The paper considers the class of matrix polytopes with a dominant vertex and the class of uncertain dynamical systems defined in discrete time and continuous time, respectively, by such polytopes. We analyze the standard concept of stability in the sense of Schur—abbreviated as SS (resp., Hurwitz—abbreviated as HS), and we develop a general framework for the investigation of the diagonal stability relative to an arbitrary Hölder $p$-norm, $1\le p\le \infty$, abbreviated as ${\text{SDS}}_p$ (resp., ${\text{HDS}}_p$). Our framework incorporates, as the particular case with $p=2$, the known condition of quadratic stability satisfied by a diagonal positive-definite matrix, i.e. ${\text{SDS}}_2$ (resp., ${\text{HDS}}_2$) means that the standard inequality of Stein (resp., Lyapunov) associated with all matrices of the polytope has a common diagonal solution. For the considered class of matrix polytopes, we prove the equivalence between SS and ${\text{SDS}}_p$ (resp., HS and ${\text{HDS}}_p$), $1\le p\le \infty$ (fact which is not true for matrix polytopes with arbitrary structures). We show that the dominant vertex provides all the information needed for testing these stability properties and for computing the corresponding robustness indices. From the dynamical point of view, if an uncertain system is defined by a polytope with a dominant vertex, then the standard asymptotic stability ensures supplementary properties for the state-space trajectories, which refer to special types of Lyapunov functions and contractive invariant sets (characterized through vector $p$-norms weighted by diagonal positive-definite matrices). The applicability of the main results is illustrated by two numerical examples that cover both discrete- and continuous-time cases for the class of uncertain dynamics studied in our paper.

An enhanced symplectic synchronization of complex chaotic systems with uncertain parameters is studied. The traditional chaos synchronizations are special cases of the enhanced symplectic synchronization. A sufficient condition is given for the asymptotical stability of the null solution of error dynamics. The enhanced symplectic synchronization may be applied to the design of secure communication. Finally, numerical simulations results are performed to verify and illustrate the analytical results.

This paper studies the resilient $L_2$-$L_{\infty }$ filtering problem for a class of uncertain Markovian jumping systems within the finite-time interval. The objective is to design such a resilient filter that the finite-time $L_2$-$L_{\infty }$ gain from the unknown input to an estimation error is minimized or guaranteed to be less than or equal to a prescribed value. Based on the selected Lyapunov-Krasovskii functional, sufficient conditions are obtained for the existence of the desired resilient $L_2$-$L_{\infty }$ filter which also guarantees the stochastic finite-time boundedness of the filtering error dynamic systems. In terms of linear matrix inequalities (LMIs) techniques, the sufficient condition on the existence of finite-time resilient $L_2$-$L_{\infty }$ filter is presented and proved. The filter matrices can be solved directly by using the existing LMIs optimization techniques. A numerical example is given at last to illustrate the effectiveness of the proposed approach.

We study the structure of the ergodic limit functions determined in random ergodic theorems. When the r random parameters are shifted by the $r_0$-shift transformation with $r_0\in \{1,2,\dots ,r\}$, the major finding is that the (random) ergodic limit functions determined in random ergodic theorems depend essentially only on the $r\text{—}{r}_0$ random parameters. Some of the results obtained here improve the earlier random ergodic theorems of Ryll-Nardzewski (1954), Gladysz (1956), Cairoli (1964), and Yoshimoto (1977) for positive linear contractions on $L_1$ and Woś (1982) for sub-Markovian operators. Moreover, applications of these results to nonlinear random ergodic theorems for affine operators are also included. Some examples are given for illustrating the relationship between the ergodic limit functions and the random parameters in random ergodic theorems.

