A non-fragile robust observer design for uncertain time-delay fractional Itô stochastic systems with input nonlinearity: An SMC approach

This article considers the problem of non-fragile observer design for uncertain fractional Itô stochastic systems. The design is based on a sliding surface whose reachability in finite time is guaranteed by introducing a novel sliding mode control law. Employing the fractional infinitesimal operator and the related lemmas, the stochastic stability of the overall closed-loop system is transformed to the problem of solving a set of linear matrix inequalities. Addressing the fragility issue, a norm-bounded term is added to the observer gain, which prevents failure of the estimation error system. The adverse effects of the input nonlinearity and time-varying delay are alleviated by the proposed approach. Furthermore, the present method is investigated for the fractional Itô stochastic systems with known states. A numerical example is presented to illustrate the effectiveness of the proposed method.

The Fundamental Nature of Time

The nature of time is intimately bound up with the nature of energy propagation, which has a long history of its philosophical understanding. Here I propose a new post-Einsteinian view of the nature of time, conceptualized as the outcome of the pure unidimensional rate of change of a process through the infinitesimal operator of differential equations. In this view, time is a local property that is generated by every individual process in the Universe rather than a fundamental dimension in which processes operate. The rate of change has an inherent “arrow of time” that does not depend on the ensemble properties of multiple processes, such as the laws of entropy, but is inherent to the function of each process, by virtue of its genesis in the Big Bang. The conventional view of time may be approximated either by aggregating the operations of large ensembles of diverse processes, or by choosing a process (such as the Atomic Clock) that has demonstrably stable temporal properties. For processes that are sufficiently nonlinear, their iterative progression may in principle lead to solutions describable as fractals, for which the integral derivation of the time variable would fractionate into a form of fractal time.

Singular perturbations for a subelliptic operator

We study some classes of singular perturbation problems where the dynamics of the fast variables evolve in the whole space obeying to an infinitesimal operator which is subelliptic and ergodic. We prove that the corresponding ergodic problem admits a solution which is globally Lipschitz continuous and it has at most a logarithmic growth at infinity. The main result of this paper establishes that, as ϵ → 0, the value functions of the singular perturbation problems converge locally uniformly to the solution of an effective problem whose operator and terminal data are explicitly given in terms of the invariant measure for the ergodic operator.

New Applications of a Kind of Infinitesimal-Operator Lie Algebra

Applying some reduced Lie algebras of Lie symmetry operators of a Lie transformation group, we obtain an invariant of a second-order differential equation which can be generated by a Euler-Lagrange formulism. A corresponding discrete equation approximating it is given as well. Finally, we make use of the Lie algebras to generate some new integrable systems including (1+1) and (2+1) dimensions.

Equilibrium Time-Consistent Strategy for Corporate International Investment Problem with Mean-Variance Criterion

We analyze a continuous-time model for corporate international investment problem (CIIP) with mean-variance criterion. Based on Nash subgame perfect equilibrium theory, we define an infinitesimal operator and directly derive an extended Hamilton-Jacobi-Bellman (HJB) equation. Besides, we also obtain the equilibrium time-consistent strategy for CIIP. In addition, we discuss two cases of risk aversion coefficient; one is constant and the other is state dependent. Finally, the simulation results are given to illustrate our conclusions and the influence of some parameters on the optimal solution.

Stabilization of Semi-Markovian Jump Systems with Uncertain Probability Intensities and Its Extension to Quantized Control

This paper concentrates on the issue of stability analysis and control synthesis for semi-Markovian jump systems (S-MJSs) with uncertain probability intensities. Here, to construct a more applicable transition model for S-MJSs, the probability intensities are taken to be uncertain, and this property is totally reflected in the stabilization condition via a relaxation process established on the basis of time-varying transition rates. Moreover, an extension of the proposed approach is made to tackle the quantized control problem of S-MJSs, where the infinitesimal operator of a stochastic Lyapunov function is clearly discussed with consideration of input quantization errors.

Almost Sure Stability of Stochastic Neural Networks with Time Delays in the Leakage Terms

The stability issue is investigated for a class of stochastic neural networks with time delays in the leakage terms. Different from the previous literature, we are concerned with the almost sure stability. By using the LaSalle invariant principle of stochastic delay differential equations, Itô’s formula, and stochastic analysis theory, some novel sufficient conditions are derived to guarantee the almost sure stability of the equilibrium point. In particular, the weak infinitesimal operator of Lyapunov functions in this paper is not required to be negative, which is necessary in the study of the traditional moment stability. Finally, two numerical examples and their simulations are provided to show the effectiveness of the theoretical results and demonstrate that time delays in the leakage terms do contribute to the stability of stochastic neural networks.

One Form of Lyapunov Operator for Stochastic Dynamic System with Markov Parameters

The form of weak infinitesimal operator of Lyapunov type on solutions of stochastic dynamic systems of random structure with constant delay which exist under the action of Markov perturbations is obtained.

Symmetry properties of fractional order transport equations

In the paper some features of applying Lie group analysis methods to fractional differential equations are considered. The problem related to point change of variables in the fractional differentiation operator is discussed and some general form of transformation that conserves the form of Riemann-Liouville fractional operator is obtained. The prolongation formula for extending an infinitesimal operator of a group to fractional derivative with respect to arbitrary function is presented. Provided simple example illustrates the necessity of considering both local and non-local symmetries for fractional differential equations in particular cases including the initial conditions. The equivalence transformation forms for some fractional differential equations are discussed and results of group classification of the wave-diffusion equation are presented. Some examples of constructing particular exact solutions of fractional transport equation are given, based on the Lie group methods and the method of invariant subspaces.