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Mathematical models, especially complex nonlinear systems, are always difficult to analyze and synthesize, and researchers need effective and suitable control methods to address these issues. In the present work, we proposed a hybrid method that combines the well-known Takagi-Sugeno fuzzy model with wavelet decomposition to investigate nonlinear systems characterized by the presence of mixed nonlinearities. Here, one nonlinearity is super-linear and convex, and other is sub-linear, concave, and singular at zero, which leads to difficulties in the analysis, as is known in PDE theory. Linear and polynomial fuzzy models were combined with wavelets to ensure an improvement in both methods for investigating such problems. The results showed a high performance compared with existing methods via error estimates and Lyapunov theory of stability. The model was applied to a prototype nonlinear Schrödinger dynamical system.

This paper focuses on the feedback global stabilization and observer construction for a sterile insect technique model. The sterile insect technique (SIT) is one of the most ecological methods for controlling insect pests responsible for worldwide crop destruction and disease transmission. In this work, we construct a feedback law that globally asymptotically stabilizes an SIT model at extinction equilibrium. Since the application of this type of control requires the measurement of different states of the target insect population, and, in practice, some states are more difficult or more expensive to measure than others, it is important to know how to construct a state estimator, which from a few well-chosen measured states, estimates the other ones, as the one we build in the second part of our work. In the last part of our work, we show that we can apply the feedback control with estimated states to stabilize the full system.

Robot manipulators play a crucial role in various industrial and research settings, requiring precise and controlled interactions with their surroundings. Achieving this goal with fewer sensors offers advantages not only in terms of cost and decreased risk of failure but also enhances accuracy and long-termreliability. In this paper,we introduce a nonlinear force/position controller that eliminates the requirement for velocity measurements. This controller provides versatility by facilitating the generation of bounded control actions during robot-environment interactions, ensuring a higher level of safety for both the robot and its environment during the execution of tasks necessitating physical contact between them. The proposed approach is underpinned by a stability analysis in the Lyapunov sense and has been validated through a series of simulation and experimental tests.

In this study, the stability and stabilization analyses are discussed for Takagi-Sugeno (T-S) fuzzy systems with input saturation. A fuzzy-based sampled-data control is designed to stabilize the T-S fuzzy systems. Based on the Lyapunov method and some integral inequality techniques, a set of sufficient conditions is obtained as linear matrix inequality (LMI) constraints to guarantee the asymptotic stability of the considered system. In this process, the linear switching method is utilized to design a controller that is dependent on the membership function, and an integral inequality is utilized. Additionally, determination of the controller parameters is achieved by resolving a series of LMI constraints. The effectiveness of these criteria is demonstrated through a real system that is modeled by the T-S system.

In this paper, an adaptive neural network control problem for nonstrict-feedback nonlinear systems with an unknown backlash-like hysteresis and bounded disturbance was presented. Radial basis function neural networks (RBFNN) were used to approximate the unknown functions and the problem of the explosion of complexity problem was handled by utilizing the command filter method. Furthermore, the influence of an unknown backlash-like hysteresis input was addressed by approximating an intermediate variable. Based on the backstepping method and the command filter technique, an adaptive neural network controller was designed via the approximation abilities of RBFNN. With the help of the Lyapunov stability theory, the proposed controller ensures that all of the signals in closed-loop systems are bounded and that the tracking error fluctuates close to the origin within a bounded area. Finally, a real-world example based on the single-link manipulator was shown to demonstrate the viability of the presented approach.

The chikungunya virus (CHIKV) infects macrophages and adherent cells and it can be transmitted via a direct contact with the virus or with an already infected cell. Thus, the CHIKV infection can have two routes. Furthermore, it can exhibit seasonal peak periods. Thus, in this paper, we consider a dynamical system model of the CHIKV dynamics under the conditions of a seasonal environment with a general incidence rate and two routes of infection. In the first step, we studied the autonomous system by investigating the global stability of the steady states with respect to the basic reproduction number. In the second step, we establish the existence, uniqueness, positivity and boundedness of a periodic orbit for the non-autonomous system. We show that the global dynamics are determined by using the basic reproduction number denoted by $ \mathcal{R}_0 $ and they are calculated using the spectral radius of an integral operator. We show the global stability of the disease-free periodic solution if $ \mathcal{R}_0 < 1 $ and we also show the persistence of the disease if $ \mathcal{R}_0 > 1 $ where the trajectories converge to a limit cycle. Finally, we display some numerical investigations supporting the theoretical findings.

