scholarly journals Legendre Polynomials Spectral Approximation for the Infinite-Dimensional Hamiltonian Systems

2011 ◽  
Vol 2011 ◽  
pp. 1-13
Author(s):  
Zhongquan Lv ◽  
Mei Xue ◽  
Yushun Wang
Keyword(s):  
Differential Operator ◽  
Hamiltonian System ◽  
Hamiltonian Systems ◽  
Legendre Polynomials ◽  
Hamiltonian Operator ◽  
Spectral Approximation ◽  
Infinite Dimensional

This paper considers a Legendre polynomials spectral approximation for the infinite-dimensional Hamiltonian systems. As a consequence, the Legendre polynomials spectral semidiscrete system is a Hamiltonian system for the Hamiltonian system whose Hamiltonian operator is a constant differential operator.

scholarly journals Homoclinic Orbits for a Class of Nonperiodic Hamiltonian Systems

2012 ◽  
Vol 2012 ◽  
pp. 1-20 ◽  
Author(s):  
Wenping Qin ◽  
Jian Zhang ◽  
Fukun Zhao
Keyword(s):  
Hamiltonian System ◽  
Hamiltonian Systems ◽  
Point Spectrum ◽  
Homoclinic Orbits ◽  
Hamiltonian Operator ◽  
Quadratic Nonlinearity ◽  
Linking Theorem ◽  
Strongly Indefinite Functionals ◽  
Superquadratic Nonlinearity

We study the following nonperiodic Hamiltonian systemż=JHz(t,z), whereH∈C1(R×R2N,R)is the formH(t,z)=(1/2)B(t)z⋅z+R(t,z). We introduce a new assumption onB(t)and prove that the corresponding Hamiltonian operator has only point spectrum. Moreover, by applying a generalized linking theorem for strongly indefinite functionals, we establish the existence of homoclinic orbits for asymptotically quadratic nonlinearity as well as the existence of infinitely many homoclinic orbits for superquadratic nonlinearity.

Design of Dynamical Controllers for Linear Hamiltonian Systems

1999 ◽  
Author(s):  
Werner Haas ◽  
Michael Krommer ◽  
Hans Irschik
Keyword(s):  
Hamiltonian Systems ◽  
Controller Design ◽  
Time Derivative ◽  
Design Parameters ◽  
Internal Dynamics ◽  
Design Algorithm ◽  
Infinite Dimensional ◽  
Infinite Dimensional Systems ◽  
Linear Hamiltonian Systems ◽  
Frequency Domain Approach

Abstract Linear Hamiltonian control systems with collocation of sensor and actuator are considered. Based on a frequency domain approach a controller design algorithm is stated. The design leads to a controller with internal dynamics which uses the output of the system and its first time derivative. The presence of internal dynamics in the controller is an extension of the usual PD–control law and a main result of the work. The design is based on the special properties of the proposed class of systems. In particular, these Hamiltonian systems are passive. It is shown that the design leads to strictly passive controllers for a certain choice of the design parameters. This is another significant result and offers a way for robust ℒ2–stabilization even in the case of infinite dimensional systems. Some features of the controller design are discussed with respect to an application, the control of a composite circular plate.

scholarly journals Integral operators, bispectrality and growth of Fourier algebras

2020 ◽  
Vol 2020 (766) ◽  
pp. 151-194 ◽  
Author(s):  
W. Riley Casper ◽  
Milen T. Yakimov
Keyword(s):  
Differential Operator ◽  
Integral Kernel ◽  
Exact Estimate ◽  
Airy Functions ◽  
Direct Computation ◽  
Infinite Dimensional ◽  
First Order ◽  
Rank 2 ◽  
Commuting Differential Operator ◽  
Algorithmic Procedure

AbstractIn the mid 1980s it was conjectured that every bispectral meromorphic function {\psi(x,y)} gives rise to an integral operator {K_{\psi}(x,y)} which possesses a commuting differential operator. This has been verified by a direct computation for several families of functions {\psi(x,y)} where the commuting differential operator is of order {\leq 6}. We prove a general version of this conjecture for all self-adjoint bispectral functions of rank 1 and all self-adjoint bispectral Darboux transformations of the rank 2 Bessel and Airy functions. The method is based on a theorem giving an exact estimate of the second- and first-order terms of the growth of the Fourier algebra of each such bispectral function. From it we obtain a sharp upper bound on the order of the commuting differential operator for the integral kernel {K_{\psi}(x,y)} leading to a fast algorithmic procedure for constructing the differential operator; unlike the previous examples its order is arbitrarily high. We prove that the above classes of bispectral functions are parametrized by infinite-dimensional Grassmannians which are the Lagrangian loci of the Wilson adelic Grassmannian and its analogs in rank 2.

