Pseudo-almost periodic solutions for some classes of nonautonomous partial evolution equations

In this paper upon making some suitable assumptions such as the Acquistapace–Terreni conditions and exponential dichotomy we obtain the existence of pseudo-almost periodic solutions to some classes of nonautonomous evolution equations of Sobolev-type. An example is given at the end of the paper to illustrate our abstract result.     

Iterative learning control for boundary tracking of uncertain nonlinear wave equations

This work presents a framework of iterative learning control (ILC) design for a class of nonlinear wave equations. The main contribution lies in that it is the first time to extend the idea of well-established ILC for lumped parameter systems to boundary tracking control of nonlinear hyperbolic distributed parameter systems (DPSs). By fully utilizing the system repetitiveness, the proposed control algorithm is capable of dealing with time-space-varying and even state-dependent uncertainties. The convergence and robustness of the proposed ILC scheme are analyzed rigorously via the contraction mapping methodology and differential/integral constraints without any system dynamics simplification or discretization. In the end, two examples are provided to show the efficacy of the proposed control scheme.     

状态依赖时滞型微分方程正周期解的存在性

本文利用Krasnoselskii不动点定理,考虑了状态依赖时滞性微分方程x′(t) = &;#8722;A(t, x(t))x(t) + B(x(t))F(x(t &;#8722;  (t, x(t))))正周期解的存在性, 得到了该方程存在与不存在正周期解的充分条件.     

Existence of three periodic solutions for a nonlinear first order functional differential equation

In this paper, we use Leggett-Williams multiple fixed point theorem to obtain different sufficient conditions for the existence of at least three nonnegative periodic solutions of the first order functional differential equation of the form

y^′(t)=-a(t)y(t)+λf(t,y(h(t))).     

Numerical solution of nonlinear equations with real and complex variables

In this paper we attempt to develop the algorithm of PIS (Perturbed Iterative Scheme) in its simplest form, yet retaining its mathematical rigor. Several applications are given and a brief comparison of PIS is made with other methods. The fundamental concept behind PIS is explained. Success of the method is basically for solutions of nonlinear systems.     

Bound of solutions to third-order nonlinear differential equations with bounded delay

A theorem is presented that contains some sufficient conditions to ensure bound of solutions to a third-order nonlinear delay differential equation. It is shown that this theorem improves the result of Ponzo [On the stability of certain nonlinear differential equations, IEEE Trans. Automatic Control AC-10 (1965) 470-472] to the bound of solutions for the equation considered.     

Periodic solutions of second-order nonlinear difference equations containing a small parameter

We extend to difference equations the classical method of harmonic balance. We show that the method can be used to obtain an approximation to the periodic solutions of a special class of second-order nonlinear difference equations containing a small parameter. Two examples illustrating the method are presented.     

Stability analysis of nonlinear control systems with characteristics equations having complex coefficients

In this paper, the stability equation method is applied to the analysis of nonlinear systems with characteristics equations having complex coefficients. Three types of systems are studied: those with unstable open-loop poles, unstable characteristics roots or an equal number of open-loop poles and zeros.     

Gradient-based neural networks for solving periodic Sylvester matrix equations

This paper considers neural network solutions of a category of matrix equation called periodic Sylvester matrix equation (PSME), which appear in the process of periodic system analysis and design. A linear gradient-based neural network (GNN) model aimed at solving the PSME is constructed, whose state is able to converge to the unknown matrix of the equation. In order to obtain a better convergence effect, the linear GNN model is extended to a nonlinear form through the intervention of appropriate activation functions, and its convergence is proved through theoretical derivation. Furthermore, the different convergence effects presented by the model with various activation functions are also explored and analyzed, for instance, the global exponential convergence and the global finite time convergence can be realized. Finally, the numerical examples are used to confirm the validity of the proposed GNN model for solving the PSME considered in this paper as well as the superiority in terms of the convergence effect presented by the model with different activation functions.     