This paper is concerned with a fractional Timoshenko system of order between one and two. We address the question of well-posedness in an appropriate space when the rotational component is viscoelastic or subject to a viscoelastic controller. To this end we use the notion of alpha-resolvent. Moreover, we prove that the memory term alone may stabilize the system in a Mittag-Leffler fashion. The system is Lyapunov stable or uniformly stable in the case of different speeds of propagation.

Background. The work is devoted to the analysis of stability in the sense Lyapunov of steady-state solutions of nonlinear differential equations in Banach spaces with time-dependent operators and with time-dependent delays. Delay differential equations model dynamic processes in many problems of physics, natural science, and technology, and methods for constructing sufficient conditions for the stability of their solutions are needed. Existing methods for finding sufficient conditions for the stability of solutions to nonlinear differential equations in Banach spaces are complex enough to be used in solving specific physical and technical problems. Of current interest is the development of methods for constructing sufficient conditions for stability, asymptotic stability, and boundedness of solutions of differential equations in Banach spaces. Materials and methods. The mathematical apparatus used in this work is the logarithmic norm and its properties. When studying the stability of solutions to nonlinear differential equations with delays in Banach spaces, a comparison is made between the norm and the logarithmic norm of the operators in the equation. The proofs of the statements formulated in the paper are carried out by the method of contradiction. Results. Algorithms are proposed that make it possible to obtain sufficient conditions for stability, asymptotic stability, and boundedness of solutions of nonlinear differential equations in Banach spaces with operators and with time-dependent delays. Sufficient stability conditions are expressed in terms of the norms and logarithmic norms of the operators entering the equations. Conclusions. A method is proposed for constructing sufficient conditions for stability, asymptotic stability, and boundedness of solutions of nonlinear differential equations in Banach spaces with time-dependent coefficients and delays. The method can be used in the study of non-stationary dynamic systems described by nonlinear differential equations with time-dependent delays.

Background. The study is devoted to the analysis of stability in the sense Lyapunov steady state solutions for systems of linear parabolic equations with coefficients depending on time, and with delays depending on time. The cases of continuous and impulsive perturbations are considered. Materials and methods. A method for studying the stability of solutions to systems of linear parabolic equations is as follows. Applying the Fourier transform to the original system of parabolic equations, we arrive at a system of non-stationary ordinary differential equations defined in the spectral region. First, the stability of the resulting system is studied by the method of frozen coefficients in the metric of the space Rn of n-dimensional vectors. Then the resulting statements are extended to the space L2 . The application of the Parseval equality allows us to return to the domain of the originals and obtain sufficient conditions for the stability of solutions to systems of linear parabolic equations. Results. An algorithm is proposed that allows one to obtain sufficient stability conditions for solutions of finite systems of linear parabolic equations with time-dependent coefficients and with time-dependent delays. Sufficient stability conditions are expressed in terms of the logarithmic norms of matrices composed of the coefficients of the system of parabolic equations. They are obtained in the metric of the space L2 . Algorithms for constructing sufficient stability conditions are efficient, as in the case continuous, and in the case of impulsive perturbations. Conclusions. A method for constructing sufficient stability conditions for solutions of finite systems of linear parabolic equations with time-dependent coefficients and delays. The method can be used in the study non-stationary dynamical systems described by systems of linear parabolic equations with delays depending from time.

In this study, we consider a system of nonlinear differential equations modeling the human immunodeficiency virus type-1 (HIV-1) in a variable environment. Infected cells were subdivided into two compartments describing both latently and productively infected cells. Thus, three routes of infection were considered including the HIV-to-cell contact, latently infected cell-to-cell contact, and actively infected cell-to-cell contact. The nonnegativity and boundedness of the trajectories of the dynamics were proved. The basic reproduction number was determined through an integral operator. The global stability of steady states is then analyzed using the Lyapunov theory together with LaSalle’s invariance principle for the case of a fixed environment. Similarly, for the case of a variable environment, we showed that the virus-free periodic solution is globally asymptotically stable once R0≤1, while the virus will persist once R0>1. Finally, some numerical examples are provided illustrating the theoretical investigations.