Hamiltonian systems of Schrödinger equations with vanishing potentials

2020 ◽  
pp. 2050074
Author(s):  
E. Toon ◽  
P. Ubilla
Keyword(s):  
Hamiltonian System ◽  
Hamiltonian Systems ◽  
Positive Solutions ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Cerami Sequence ◽  
Cerami Sequences ◽  
Vanishing Potentials

In this paper, by means of minimax techniques involving Cerami sequences, we prove the existence of at least one pair of positive solutions for a Hamiltonian system of Schrödinger equations in [Formula: see text] with potentials vanishing at infinity and subcritical nonlinearities which are superlinear at the origin and at infinity. We establish new estimates to prove the boundedness of a Cerami sequence.

scholarly journals The number of limit cycles for a class of quintic Hamiltonian systems under quintic perturbations

2002 ◽  
Vol 73 (1) ◽  
pp. 37-54 ◽  
Author(s):  
Guowei Chen ◽  
Yongbin Wu ◽  
Xinan Yang
Keyword(s):  
Hopf Bifurcation ◽  
Hamiltonian System ◽  
Hamiltonian Systems ◽  
Limit Cycles ◽  
Homoclinic Bifurcation

AbstractThe Hopf bifurcation and homoclinic bifurcation of the quintic Hamiltonian system is analyzed under quintic perturbations by using unfolding theory in this paper. We show that a quintic system can have at least 29 limit cycles.

scholarly journals Higher Order Hamiltonian Systems with Generalized Legendre Transformation

Mathematics ◽  
2018 ◽  
Vol 6 (9) ◽  
pp. 163
Author(s):  
Dana Smetanová
Keyword(s):  
Hamiltonian System ◽  
Hamiltonian Systems ◽  
Higher Order ◽  
Second Order ◽  
Legendre Transformation ◽  
Lagrange Equations ◽  
Hamilton Equations ◽  
Strongly Regular

The aim of this paper is to report some recent results regarding second order Lagrangians corresponding to 2nd and 3rd order Euler–Lagrange forms. The associated 3rd order Hamiltonian systems are found. The generalized Legendre transformation and geometrical correspondence between solutions of the Hamilton equations and the Euler–Lagrange equations are studied. The theory is illustrated on examples of Hamiltonian systems satisfying the following conditions: (a) the Hamiltonian system is strongly regular and the Legendre transformation exists; (b) the Hamiltonian system is strongly regular and the Legendre transformation does not exist; (c) the Legendre transformation exists and the Hamiltonian system is not regular but satisfies a weaker condition.

Generalized Inverse of Upper Triangular Infinite Dimensional Hamiltonian Operators

2013 ◽  
Vol 20 (03) ◽  
pp. 395-402
Author(s):  
Junjie Huang ◽  
Xiang Guo ◽  
Yonggang Huang ◽  
Alatancang
Keyword(s):  
Explicit Expression ◽  
Generalized Inverse ◽  
Hamiltonian Operator ◽  
Operator Matrix ◽  
Infinite Dimensional ◽  
Symplectic Spaces ◽  
Hamiltonian Operators

In this paper, we deal with the generalized inverse of upper triangular infinite dimensional Hamiltonian operators. Based on the structure operator matrix J in infinite dimensional symplectic spaces, it is shown that the generalized inverse of an infinite dimensional Hamiltonian operator is also Hamiltonian. Further, using the decomposition of spaces, an upper triangular Hamiltonian operator can be written as a new operator matrix of order 3, and then an explicit expression of the generalized inverse is given.

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