Upper bounds on the Lyapunov exponents for dissipative perturbations of infinite dimensional Hamiltonian systems

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.

Download Full-text

Crossing Limit Cycles of Planar Piecewise Linear Hamiltonian Systems without Equilibrium Points

Mathematics ◽

10.3390/math8050755 ◽

2020 ◽

Vol 8 (5) ◽

pp. 755

Author(s):

Rebiha Benterki ◽

Jaume LLibre

Keyword(s):

Hamiltonian Systems ◽

Limit Cycles ◽

Piecewise Linear ◽

Equilibrium Points ◽

Upper Bounds ◽

Linear Hamiltonian Systems ◽

Straight Lines

In this paper, we study the existence of limit cycles of planar piecewise linear Hamiltonian systems without equilibrium points. Firstly, we prove that if these systems are separated by a parabola, they can have at most two crossing limit cycles, and if they are separated by a hyperbola or an ellipse, they can have at most three crossing limit cycles. Additionally, we prove that these upper bounds are reached. Secondly, we show that there is an example of two crossing limit cycles when these systems have four zones separated by three straight lines.

Download Full-text

On the control by interconnection and exponential stabilisation of infinite dimensional port-Hamiltonian systems

2016 IEEE 55th Conference on Decision and Control (CDC) ◽

10.1109/cdc.2016.7798739 ◽

2016 ◽

Cited By ~ 2

Author(s):

Alessandro Macchelli

Keyword(s):

Hamiltonian Systems ◽

Infinite Dimensional

Download Full-text

The Cyclicity of Period Annulus of Degenerate Quadratic Hamiltonian Systems with Polycycles S(2) or S(3) Under Perturbations of Piecewise Smooth Polynomials

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127420502302 ◽

2020 ◽

Vol 30 (15) ◽

pp. 2050230

Author(s):

Jiaxin Wang ◽

Liqin Zhao

Keyword(s):

Hamiltonian Systems ◽

Limit Cycles ◽

Upper Bounds ◽

Melnikov Function ◽

Piecewise Smooth ◽

Bifurcation Of Limit Cycles ◽

First Order ◽

Period Annulus ◽

Quadratic Hamiltonian

In this paper, by using Picard–Fuchs equations and Chebyshev criterion, we study the bifurcation of limit cycles for degenerate quadratic Hamilton systems with polycycles [Formula: see text] or [Formula: see text] under the perturbations of piecewise smooth polynomials with degree [Formula: see text]. Roughly speaking, for [Formula: see text], a polycycle [Formula: see text] is cyclically ordered collection of [Formula: see text] saddles together with orbits connecting them in specified order. The discontinuity is on the line [Formula: see text]. If the first order Melnikov function is not equal to zero identically, it is proved that the upper bounds of the number of limit cycles bifurcating from each of the period annuli with the boundary [Formula: see text] and [Formula: see text] are respectively [Formula: see text] and [Formula: see text] (taking into account the multiplicity).

Download Full-text