The main goal of this work is to construct an algorithm for modeling chaotic processes using special neural ODEs with antisymmetric matrices (antisymmetric neural ODEs) and power activation functions (PAFs). The central part of this algorithm is to design a neural ODEs architecture that would guarantee the generation of a stable limit cycle for a known time series. Then, one neuron is added to each equation of the created system until the approximating properties of this system satisfy the well-known Kolmogorov theorem on the approximation of a continuous function of many variables. In addition, as a result of such an addition of neurons, the cascade of bifurcations that allows generating a chaotic attractor from stable limit cycles is launched. We also consider the possibility of generating a homoclinic orbit whose bifurcations lead to the appearance of a chaotic attractor of another type. In conclusion, the conditions under which the found attractor adequately simulates the chaotic process are discussed. Examples are given.

Abstract The fuzzy set theory enables us to represent our knowledge under multiple interpretations and axiomatic foundations from linguistic to computational representations. While the intuitionistic fuzzy set, as a generalization of the fuzzy set, cannot only represent the tolerance levels, but also the intolerance levels which a decision maker can tolerate and cannot tolerate in the accomplishment of a linguistic interpretation, in this paper, we introduce the generalized variational inequalities for linguistic interpretations using intuitionistic fuzzy relations. It is shown that such problems can be transformed into the classical (nonfuzzy) generalized variational inequalities by means of level sets of the intuitionistic fuzzy relation. Furthermore, the equivalence between the generalized variational inequalities with intuitionistic fuzzy relations and the fuzzy fixed point problems is established. Finally, based on the projection method, we propose an iterative algorithm and a projected neural network model for the generalized variational inequalities with intuitionistic fuzzy relations, and the stability of the proposed projected dynamical system is also investigated.

This paper formulates and analyzes two mathematical models that describe the within-host dynamics of human immunodeficiency virus type 1 (HIV-1) with impairment of both cytotoxic T lymphocytes (CTLs) and B cells. Both viral transmission (VT) and cellular infection (CT) mechanisms are considered. The second model is a generalization of the first model that includes distributed time delays. For the two models, we establish the non-negativity and boundedness of the solutions, find the basic reproductive numbers, determine all possible steady states and establish the global asymptotic stability properties of all steady states by means of the Lyapunov method. We confirm the theoretical results by conducting numerical simulations. We conduct a sensitivity analysis to show the effect of the values of the parameters on the basic reproductive number. We discuss the results, showing that impaired B cells and CTLs, time delay and latent CT have significant effects on the HIV-1 dynamics.

In this paper, we introduce a numerical integration method for hyperbolic systems problems known as the splitting method, which serves as an effective tool for solving complex multidimensional problems in mathematical physics. The exponential stability of the upwind explicit–implicit difference scheme split into directions is established for the mixed problem of a linear two-dimensional symmetric t-hyperbolic system with variable coefficients and lower-order terms. It is noteworthy that there are control functions in the dissipative boundary conditions. A discrete quadratic Lyapunov function was devised to address this issue. A condition for the problem’s boundary data, ensuring the exponential stability of the difference scheme with directional splitting for the mixed problem in the l2 norm, has been identified.

In this work, a pair of observers are proposed for a class of nonlinear systems whose dynamics involve a generalized differential operator that encompasses the conformable derivatives. A generalized conformable exponential stability function, based on this derivative, is introduced in order to prove some Lyapunov-like theorems. These theorems help to verify the stability of the observers proposed, which is exponential in a generalized sense. The performance of the observation scheme is evaluated by means of numerical simulations. Moreover, a comparison of the results obtained with integer, fractional, and generalized conformable derivatives is made.

Active air spring suspensions can improve the vehicle ride comfort and meanwhile realize the vehicle height regulation, and therefore, they have been widely used and studied. However, to achieve better ride comfort and a satisfactory vehicle body height adjustment, the active air suspension controller becomes an indispensable and significant part of the system. Since the nonlinear suspension system possesses uncertainties, it is difficult to take into account both ride comfort and height regulation. This study innovatively proposes an adaptive control algorithm to specifically address the problem of vehicle height regulation and ride comfort for nonlinear active air suspension systems with uncertainties. The accurate tracking to reference vehicle body height curves is realized, and the ride comfort is also improved. Through simulations with two scenarios, it is illustrated that the active air suspension controller owns better control effectiveness than the PID controller. Compared with the PID controller, the designed controller can track the reference vehicle body height curves faster and more accurately. The result also verifies the priority of the designed controller.