Existence of solutions for nonlinear fractional q-difference integral equations with two fractional orders and nonlocal four-point boundary conditions

We investigate the existence of solutions for a nonlinear fractional q-difference integral equation (q-variant of the Langevin equation) with two different fractional orders and nonlocal four-point boundary conditions. Our results are based on some classical fixed point theorems. An illustrative example is also presented.     

Global asymptotic stability and symmetric periodic solutions

The global asymptotic stability of “second-order” relay control systems with negative eigenvalues is shown to be a consequence of the nonexistence of symmetric periodic solutions. The result shows further that, except for a special case, the solutions reach the origin in finite time.     

A finite iterative algorithm for the general discrete-time periodic Sylvester matrix equations

The purpose of this paper is to present an iterative algorithm for solving the general discrete-time periodic Sylvester matrix equations. It is proved by theoretical analysis that this algorithm can get the exact solutions of the periodic Sylvester matrix equations in a finite number of steps in the absence of round-off errors. Furthermore, when the discrete-time periodic Sylvester matrix equations are consistent, we can obtain its unique minimal Frobenius norm solution by choosing appropriate initial periodic matrices. Finally, we use some numerical examples to illustrate the effectiveness of the proposed algorithm.     

Stability criteria for certain classes of nonlinear difference equations

The general mth order difference equation X(n+m)+a₁X(n+m?1)+…+a_mX(n) = F[n,X(n),…,X(n+m?1)] is considered. The stability properties of its solutions are studied using the discrete form of Liapunov's direct method. A quadratic form is selected as a possible Liapunov function V(n,X) and a scheme is developed for determining appropriate conditions on this function to insure that its total difference ΔV(n,X) is negative semi-definite or negative definite with respect to the difference equation. The approach is applied to the fourth-order difference equation in full detail to illustrate the method for determining the conditions which imply either uniform stability or uniform asymptotic stability and specific results are obtained. Several comments on, and extensions of, the work done by Puri and Drake for the cases m = 2 and m = 3 are presented.The results of the present approach in the homogeneous case where F[n,X(n),…,X(n+m?1)] = 0 are compared with the usual Schur-Cohn criteria and are shown to be at least as good.     

An iterative algorithm for generalized periodic multiple coupled Sylvester matrix equations

The paper is indicated to constructing a modified conjugate gradient iterative (MCG) algorithm to solve the generalized periodic multiple coupled Sylvester matrix equations. It can be proved that the proposed approach can find the solution within finite iteration steps in the absence of round-off errors. Furthermore, we provide a method for choosing the initial matrices to obtain the least Frobenius norm solution of the system. Some numerical examples are illustrated to show the performance of the proposed approach and its superiority over the existing method CG.     

Gradient based approach for generalized discrete-time periodic coupled Sylvester matrix equations

The paper is dedicated to solving the generalized periodic discrete-time coupled Sylvester matrix equation, which is frequently encountered in control theory and applied mathematics. The solvable condition and a iterative algorithm for this equation are presented. The proposed method is developed from a point of least squares method. The rationality of the method is testified by theoretical analysis, which shows that the algorithm can solve the problem within finite number of iterations. The presented approach is numerically reliable and requires less computation. A numerical example illustrates the effectiveness of the raised result.     

Alternative convergence criteria for iterative methods of solving nonlinear equations

Let χ_m+1=T(χ_m) or even χ_m+1=T(χ_m,χ_m?1, …, χ_m?q), m=1,2,3 … be an iteration method for solving the nonlinear problem F(χ)=0, where F(χ) and its derivatives possess all of the properties required by T(χ_m). Then if it can be established that for the problem at hand ∥F(χ_m+1)∥?β_m∥F(χ_m)∥, ? m > M₀ (M₀<∞) and 0?β_m<1 , definitions are established and theorems proven concerning convergence, uniqueness and bounds on the error after ‘m’ successive iterations of a new approach to convergence properties T(χ_m). These charateristics are referred to as “alternate” (local, global) convergence properties and none of the proofs given are restricted to any specific type of method such as, e.g. contraction mapping types. Application of results obtained are illustrated using Newton's method as well as the general concept of Newton-like methods.