Fourier expansion based recursive algorithms for periodic Riccati and Lyapunov matrix differential equations

Combining Fourier series expansion with recursive matrix formulas, new reliable algorithms to compute the periodic, non-negative, definite stabilizing solutions of the periodic Riccati and Lyapunov matrix differential equations are proposed in this paper. First, periodic coefficients are expanded in terms of Fourier series to solve the time-varying periodic Riccati differential equation, and the state transition matrix of the associated Hamiltonian system is evaluated precisely with sine and cosine series. By introducing the Riccati transformation method, recursive matrix formulas are derived to solve the periodic Riccati differential equation, which is composed of four blocks of the state transition matrix. Second, two numerical sub-methods for solving Lyapunov differential equations with time-varying periodic coefficients are proposed, both based on Fourier series expansion and the recursive matrix formulas. The former algorithm is a dimension expanding method, and the latter one uses the solutions of the homogeneous periodic Riccati differential equations. Finally, the efficiency and reliability of the proposed algorithms are demonstrated by four numerical examples.     

A note on the Lyapunov stability of periodic discrete-time systems

We characterize the stability of discrete-time Lyapunov equations with periodic coefficients. The characterization can be seen as the analog of the classical stability theorem of Lyapunov equations with constant coefficients. It involves quantities readily computable with good accuracy.     

Converse Lyapunov theorems for nonautonomous discrete-time systems

This paper presents converse Lyapunov theorems for exponential stability of nonautonomous discrete-time systems with disturbances and free of disturbances, respectively. It is shown that Lyapunov functions exist for discrete-time systems if the systems are exponentially stable. Moreover, in the periodic case, we explicitly construct a Lyapunov function for systems with disturbances. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 61, Optimal Control, 2008.     

On almost periodic processes in impulsive competitive systems with delay and impulsive perturbations

In the present paper we investigate the existence of almost periodic processes of ecological systems which are presented with the general impulsive nonautonomous Lotka–Volterra system of integro-differential equations with infinite delay. The impulses are at fixed moments of time, and by using the techniques of piecewise continuous Lyapunov’s functions, new sufficient conditions for the global exponential stability of the unique positive almost periodic solutions of these systems are given.     

On existence of periodic solutions for gradient systems with applications

In this paper we prove the existence of periodic solutions for gradient systems in finite and infinite dimensional spaces. The techniques of the proofs are based on the application of a global inverse functions theorem, the Schäefer fixed point theorem and the Faedou–Galerkin method. We apply our results in order to solve nonlinear reaction–diffusion equations with Dirichlet and Neumann boundary conditions.     

Bifurcations from nondegenerate families of periodic solutions in Lipschitz systems

The paper addresses the problem of bifurcation of periodic solutions from a normally nondegenerate family of periodic solutions of ordinary differential equations under perturbations. The approach to solve this problem can be described as transforming (by a Lyapunov–Schmidt reduction) the initial system into one which is in the standard form of averaging, and subsequently applying the averaging principle. This approach encounters a fundamental problem when the perturbation is only Lipschitz (nonsmooth) as we do not longer have smooth Lyapunov–Schmidt projectors. The situation of Lipschitz perturbations has been addressed in the literature lately and the results obtained conclude the existence of the bifurcated branch of periodic solutions. Motivated by recent challenges in control theory, we are interested in the uniqueness problem. We achieve this in the case when the Lipschitz constant of the perturbation obeys a suitable estimate.     

Justification of the coupled-mode approximation for a nonlinear elliptic problem with a periodic potential

Coupled-mode systems are used in physical literature to simplify the nonlinear Maxwell and Gross–Pitaevskii equations with a small periodic potential and to approximate localized solutions called gap solitons by analytical expressions involving hyperbolic functions. We justify the use of the 1D stationary coupled-mode system for a relevant elliptic problem by employing the method of Lyapunov–Schmidt reductions in Fourier space. In particular, existence of periodic/anti-periodic and decaying solutions is proved and the error terms are controlled in suitable norms. The use of multi-dimensional stationary coupled-mode systems is justified for analysis of bifurcations of periodic/anti-periodic solutions in a small multi-dimensional periodic potential.     

Positive operator based iterative algorithms for solving Lyapunov equations for Itô stochastic systems with Markovian jumps

This paper studies the iterative solutions of Lyapunov matrix equations associated with Itô stochastic systems having Markovian jump parameters. For the discrete-time case, when the associated stochastic system is mean square stable, two iterative algorithms with one in direct form and the other one in implicit form are established. The convergence of the implicit iteration is proved by the properties of some positive operators associated with the stochastic system. For the continuous-time case, a transformation is first performed so that it is transformed into an equivalent discrete-time Lyapunov equation. Then the iterative solution can be obtained by applying the iterative algorithm developed for discrete-time Lyapunov equation. Similar to the discrete-time case, an implicit iteration is also proposed for the continuous case. For both discrete-time and continuous-time Lyapunov equations, the convergence rates of the established algorithms are analyzed and compared. Numerical examples are worked out to validate the effectiveness of the proposed algorithms.     

A periodic dividend problem with inconstant barrier in Markovian environment

Periodic dividend problem is a meaningful issue. Based on a compound binomial model with periodic dividend, we use a homogeneous, ergodic and irreducible discrete-time Markov chain to express the evolution from one period to the subsequent of the economic or the environmental and climatic conditions. We derive some properties about the model. A system of integral equations for the expectation and the r-th moment of discounted dividends until ruin time are obtained respectively. Moreover, by using of Contraction Mapping Principle, we solve the equation system and obtain the explicit expression.     

The existence and attractivity of time periodic solutions for evolution equations with delays

In this paper, we investigate the existence and global attractivity of time periodic solutions for evolution equations with several delays under some reasonable assumptions. The key-steps are constructing some suitable Lyapunov functionals and establishing the prior bound for all possible periodic solutions. Some illustrative examples are presented in the last section.     

Long periodic shadowing

A general theorem for establishing the existence of a true periodic orbit near a numerically computed pseudoperiodic orbit of an autonomous system of ordinary differential equations is presented. For practical applications, a Newton method is devised to compute appropriate pseudoperiodic orbits. Then numerical considerations for checking the hypotheses of the theorem in terms of quantities which can be computed directly from the pseudoperiodic orbit and the vector field are addressed. Finally, a numerical method for estimating the Lyapunov exponents of the true periodic orbit is given. The theory and computations are designed to be applicable for unstable periodic orbits with long periods. The existence of several such periodic orbits of the Lorenz equations is exhibited. This revised version was published online in June 2006 with corrections to the Cover Date.