In this paper, we propose a new approach to investigate the chaos projective synchronization of the modified fractional-order hyperchaotic Rossler system and its application in secure communication. The proposed communication system consists of four main elements including: modulation, master system, slave system and demodulation. The main idea of this approach is to inject the bounded or unbounded message into one of the parameters of the proposed system using the exponential function. However, the way of injecting the message in the modulation parameter must not remove the hyperchaotic character of the signal sent to the slave system. The slave system adaptively synchronizes with the master system, and the information signal can be recovered. Based on the Lyapunov stability theory, an adaptation laws and adaptive control are designed to achieve projection synchronization of the modified system. Numerical simulations are performed to show the feasibility of the proposed secure communication scheme.

A virus dynamics model with two state-dependent delays and logistic growth term is investigated. A general class of nonlinear incidence rates is considered. The model describes the in-host interplay between viral infection and CTL (cytotoxic T lymphocytes) and antibody immune responses. The wellposedness of the model proposed and Lyapunov stability properties of interior infection equilibria which describe the cases of a chronic disease are studied. We choose a space of merely continuous initial functions which is appropriate for therapy, including drug administration.

Infectious diseases include all diseases caused by the transmission of a pathogenic agent such as bacteria, viruses, parasites, prions, and fungi. They, therefore, cover a wide spectrum of benign pathologies such as colds or angina but also very serious ones such as AIDS, hepatitis, malaria, or tuberculosis. Many epidemic diseases exhibit seasonal peak periods. Studying the population behaviours due to seasonal environment becomes a necessity for predicting the risk of disease transmission and trying to control it. In this work, we considered a five-dimensional system for a fatal disease in a seasonal environment. We studied, in the first step, the autonomous system by investigating the global stability of the steady states. In a second step, we established the existence, uniqueness, positivity, and boundedness of a periodic orbit. We showed that the global dynamics are determined using the basic reproduction number denoted by R0 and calculated using the spectral radius of an integral operator. The global stability of the disease-free periodic solution was satisfied if R0<1, and we show also the persistence of the disease once R0>1. Finally, we displayed some numerical investigations supporting the theoretical findings, where the trajectories converge to a limit cycle if R0>1.

Abstract In this paper, we focus on the problem of synchronization for chaotic neural networks with stochastic disturbances. Firstly, we provide a basic result that the systems including the drive system, response system, and error system have a unique solution on the whole time horizon. Based on this result, we design a new control law such that the response system can be synchronized with the drive chaotic system in finite time. Furthermore, we show that the settling time is independent of the initial data under some proper conditions, which hints that the fixed-time synchronization of chaotic neural networks can be realized by our proposed method. Finally, we give simulations to verify the theoretical analysis for our main results.

The article presents a study of solutions of ODEs system with a special nonlinear part, which is a continuous analogue of an arbitrary recurrent neural network (neural ODEs). As a nonlinear part of the mentioned system of differential equations, we used sums of piecewise continuous functions, where each term is a power function. (These are activation functions.) The use of power activation functions (PAF) in neural networks is a generalization of well-known the rectified linear units (ReLU). In the present time ReLU are commonly used to increase the depth of trained of a neural network. Therefore, the introduction of PAF into neural networks significantly expands the possibilities of ReLU. Note that the purpose of introducing power activation functions is that they allow one to obtain verifiable Lyapunov stability conditions for solutions of the system differential equations simulating the corresponding dynamic processes. In turn, Lyapunov stability is one of the guarantees of the adequacy of the neural network model for the process under study. In addition, from the global stability (or at least the boundedness) of continuous analog solutions it follows that learning process of the corresponding neural network will not diverge for any training sample.

In this paper, we investigate the projected dynamical system associated with the generalized variational inequality with hesitant fuzzy relation. We establish the equivalence between the generalized variational inequality with hesitant fuzzy relation and the fuzzy fixed point problem. And we analyze the existence theorem and iterative algorithm of solutions to such problem. Furthermore, using the projection method, we propose a projection neural network for solving the generalized variational inequality with hesitant fuzzy relation and discuss the stability of the proposed projected dynamical system.

We analyze the stability of the geodesic curves in the geometry of the Gibbons–Maeda–Garfinkle–Horowitz–Strominger black hole, describing the space time of a charged black hole in the low energy limit of the string theory. The stability analysis is performed by using both the linear (Lyapunov) stability method, as well as the notion of Jacobi stability, based on the Kosambi–Cartan–Chern theory. Brief reviews of the two stability methods are also presented. After obtaining the geodesic equations in spherical symmetry, we reformulate them as a two-dimensional dynamic system. The Jacobi stability analysis of the geodesic equations is performed by considering the important geometric invariants that can be used for the description of this system (the nonlinear and the Berwald connections), as well as the deviation curvature tensor, respectively. The characteristic values of the deviation curvature tensor are specifically calculated, as given by the second derivative of effective potential of the geodesic motion. The Lyapunov stability analysis leads to the same results. Hence, we can conclude that, in the particular case of the geodesic motion on circular orbits in the Gibbons–Maeda–Garfinkle–Horowitz–Strominger, the Lyapunov and the Jacobi stability analysis gives equivalent results.

