Vol. 8, No. 2 (2019): Regular Issue

This paper presents an extensive tutorial and a complete review on the state-dependent Riccati equation (SDRE). The review covers contributions from the beginning to (near the end of 2017) current trend and works, categorized according to theoretical and practical impact. The tutorial section presents the fundamental relations, derivation and necessary conditions of the SDRE as a controller, observer and estimator. The state-dependent Riccati equation serves as a continuous and discrete time controller, observer, filter and estimator in the field of control engineering. The nature of the SDRE is in nonlinear optimal control domain; and nowadays it plays a vital role in aerospace, robotics, control of unmanned aerial vehicle, aircraft, autonomous underwater vehicle, surface vessel and other nonlinear plants. Optimality, robustness, asymptotic stability, flexibility in design and a systematic procedure are some of the advantages of this method. The capability to combine the SDRE with other methods (such as sliding mode control, fuzzy, genetic, neural network and etc.) is another important feature of the approach. The mentioned characteristics, advantages and combination of the SDRE with other techniques are reviewed comprehensively in this work.

This paper investigates the dynamic response of a simply-supported uniform beam resting on elastic subgrade and subjected to travelling distributed load. A robust mathematical procedure is developed to treat this vibrating system problem. In particular, the Mindlin and Goodman’s technique in the first instance is used to transform the non-homogeneous fourth order partial differential equations with governing non-homogeneous boundary conditions into non-homogeneous fourth order partial differential equations with homogeneous boundary conditions and then the resulting transformed equation is then further treated using the versatile generalized integral transform with the series representation of the Heaviside function and a modification of Struble’s asymptotic method. Analytical solution was obtained for the dynamic response of the uniform elastic beam. Analytical and Numerical calculations give important results which is very useful in structural design and engineering analysis.

We discuss the problem of establishing dissipative estimates for certain differential equations for which the usual methods do not work. The aim of this paper are some new nonlinear integral inequalities leading to suitable uniform (with respect to time and the parameter ε ) bounds on the solutions to problems x(t) = f (t,x(t))+gε(t,x(t)), t ≥ 0 where f ,gε : R+ ×Rn →Rn are supposed to be piecewise continuous in time, locally Lipschitz in x, for any fixed ε ≥ 0. These problems are seen as perturbation to x(t) = f (t,x(t)), t ≥ 0. Furthermore, some examples are given to illustrate the applicability of the obtained results. ©2019 L&H Scientific Publishing, LLC. All rights reserved.

Existence and local stability of a coexistence point are investigated for a class of nonlinear polynomial differential systems. The studied dynamical system may be used to represent dynamics of species populations in a trophic food chain with n distinguished populations. The presented model incorporates stage structured population dynamics and captures both predation and competition phenomena. Local stability is studied via introducing a change of variables and then applying the Lyapunov direct method. An example is given for clarification. The investigated model, enhances the previously presented differential equations for population dynamics via incorporating competition and stage structures in the general predator-prey model.

This paper establishes a set of sufficient conditions for the nonlinear fractional delay dynamical systems with control delay of order 1 < α < 2, and the delays are in state variable as well as control variable. The solution representations are provided. The main tool are the Mittag-Leffler matrix function and the Schaefer’s fixed point theorem. Examples are presented to illustrate the results.

Dynamics of population density of species in a trophic food chain with n distinguished members are investigated. A nonlinear feedback harvesting law is proposed to stabilize the coexistence equilibrium point of the system. A method is also proposed to derive the feedback parameters. The proposed dynamical model captures predation, competition, logistic growth and carrying capacity phenomena. An interesting real-world example is included to show the counter-intuitive behavior of the studied system and the effectiveness of the proposed harvesting method.

We pay our attention to a non-linear differential equation of first order with multiple two variable advanced arguments. We find sufficient conditions satisfying the convergence (C) and exponential convergence (EC) of solutions of the considered non-linear advanced differential equation (NADE) by contraction mapping principle (CMP). The obtained results improve and extend the results can be found in the relevant literature from a case of linear advanced differential equation (LADE) of first order to a case of (NADE) of first order with multiple two variable advanced arguments. We give examples for illustrations by applying MATLAB-Simulink. It is also clearly shown the behaviors of the orbits for the special cases of the considered (NADE).

In this paper, we discuss the existence, uniqueness and stability of boundary value problem for differential equations with ψ-Hilfer fractional derivative. The arguments are based upon Schaefer’s fixed point theorem, Banach contraction principle and ulam type stability.

In this article, the authors have investigated a novel scheme of dual combination synchronization among six different complex chaotic systems using nonlinear control technique. The important feature of the considered dual combination synchronization is various synchronization processes viz., complete synchronization, combination synchronization, projective synchronization, dual synchronization, chaos control problem become particular cases of the said scheme. Since complex chaotic systems have additional variables and complexity in behaviours, the scheme is expected to be more secure for transmitting and receiving signals in communication theory. The stability analysisof the scheme is achieved by using nonlinear control method based on Lyapunov stability theory. The corresponding theoretical results are being simulated using fourth order Runge-Kutta algorithm taking complex Lorenz system, complex Lu system, complex T system, complex Chen system as drive systems and complex two coupled dynamos system and a new nonlinear complex chaotic system as response systems. The results are depicted through graphical presentations for different particular cases.

We pay our attention to a number of nonlinear Volterra integrodifferential equations of first order, VIDEs, with constant retardation. The uniform boundedness, UB, of the solutions, to that VIDEs is investigated via the Lyapunov functionals, LFs, method. The results obtained improve and complement a number of works found in the literature. The novelty and originality of this article is that it improves and extends the results found in the literature from the cases of the without delay to the more general cases with delay by constructing some new LFs.

In this paper, we discuss the approximate controllability of the second order neutral impulsive stochastic evolution equations of Sobolev type with unbounded delay in Hilbert Spaces. A set of sufficient conditions are established for the existence and approximate controllability of the mild solutions using Krasnoselskii-Schaefer-type fixed point theorem and stochastic analysis theory. An application involving nonlinear differential equation with unbounded delay is addressed.

A classical glacial climate model previously developed by Saltzman et al. [1–4] and Nicollis et al. [5] is analysed when parametric forcing is present. This particular model was actually used in the study of glacier formation. Our analysis mainly focusses on the issues due to the periodic or quasiperiodic variation of these parameters with time. It is observed that this quasiperiodically driven Saltzman model leads to the generation of Strange Nonchaotic Attractor (SNA), which is very interesting because it is neither a periodic state nor a fully chaotic one. Due to this fact, it is usually very difficult to pinpoint its existence and generation. In particular, we focus on an intermittency transitions of type I and on subharmonic bifurcation leading to type III intermittency. The properties of the attractor are characterised by the finite time Lyapunov exponents, its variance and their distributions along with Poincar´e sections. The zone of existence of SNA for different parameter values have been found.