In the first part of this paper, we show that an AH algebra $A=\underrightarrow{\lim }(A_i,{\varphi }_i)$ has the LP property if and only if every element of the centre of $A_i$ belongs to the closure of the linear span of projections in $A$. As a consequence, a diagonal AH-algebra has the LP property if it has small eigenvalue variation in the sense of Bratteli and Elliott. The second contribution of this paper is that for an inclusion of unital $C^{*}$-algebras $P\subset A$ with a finite Watatani index, if a faithful conditional expectation $E:A\to P$ has the Rokhlin property in the sense of Kodaka et al., then $P$ has the LP property under the condition that$A$ has the LP property. As an application, let $A$ be a simple unital $C^{*}$-algebra with the LP property, $\alpha$ an action of a finite group $G$ onto $\text{Aut}(A)$. If $\alpha$ has the Rokhlin property in the sense of Izumi, then the fixed point algebra $A^G$ and the crossed product algebra $\mathrm{A\hspace{0.17em}}{⋊}_{\alpha }\mathrm{\hspace{0.17em}G}$ have the LP property. We also point out that there is a symmetry on the CAR algebra such that its fixed point algebra does not have the LP property.

We propose a method to approximate the solutions of fully fuzzy linear system (FFLS), the so-called general solutions. So, we firstly solve the 1-cut position of a system, then some unknown spreads are allocated to each row of an FFLS. Using this methodology, we obtain some general solutions which are placed in the well-known solution sets like Tolerable solution set (TSS) and Controllable solution set (CSS). Finally, we solved two examples in order to demonstrate the ability of the proposed method.

We expose the chaotic attractors of time-reversed nonlinear system, further implement its behavior on electronic circuit, and apply the pragmatical asymptotically stability theory to strictly prove that the adaptive synchronization of given master and slave systems with uncertain parameters can be achieved. In this paper, the variety chaotic motions of time-reversed Lorentz system are investigated through Lyapunov exponents, phase portraits, and bifurcation diagrams. For further applying the complex signal in secure communication and file encryption, we construct the circuit to show the similar chaotic signal of time-reversed Lorentz system. In addition, pragmatical asymptotically stability theorem and an assumption of equal probability for ergodic initial conditions (Ge et al., 1999, Ge and Yu, 2000, and Matsushima, 1972) are proposed to strictly prove that adaptive control can be accomplished successfully. The current scheme of adaptive control—by traditional Lyapunov stability theorem and Barbalat lemma, which are used to prove the error vector—approaches zero, as time approaches infinity. However, the core question—why the estimated or given parameters also approach to the uncertain parameters—remains without answer. By the new stability theory, those estimated parameters can be proved approaching the uncertain values strictly, and the simulation results are shown in this paper.

This paper shows the controller design for a second-order plant with unknown varying behavior in the parameters and in the disturbance. The state adaptive backstepping technique is used as control framework, but important modifications are introduced. The controller design achieves mainly the following two benefits: upper or lower bounds of the time-varying parameters of the model are not required, and the formulation of the control and update laws and stability analysis are simpler than closely related works that use the Nussbaum gain method. The controller has been developed and tested for a DC motor speed control and it has been implemented in a Rapid Control Prototyping system based on Digital Signal Processing for dSPACE platform. The motor speed converges to a predefined desired output signal.

We present a stochastic simple chemostat model in which the dilution rate was influenced by white noise. The long time behavior of the system is studied. Mainly, we show how the solution spirals around the washout equilibrium and the positive equilibrium of deterministic system under different conditions. Furthermore, the sufficient conditions for persistence in the mean of the stochastic system and washout of the microorganism are obtained. Numerical simulations are carried out to support our results.

This paper proposes a chaos-based secure direct-sequence/spread-spectrum (DS/SS) communication system which is based on a novel combination of the conventional DS/SS and chaos techniques. In the proposed system, bit duration is varied according to a chaotic behavior but is always equal to a multiple of the fixed chip duration in the communication process. Data bits with variable duration are spectrum-spread by multiplying directly with a pseudonoise (PN) sequence and then modulated onto a sinusoidal carrier by means of binary phase-shift keying (BPSK). To recover exactly the data bits, the receiver needs an identical regeneration of not only the PN sequence but also the chaotic behavior, and hence data security is improved significantly. Structure and operation of the proposed system are analyzed in detail. Theoretical evaluation of bit-error rate (BER) performance in presence of additive white Gaussian noise (AWGN) is provided. Parameter choice for different cases of simulation is also considered. Simulation and theoretical results are shown to verify the reliability and feasibility of the proposed system. Security of the proposed system is also discussed.