The fear response is an important anti-predator adaptation that can significantly reduce prey's reproduction by inducing many physiological and psychological changes in the prey. Recent studies in behavioral sciences reveal this fact. Other than terrestrial vertebrates, aquatic vertebrates also exhibit fear responses. Many mathematical studies have been done on the mass mortality of pelican birds in the Salton Sea in Southern California and New Mexico in recent years. Still, no one has investigated the scenario incorporating the fear effect. This work investigates how the mass mortality of pelican birds (predator) gets influenced by the fear response in tilapia fish (prey). For novelty, we investigate a modified fractional-order eco-epidemiological model by incorporating fear response in the prey population in the Caputo-fractional derivative sense. The fundamental mathematical requisites like existence, uniqueness, non-negativity and boundedness of the system's solutions are analyzed. Local and global asymptotic stability of the system at all the possible steady states are investigated. Routh-Hurwitz criterion is used to analyze the local stability of the endemic equilibrium. Fractional Lyapunov functions are constructed to determine the global asymptotic stability of the disease-free and endemic equilibrium. Finally, numerical simulations are conducted with the help of some biologically plausible parameter values to compare the theoretical findings. The order $\alpha$ of the fractional derivative is determined using Matignon's theorem, above which the system loses its stability via a Hopf bifurcation. It is observed that an increase in the fear coefficient above a threshold value destabilizes the system. The mortality rate of the infected prey population has a stabilization effect on the system dynamics that helps in the coexistence of all the populations. Moreover, it can be concluded that the fractional-order may help to control the coexistence of all the populations.

The present work studies the Lyapunov instability for discontinuous differential equations through the use of the notion of Carathéodory solution to differential equations. From Lyapunov's first instability theorem and Chetaev's instability theorem, which deal with instability to ordinary differential equations, two Lyapunov instability results for discontinuous differential equations are obtained.

In this research work, time-delay adaptive synchronization and adaptive anti-synchronization of chaotic fractional order systems are analyzed via the Caputo fractional derivative, and the prob-lem of synchronization and anti-synchronization of chaotic systems of variable fractional order is solved by using the fractional order PID control law, the adaptive laws of variable-order frac-tional calculus, and a control law deduced from Lyapunov’s theory extended to systems of time-delay variable-order fractional calculus. In this research work, two important problems are solved in the control area: The first problem is described in which deals with syn-chro-nization of chaotic systems of adaptive fractional order with time delay, this problem is solved by using the fractional order PID control law and adaptative laws. The second problem is de-scribed in which deals with anti-synchronization of chaotic systems of adaptive frac-tional order with time delay, and this problem is solved by using the fractional order PID con-trol law and adaptative laws.

One of the most interesting problems is the investigation of the boundaries of chaotic or hyperchaotic systems. In addition to estimating the Lyapunov and Hausdorff dimensions, it can be applied in chaos control and chaos synchronization. In this paper, by means of the analytical optimization, comparison principle, and generalized Lyapunov function theory, we find the ultimate bound set for a new six-dimensional hyperchaotic system and the globally exponentially attractive set for a new four-dimensional Lorenz- type hyperchaotic system. The novelty of this paper is that it not only shows the 4D hyperchaotic system is globally confined but also presents a collection of global trapping regions of this system. Furthermore, it demonstrates that the trajectories of the 4D hyperchaotic system move at an exponential rate from outside the trapping zone to its inside. Finally, some numerical simulations are shown to demonstrate the efficacy of the findings.

In this paper, the H∞ and H2 time-varying formation tracking problems for multi-agent systems with directed topologies in the presence of external disturbances are investigated. The followers need to achieve the desired time-varying formation during movement and simultaneously track the state trajectory generated by the leader. First, a distributed consensus protocol based on the local state information of neighbors of the agents for solving H∞ and H2 time-varying formation tracking problems are proposed without utilizing global information about the entire agents. The conditions to achieve H∞ and H2 time-varying formation tracking in the presence of external disturbances are suggested respectively. Then, to determine the parameters of the designed protocol which satisfy suitable conditions, algorithms for H∞ and H2 time-varying formation tracking in the form of pseudo-code are presented, respectively. Furthermore, the proofs of the proposed theorems are derived by utilizing algebraic graph theory and Lyapunov analysis theory tools to demonstrate the closed-loop stability of the system in the presence of external disturbances. Finally, the usefulness and effectiveness of the approaches proposed are demonstrated by numerical simulation examples.

