded in power series of the vertical coordinate. By taking the kinetic thickness of lower fluid_layer and the reduced kinetic thickness of upper fluid_layer as the generalized displacements, choosing the velocity potentials at the interface and free surface as the generalized momenta and using Hamilton's principle, the Hamiltonian canonical equations for the system are derived with the Legendre transformation under the shallow water assumption. Hence the results for single_layer fluid are extended to the case of stratified fluid.

Variational Principles and Hamiltonian Formulation for Nonlinear Water Waves

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Instability of Standing Waves for Hamiltonian Wave Equations

This paper deals with the standing wave for a Hamiltonian nonlinear wave equation which can be viewed as a representative of the class of equations of interest.On the one hand,by proving a compactness lemma and solving a variational problem,the existence of the standing wave with ground state for the aforementioned equation is proved.On the other hand,the authors derive the instability of the standing wave by applying the potential well argument,the concavity method and an invariant region under the solution flow of the Cauchy problem for the equation under study,and the invariance of the region aforementioned can be shown by introducing an auxiliary functional and a supplementary constrained variational problem.

Explicit Multi-Symplectic Methods for Hamiltonian Wave Equations

In this paper,based on the multi-symplecticity of concatenating symplectic Runge-Kutta-Nystrom(SRKN)methods and symplectic Runge-Kutta-type methods for numerically solving Hamiltonian PDEs,explicit multi-symplectic schemes are constructed and investigated,where the nonlinear wave equation is taken as a model problem.Numerical comparisons are made to illustrate the effectiveness of our newly derived explicit multi-symplectic integrators.

Dynamics of cantilever plates and its hybrid vibration control

Applying Lagrange-Germain＇s theory of elas- tic thin plates and Hamiltonian formulation, the dynamics of cantilever plates and the problem of its vibration control are studied, and a general solution is finally given. Based on Hamiltonian and Lagrangian density function, we can obtain the flexural wave equation of the plate and the relationship between the transverse and the longitudinal eigenvalues. Based on eigenfunction expansion, dispersion equations of propagation mode of cantilever plates are deduced. By satisfying the boundary conditions of cantilever plates, the natural frequencies of the cantilever plate structure can be given. Then, analytic solution of the problem in plate structure is obtained. An hybrid wave/mode control approach, which is based on both independent modal space control and wave control methods, is described and adopted to analyze the active vibration control of cantilever plates. The low-order （controlled by modal control） and the high-order （controlled by wave control） frequency response of plates are both improved. The control spillover is avoided and the robustness of the system is also improved. Finally, simulation results are analyzed and discussed.

Efficient Energy-preserving Methods for General Nonlinear Oscillatory Hamiltonian System

In this paper, we propose and analyze two kinds of novel and symmetric energy-preservmg formulae for the nonlinear oscillatory Hamiltonian system of second-order differential equations Aq＂ （t）＋ Bq（t） = f（q（t））, where A ∈ R^m×m is a symmetric positive definite matrix, B ∈ R^m×m is a symmetric positive semi-definite matrix that implicitly contains the main frequencies of the problem and f（q） = -VqV（q） for a real-valued function V（q）. The energy-preserving formulae can exactly preserve the Hamiltonian H（q＇,q） = 1/2q＇^TAq＇＋ 1/2q^TBq - V（q）. We analyze the properties of energy-preserving and convergence of the derived energy-preserving formula and obtain new efficient energy-preserving integrators for practical computation. Numerical experiments are carried out to show the efficiency of the new methods by the nonlinear Hamiltonian systems.

Gravitational Field as a Pressure Force from Logarithmic Lagrangians and Non-Standard Hamiltonians：The Case of Stellar Halo of Milky Way

Recently,the notion of non-standard Lagrangians was discussed widely in literature in an attempt to explore the inverse variational problem of nonlinear differential equations.Different forms of non-standard Lagrangians were introduced in literature and have revealed nice mathematical and physical properties.One interesting form related to the inverse variational problem is the logarithmic Lagrangian,which has a number of motivating features related to the Li′enard-type and Emden nonlinear differential equations.Such types of Lagrangians lead to nonlinear dynamics based on non-standard Hamiltonians.In this communication,we show that some new dynamical properties are obtained in stellar dynamics if standard Lagrangians are replaced by Logarithmic Lagrangians and their corresponding non-standard Hamiltonians.One interesting consequence concerns the emergence of an extra pressure term,which is related to the gravitational field suggesting that gravitation may act as a pressure in a strong gravitational field.The case of the stellar halo of the Milky Way is considered.

Indefinite stochastic linear-quadratic optimal control problems with random jumps and related stochastic Riccati equations

We discuss the stochastic linear-quadratic(LQ) optimal control problem with Poisson processes under the indefinite case. Based on the wellposedness of the LQ problem, the main idea is expressed by the definition of relax compensator that extends the stochastic Hamiltonian system and stochastic Riccati equation with Poisson processes(SREP) from the positive definite case to the indefinite case. We mainly study the existence and uniqueness of the solution for the stochastic Hamiltonian system and obtain the optimal control with open-loop form. Then, we further investigate the existence and uniqueness of the solution for SREP in some special case and obtain the optimal control in close-loop form.

NEW EXACT TRAVELLING WAVE SOLUTIONS OF NONLINEAR WAVE EQUATIONS USING THE DYNAMICAL SYSTEM METHOD

By the method of dynamical system, we construct the exact travelling wave solu- tions of a new Hamiltonian amplitude equation and the Ostrovsky equation. Based on this method, the new exact travelling wave solutions of the new Hamiltonian am- plitude equation and the Ostrovsky equation, such as solitary wave solutions, kink and anti-kink wave solutions and periodic travelling wave solutions, are obtained, respectively.

Regularity of the Attractor for the Dissipative Hamiltonian Amplitude Equation Governing Modulated Wave Instabilities

In the present paper, the existence of global attractor for dissipative Hamiltonian amplitude equation governing the modulated wave instabilities in E0 is considered. By a decomposition of solution operator, it is shown that the global attractor in E0 is actually equal to a global attractor in E1.

Nonconservative mechanical systems with nonholonomic constraints

A geometric setting for generally nonconservative mechanical systems on fibred manifolds is proposed. Emphasis is put on an explicit formulation of nonholonomic mechanics when an unconstrained Lagrangian system moves in a generally non-potential force field depending on time, positions and velocities, and the constraints are nonholonomic, not necessarily linear in velocities. Equations of motion, and the corresponding Harniltonian equations in intrinsic form are given. Regularity conditions are found and a nonholonomic Legendre transformation is proposed, leading to a canonical form of the nonholonomic Hamiltonian equations for nonconservative systems.

THE COMPLETELY INTEGRABLE SYSTEMS OF THE COMPLEX FORM AND THE COUPLED AKNS-KAUP-NEWELL EQUATION HIERARCHY

In this paper, the coupled AKNS-Kaup-Newell equation hierarchy are obtained by means of the new spectral problem. By means of the complex representation of the standard symplect form on R4N, and the constraint relations between the potential and the wave functions, the new completely integrable systems of the complex form are got. Therefore, the involutive solutions of the coupled AKNS-Kaup-Newell equation hierarchy are given.