Existence of triple positive solutions of two-point right focal boundary value problems on time scales

We consider the following boundary value problem, (−1)ⁿ⁻¹y^Δn(t)=(−1)^p+1F(t,y(σⁿ⁻¹(t))),t[a,b]∩T, y^Δn(a)=0,0≤i≤p−1, y^Δn(σ(b))=0,p≤i≤n−1,where n ≥ 2, 1 ≤ p ≤ n - 1 is fixed and T is a time scale. By applying fixed-point theorems for operators on a cone, existence criteria are developed for triple positive solutions of the boundary value problem. We also include examples to illustrate the usefulness of the results obtained.     

Necessary conditions of optimality for a class of optimal control problems on time scales

A class of optimal control problems of a system governed by linear dynamic equations on time scales with quadratic cost functional is considered. By the Lebesgue Δ-integral theory and the Sobolev-type space H¹ on time scales the weak solution of linear dynamic equations on time scales for both initial value problem and backward problem are introduced, therefore the necessary conditions of optimality are presented. Some typical examples are given for demonstration.     

Robust stability of delayed reaction-diffusion recurrent neural networks with Dirichlet boundary conditions on time scales

In this paper, the global robust exponential stability of equilibrium solution to delayed reaction-diffusion recurrent neural networks with Dirichlet boundary conditions on time scales is studied. Using topological degree theory, M-matrix method, Lyapunov functional and inequality skills, we establish some sufficient conditions for the existence, uniqueness and global robust exponential stability of equilibrium solution to delayed reaction-diffusion recurrent neural networks with Dirichlet boundary conditions on time scales. One example is given to illustrate the effectiveness of our results.     

Oscillation criteria for second order quasilinear neutral delay differential equations on time scales

In this paper, we consider the second-order quasilinear neutral dynamic equation

(r(t)|Z^Δ(t)|^α−1Z^Δ(t))^Δ+q(t)|x(δ(t))|^β−1x(δ(t))=0,     

Stability for first order delay dynamic equations on time scales

In this paper, we establish some sufficient conditions for the uniform stability and the uniformly asymptotical stability of the first order delay dynamic equation

where is a time scale, p(.) is rd-continuous and positive, the delay function . Our results unify the corresponding ones for differential and difference equations. To the best of our knowledge, this is the first time to discuss the asymptotical behavior of delay dynamic equations on time scales.     

Satisfaction of boundary conditions for Chebyshev collocation methods in cuboidal domains

A collocation strategy for the satisfaction of boundary conditions in the application of Chebyshev spectral methods to Poisson and biharmonic-type problems in cuboidal domains is presented. This strategy leads, in both cases, to nonsingular linear systems for the determination of the unknown coefficients.     

Stability in terms of two measures of dynamic system on time scales

By using a notion of upper quasi-monotone nondecreasing, this paper presents a new comparison principle which connects the solutions of two higher-dimensional dynamic systems on time scales. Then the stability criteria of a solution of a dynamic system in terms of two measures are obtained. Finally, two examples are provided to illustrate our results.     

First order conditions for generalized variational problems over time scales

In this paper, we derive the weak Pontryagin principle for generalized optimal control problems over time scales. Three types of problems are considered, namely (i) the problems involving the values of the state endpoints in the Lagrangian and the dynamics, (ii) the problems with an integral of the state in the Lagrangian and the dynamics, and (iii) the isoperimetric problems. As special cases, we obtain the first order optimality conditions for the corresponding calculus of variations problems, which have been of a recent interest in the literature. Our method is based on transforming the generalized optimal control or calculus of variations problem into a traditional optimal control problem on time scales, to which the known weak Pontryagin principle can be applied.     

Stability radius of implicit dynamic equations with constant coefficients on time scales

This paper deals with the stability radii of implicit dynamic equations on time scales when the structured perturbations act on both the coefficient of derivative and the right-hand side. Formulas of the stability radii are derived as a unification and generalization of some previous results. A special case where the real stability radius and the complex stability radius are equal is studied. Examples are derived to illustrate results.     

Stability of moving invariant sets and uncertain dynamic systems on time scales

In this paper, we study the stability of moving invariant sets and uncertain dynamic systems on time scale. A control application is considered.     