The permanent magnet synchronous motors (PMSMs) as the completely symmetrical three-phase machines, which are usually driven by symmetrical voltage signals. Unfortunately, a PMSM system usually suffers from the different lumped disturbances, such as internal parametric perturbations and external load torques, the speed regulation problem should be addressed within the different operation situations. Characterizing by the current variation speed of the motor winding is much faster than that of the mechanical dynamic velocity, a dual-time-scale sliding mode control (SMC) method for the surface-mounted PMSMs is proposed in this paper. Firstly, the mathematical model of PMSMs is established in the two-phase synchronous rotating orthogonal reference coordinate system, and the slow and fast variation subsystems are obtained based on the quasi-steady-state theory. Secondly, a tracking differentiator (TD)-based and exponential reaching law-based sliding mode controllers are individually designed within dual-time-scale, respectively. As a result, the eventual SMC strategy is presented, and the stability of control system is analyzed by applying the Lyapunov stability theory. The main contribution of this study is to present an alternative control framework for the PMSMs servo system, where the dual-time-scale characteristic is involved, and thus a non-cascade control structure that different from the traditional drive strategy is proposed in the motor community. Finally, the model of whole system is built and carried out on the simulation platform. Research results demonstrate that the presented servo control system can accurately track the reference angle velocity signal, while the strong robustness and fast response performance are guaranteed in the presence of external disturbances. In addition, the three-phase current transient response values are completely symmetrical with the rapid adjustment characteristic.

In this paper, the global asymptotic stability and strict passivity of three types of nonlinear RLC circuits are investigated by utilizing the Lyapunov direct method. The stability conditions are obtained by constructing appropriate Lyapunov function, which demonstrates the practical application of the Lyapunov theory with a clear perspective. The meaning of Lyapunov functions is not clear by many specialists whose studies based on Lyapunov theory. They construct Lyapunov functions by using some properties of Lyapunov functions with much trial and errors or for a system choose candidate Lyapunov functions. So, for a given system Lyapunov function is not unique. But we insist that Lyapunov (energy) function is unique for a given physical system. In this study we highly simplified Lyapunov’s direct method with suitable tools. Our approach constructing energy function based on power-energy relationship that also enable us to take the derivative of integration of energy function. These aspects have not been addressed in the literature. This paper is an attempt towards filling this gap. The results are provided within and are of central importance for the analysis of nonlinear electrical, mechanical, and neural systems which based on the system energy perspective. The simulation results given from Matlab successfully verifies the theoretical predictions.

In this paper, we present a new fractional-order financial system (FOFS) with the new parameters. We study the synchronization for commensurate order of the fractional-order financial system with disturbance observer (FOFSDO) on the new parameters. Also, the sensitivity analysis of the synchronization error was investigated by using the feedback control technique for the FOFSDO. The stability of the used method demonstrates by Lyapunov stability theorem. Numerical simulations are presented to ensure the validity and inuence of the target feedback control design in the presence of extrinsic bounded unknown disturbance. [ABSTRACT FROM AUTHOR]

This paper deals with a recently reported mathematical model formulated by five first-order ordinary differential equations that describe glucoregulatory dynamics. As main contributions, we found a localization domain with all compact invariant sets; we settled on sufficient conditions for the existence of a bounded positively-invariant domain. We applied the localization of compact invariant sets and Lyapunov’s direct methods to obtain these results. The localization results establish the maximum cell concentration for each variable. On the other hand, Lyapunov’s direct method provides sufficient conditions for the bounded positively-invariant domain to attract all trajectories with non-negative initial conditions. Further, we illustrate our analytical results with numerical simulations. Overall, our results are valuable information for a better understanding of this disease. Bounds and attractive domains are crucial tools to design practical applications such as insulin controllers or in silico experiments. In addition, the model can be used to understand the long-term dynamics of the system.

