Infinitely many solutions for a class of nonlinear impulsive differential equations with periodic boundary conditions

In this paper, we consider the existence of solutions for a class of nonlinear impulsive problems with periodic boundary conditions. By using critical point theory, we obtain some existence theorems of infinitely many solutions for the nonlinear impulsive problem when the impulsive functions are superlinear. We extend and improve some recent results.     

The fibering map approach to a nonlocal problem involving p(x)-Laplacian

We study the existence of positive solutions for a $p\left(x\right)$-Kirchhoff problem. The main tool used is the fibering map approach for the corresponding Nehari manifold.     

Extended invariance principle for nonautonomous switched systems

Stability analysis is developed for nonlinear nonautonomous switched systems, trajectories of which admit, generally speaking, a nonunique continuation on the right. For these systems Krasovskii-LaSalle's invariance principle is extended in such a manner to remain true even in the nonautonomous case. In addition to a nonsmooth Lyapunov function with negative-semidefinite time derivatives along the system trajectories, the extended invariance principle involves a coupled indefinite function to guarantee asymptotic stability of the system in question. As an illustration of the capabilities of this principle, a switched regulator of a fully-actuated manipulator with frictional joints is constructed.     

(2,1)-Method for solving stiff nonautonomous problems

An L-stable (2,1)-method with second-order accuracy is developed for solving stiff nonautonomous problems. At each step, the right-hand side of the system, as well as Jacobian matrix decomposition are evaluated only once. Efficiency of the integration method is illustrated using ring modulator calculation as an example.     

A theorem for UGAS and ULES of (passive) nonautonomous systems: robust control of mechanical systems and ships

The main contribution of this paper is a theorem to guarantee uniform global asymptotic stability (UGAS) and uniform local exponential stability (ULES) for a class of nonlinear non‐autonomous systems which includes passive systems. These properties (and a uniform local Lipschitz condition) guarantee robustness of stability while weaker properties, like uniform global stability plus global convergence, do not. Our main result is then used in the tracking control problem of mechanical systems and ships. We use an adaptive backstepping design and prove UGAS of the closed‐loop tracking error system, in particular, we obtain that both the tracking and parameter estimation errors converge uniformly globally to zero. Copyright © 2001 John Wiley & Sons, Ltd.     

A new stability test for nonlinear nonautonomous systems

We consider a class of nonlinear nonautonomous systems, whose linear parts are slowly varying matrices. Stability conditions and estimates for the region of attraction of the zero solution are derived. These conditions are formulated in terms of the determinants of the variable matrices.     

Formal energy of symplectic scheme for Hamiltonian systems and its applications (II)

In this paper, for 4-fold and 8-fold compositions of symplectic schemes, the authors obtain the formulae for calculation of the first three terms of the power series in stepsize of their formal energies. Utilizing the special properties of revertible schemes, the authors construct higher order revertible symplectic schemes for general Hamiltonian systems only through formal energies. From any order s (even) to order s + 2, a determining conclusion is obtained. And from any order s (even) to order s + 4, an algebraic equation system, when s is evaluated whose solution gives a rise by 4 in order, is about to be set up. As examples, for the cases of s = 2 and s = 4, the numerical results are to be gained.     

Uniform persistence and existence of strictly positive solutions in nonautonomous Lotka-Volterra competitive systems with delays

In this paper, we establish some new sufficient conditions for uniform persistence and existence of strictly positive solutions of the general nonautonomous N-species competitive Lotka-Volterra systems with delays. These results are different to those in [1–4].     

Observer-based Hamiltonian identification for quantum systems

A symmetry-preserving observer-based parameter identification algorithm for quantum systems is proposed. Starting with a 2-level quantum system (qubit), where the unknown parameters consist of the atom-laser frequency detuning and coupling constant, we prove an exponential convergence result. The analysis is inspired by Lyapunov and adaptive control techniques and is based on averaging theory. The observer is then extended to the multi-level case where all the atom-laser coupling constants are unknown. The extension of the convergence analysis is discussed through some heuristic arguments. The relevance and the robustness with respect to various noises are tested through numerical simulations.     

Improved potential energy-shaping for port-controlled Hamiltonian systems

This paper investigates the asymptotical stabilization of port-controlled Hamiltonian(PCH)systems via the improved potential energy-shaping(IPES)method.First,a desired potential energy introduced by a transitive Hamiltonian function is added to the original kinetic energy to yield a desired Hamiltonian function.Second,an asymptotically stabilized controller is designed based on a new matching equation with the obtained Hamiltonian function.Finally,a numerical example is given to show the effectiveness of the proposed method.     

Input-output decoupling with stability for Hamiltonian systems

The input-output decoupling problem with stability for Hamiltonian systems is treated using decoupling feedbacks, all of which make the system maximally unobservable. Using the fact that the dynamics of the maximal unobservable subsystem are again Hamiltonian, an easily checked condition for input-output decoupling with (critical) stability is deduced.     

On stabilization of energy for Hamiltonian systems

We study post-stabilization of numerical integrators for an autonomous Hamiltonian system in the form H=H₁+H₂, where H₁ and H₂ are two constants of motion. It is better to stabilize the two energies H₁ and H₂, respectively. As a particular case, the effectiveness of post-stabilization by the total energy is equivalent to one by the two independent energies when the splitting Hamiltonian becomes isotropic.     

Energy-based control for hybrid port-controlled Hamiltonian systems

In this paper we develop an energy-based hybrid control framework for hybrid port-controlled Hamiltonian systems. In particular, we obtain constructive sufficient conditions for hybrid feedback stabilization that provide a shaped energy function for the closed-loop system, while preserving a hybrid Hamiltonian structure at the closed-loop level. Furthermore, an inverse optimal hybrid feedback control framework is developed that characterizes a class of globally stabilizing energy-based controllers that guarantee hybrid sector and gain margins to multiplicative input uncertainty of hybrid Hamiltonian systems.     

Lyapunov inequalities for discrete linear Hamiltonian systems

In this paper, we present some Lyapunov type inequalities for discrete linear scalar Hamiltonian systems when the coefficient c(t) is not necessarily nonnegative valued and when the end-points are not necessarily usual zeros, but rather, generalized zeros. Applying these inequalities, we obtain some disconjugacy and stability criteria for discrete Hamiltonian systems