A control synthesis approach for time discrete event systems

In this paper, we introduce a control synthesis method for discrete event systems whose behavior is dependent on explicit values of time. Our goal is to control the occurrence dates of the controllable events so that the functioning of the system respects given specifications. The system to be controlled is modeled by a time Petri net. In a previous work we proposed a systematic method to build the timed automaton which models the exact behavior of a time Petri net. Furthermore, the forbidden behaviors of the system are modeled by forbidden timed automaton locations. This paper focuses on the control synthesis method, which consists in computing new firing conditions for the timed automaton transitions so that the forbidden locations are no longer reachable.     

Global exponential stability of neural networks with discrete and distributed delays and general activation functions on time scales

By employing time scale calculus theory, free weighting matrix method and linear matrix inequality (LMI) approach, several delay-dependent sufficient conditions are obtained to ensure the existence, uniqueness and global exponential stability of the equilibrium point for the neural networks with both infinite distributed delays and general activation functions on time scales. Both continuous-time and discrete-time neural networks are described under the same framework by the reported method. Illustrated numerical examples are given to show the effectiveness of the theoretical analysis. It is noteworthy that the activation functions are assumed to be neither bounded nor monotone.     

Convergence and instability in PCG methods for bordered systems

Bordered almost block diagonal systems arise from discretizing a linearized first-order system of n ordinary differential equations in a two-point boundary value problem with nonseparated boundary conditions. The discretization may use spline collocation, finite differences, or multiple shooting. After internal condensation, if necessary, the bordered almost block diagonal system reduces to a standard finite difference structure, which can be solved using a preconditioned conjugate gradient method based on a simple matrix splitting technique. This preconditioned conjugate gradient method is “guaranteed” to converge in at most 2n + 1 iterations. We exhibit a significant collection of two-point boundary value problems for which this preconditioned conjugate gradient method is unstable, and hence, convergence is not achieved.     

Creases and boundary conditions for subdivision curves

Our goal is to find subdivision rules at creases in arbitrary degree subdivision for piece-wise polynomial curves, but without introducing new control points e.g. by knot insertion. Crease rules are well understood for low degree (cubic and lower) curves. We compare three main approaches: knot insertion, ghost points, and modifying subdivision rules. While knot insertion and ghost points work for arbitrary degrees for B-splines, these methods introduce unnecessary (ghost) control points.The situation is not so simple in modifying subdivision rules. Based on subdivision and subspace selection matrices, a novel approach to finding boundary and sharp subdivision rules that generalises to any degree is presented. Our approach leads to new higher-degree polynomial subdivision schemes with crease control without introducing new control points.     

Oscillation theorems for second-order half-linear delay dynamic equations with damping on time scales

By using the generalized Riccati transformation and the inequality technique, we establish a few new oscillation criteria for certain second-order half-linear delay dynamic equations with damping on a time scale. Our results extend and improve some known results, but also unify the oscillation of the second-order half-linear delay differential equation with damping and the second-order half-linear delay difference equation with damping.     

Improved stability criteria for linear time-varying systems on time scales

ABSTRACT

In this paper, asymptotic stability problems of linear time-varying (LTV) systems on time scales are considered based on a less conservative Lyapunov inequality, whose right side is not required to be necessarily negative. It is shown that the Lyapunov inequality covers not only the corresponding trivial (continuous and discrete) ones but also nontrivial ones. Based on this inequality, some necessary and sufficient conditions for asymptotic stability, exponential stability, uniformly exponential stability of LTV systems on time scales are obtained. An example about nontrivial systems is given for illustrating the effectiveness of the proposed results.     

Stability radii for positive linear time-invariant systems on time scales

We deal with dynamic equations on time scales, where we characterize the positivity of a system. Uniform exponential stability of a system is determined by the spectrum of its matrix. We investigate the corresponding stability radii with respect to structured perturbations and show that, for positive systems, the complex and the real stability radius coincide.     

Representations and Ostrowski type inequalities on time scales

Here, we give univariate and multivariate representations of the Montgomery type for hybrid functions on time scales. Based on these, we establish univariate and multivariate Ostrowski type inequalities on time scales domains. These compare the average of a function to its values. The estimates involve the higher order delta and nabla derivatives and partial derivatives. We finish with applications on the time scales R and Z.