We investigated the finite/fixed-time stability (FNTS/FXTS) of discontinuous differential equations (DDEs) in this paper. To cope with differential equations that were discontinuous on the right-hand side, we utilized the Filippov solution, which is widely used in engineering. Under the framework of the Filippov solution, we transformed this issue into an FNTS/FXTS problem in the corresponding functional differential inclusion. We proposed some new FNTS/FXTS criteria, which will have important applications in the field of control engineering. It is worth mentioning that the coefficient function in the inequality satisfied by the Lyapunov function (LF) could be indefinite. Moreover, our paper gave a new estimation for the settling time (ST). Finally, two illustrative examples were given to demonstrate the validity and feasibility of the proposed criteria.

This paper presents a robust nonlinear control scheme with time-varying output constraint for the electro-hydraulic force control system (EHFCS). Two typical double-rod symmetrical hydraulic cylinders are employed to simulate force environments in the EHFCS. Therefore, in order to improve the performance of the EHFCS, firstly, the model of the EHFCS is established with taking external disturbances, parameter uncertainties as well as structural vibrations into consideration. Secondly, in order to estimate external disturbances, parameter uncertainties and structural vibrations in the EHFCS and compensate them in the following robust controller design, two disturbance observers (DOs) are designed according to the nonlinear system model. Thirdly, with two estimation values from two DOs, a time-varying constraint-based robust controller (TVCRC) is presented in detail. Moreover, the stability of the proposed controller is analyzed by defining a proper Lyapunov functions. Finally, in order to validate the performance of the proposed controller, a series of simulation studies are conducted using the MATLAB/Simulink software. These simulation results give a fine proof of the efficiency of the proposed controller. What’s more, an experimental setup of the EHFCS is established to further validate the performance. Comparative experimental results show that the proposed controller exhibits better performance than the TVCRC without two DOs and a conventional proportional integral (PI) controller.

In this paper, position control using both a nonlinear position controller and a current controller with an augmented observer is proposed for a Brushless DC motor. The nonlinear position controller is designed to improve the position tracking performance based on the tracking error dynamics. The current controller is developed to track the desired currents generated from the desired torque, which is calculated based on the nonlinear position controller. The augmented observer is designed to obtain the knowledge of both state variables and disturbance. Closed-loop stability is proven through the Lyapunov theorem. Simulations were performed to evaluate the effectiveness of the proposed method.

In this article, we examine the dynamics of a Chikungunya virus (CHIKV) infection model with two routes of infection. The model uses four categories, namely, uninfected cells, infected cells, the CHIKV virus, and antibodies. The equilibrium points of the model, which consist of the free point for the CHIKV and CHIKV endemic point, are first analytically determined. Next, the local stability of the equilibrium points is studied, based on the basic reproduction number (R0) obtained by the next-generation matrix. From the analysis, it is found that the disease-free point is locally asymptotically stable if R0≤1, and the CHIKV endemic point is locally asymptotically stable if R0>1. Using the Lyapunov method, the global stability analysis of the steady-states confirms the local stability results. We then describe our design of an optimal recruitment strategy to minimize the number of infected cells, as well as a nonlinear optimal control problem. Some numerical simulations are provided to visualize the analytical results obtained.

In this study, a novel fast terminal sliding mode control technique based on the disturbance observer is recommended for the stabilization of underactuated robotic systems. The finite time disturbance observer is employed to estimate the exterior disturbances of the system and develop the finite time control law. The proposed controller can regulate the state trajectories of the underactuated systems to the origin within a finite time in the existence of external disturbances. The stability analysis of the proposed control scheme is verified via the Lyapunov stabilization theory. The designed control law is enough to drive a switching surface achieving the fast terminal sliding mode against severe model nonlinearities with large parametric uncertainties and external disturbances. Illustrative simulation results and experimental validations on a cart-inverted pendulum system are provided to display the success and efficacy of the offered method.

Abstract In this paper, we investigate the control of 4-D nonautonomous fractional-order uncertain model of a PI speed-regulated current-driven induction motor (FOIM) using a fractional-order adaptive sliding mode controller (FOASMC). First, we derive a dimensionless fractional-order model of the induction motor from the well-known integer -model of the induction motor. Various dynamic properties of the fractional-order induction motor, such as stability of the equilibrium points, Lyapunov exponents, bifurcation, and bicoherence, are investigated. An adaptive controller is derived to suppress the chaotic oscillations of the fractional-order model of the induction motor. Numerical simulations of the adaptive chaos suppression methodology are depicted for the fractional-order uncertain model of the induction motor to validate the analytical results of this work. A genetically optimized fractional-order PID (FOPID) controller is also derived to stabilize the states of the FOIM system. FPGA implementation of the proposed FOASMC is also presented to show that the proposed controller is hardware realizable.

In this paper, we investigate the global uniform practical exponential stability for a class of uncertain Takagi-Sugeno fuzzy systems. The uncertainties are supposed uniformly to be bounded by some known integrable functions to obtain an exponential convergence toward a neighborhood of the origin. Therefore, we use common quadratic Lyapunov function (CQLF) and parallel distributed compensation (PDC) controller techniques to show the global uniform practical exponential stability of the closed-loop system. Numeric simulations are given to validate the proposed approach.

Abstract The boundedness of chaotic systems plays an important role in investigating the stability of the equilibrium, estimating the Lyapunov dimension of attractors, the Hausdorff dimension of attractors, the existence of periodic solutions, chaos control, and chaos synchronization. However, as far as the authors know, there are only a few papers dealing with bounds of high-order chaotic systems due to their complex algebraic structure. To sort this out, in this paper, we study the bounds of a high-order Lorenz-Stenflo system arising in mathematical physics. Based on Lyapunov stability theory, we show that there exists a globally exponential attractive set for this system. The innovation of the paper is that we not only prove that this system is globally bounded for all the parameters, but also give a family of mathematical expressions of global exponential attractive sets of this system with respect to its parameters. We also study some other dynamical characteristics of this chaotic system such as invariant sets and chaotic behaviors. To justify the theoretical analysis, we carry out detailed numerical simulations.

Ebola virus disease (EVD) struck West Africa in 2013–2016 in an epidemic of unprecedented scope, with over 28000 cases and 11000 fatalities in the affected region. The protracted duration of the outbreak – more than two-and-one-half years of active transmission – raises questions about the persistence of EVD. In this brief paper, we qualitatively examine conditions supporting long-running EVD epidemics via a susceptible – exposed – infectious – recovered – deceased-infectious differential equations model that incorporates births and non disease-related deaths. We define an ‘effective epidemiological population’ to include contagious individuals recently deceased from the disease. Under a constant effective epidemiological population condition, we consider the basic reproductive number $ \mathcal R _{0} $ and use Lyapunov function arguments to establish conditions in the parameter space supporting an exchange of stability from the disease-free equilibrium to an endemic equilibrium.

In this paper, a robust sliding mode control (SMC) based on backstepping technique is studied for a microgyroscope in the presence of unknown model uncertainties and external disturbances using adaptive fuzzy compensator and fractional calculus. At first, the dynamic of microgyroscope is transformed into analogically cascade system to guarantee the application of backstepping design. Then a novel fractional differential sliding surface is proposed which integrates the capacities of the fractional calculus and SMC. In order to reduce the chattering in SMC, a fuzzy logical system is utilized to approximate the external disturbances. In addition, fractional order adaptive laws are derived to estimate the damping and stiffness coefficients and angular velocity online based on Lyapunov stability theory which also guarantees the stability of the closed loop system. Finally, simulation results signify the robustness and effectiveness of the proposed control schemes and the comparison of root mean square error under different fractional orders and integer order are given to demonstrate the better performance of proposed controller. [ABSTRACT FROM AUTHOR]

We establish Anderson localization for quasiperiodic operator families of the form... for all coupling constantsλ > 0 and all Diophantine frequencies α, provided that v is a 1-periodic function satisfying a Lipschitz monotonicity condition on [0; 1). The localization is uniform on any energy interval on which the Lyapunov exponent is bounded from below. [ABSTRACT FROM AUTHOR]

This paper presents a new formulation of the optimal control problem with uncertainty, in which an additive bounded function is considered as uncertainty. The purpose of the control is to ensure the achievement of terminal conditions with the optimal value of the quality functional, while the uncertainty has a limited impact on the change in the value of the functional. The article introduces the concept of feasibility of the mathematical model of the object, which is associated with the contraction property of mappings if we consider the model of the object as a one-parameter mapping. It is shown that this property is sufficient for the development of stable practical systems. To find a solution to the stated problem, which would ensure the feasibility of the system, the synthesized optimal control method is proposed. This article formulates the theoretical foundations of the synthesized optimal control. The method consists in making the control object stable relative to some point in the state space and to control the object by changing the position of the equilibrium points. The article provides evidence that this approach is insensitive to the uncertainties of the mathematical model of the object. An example of the application of the method for optimal control of a group of robots is given. A comparison of the synthesized optimal control method with the direct method on the model without disturbances and with them is presented.