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In this paper, we develop a non-autonomous delay differential equation model for mosquito population suppression. After establishing the positiveness and boundedness of the solutions, we study the dynamical behaviours of the model with or without Wolbachia-infected male mosquitoes. More specifically, for the model without infected male mosquitoes, we analyse the asymptotic stability of the equilibria and demonstrate that the model undergo Hopf bifurcations under certain conditions. For the model incorporating infected male mosquitoes, we derive sufficient conditions for the global asymptotic stability of the origin. Numerical examples are provided to illustrate and support our theoretical findings. [ABSTRACT FROM AUTHOR]

Alopecia areata (AA) is an autoimmune disease whose clinical phenotype is characterized by the formation of distinct hairless patterns on the scalp or other parts of the body. In this paper, we study a three-component chemotaxis model for AA, which describes the complex interactions among CD 4 + T cells, CD 8 + T cells and interferon-gamma (IFN- γ ). Our first purpose is to establish the uniform boundedness of classical solutions for the model by self-map method, which extends the corresponding results of Lou and Tao (J Differ Equ 305:401–427, 2021, JDE) and Zhang et al. (Math Biosci Eng 20(5):7922–7942, 2023, MBE) to the case of arbitrary spatial dimensions and non-equidiffusive coefficients. Another purpose is to consider the globally asymptotic stability and convergence rate of the positive equilibrium under either (i) small proliferation rate and large degradation parameters or (ii) weak chemoattractive effect or strong random motions. It is shown under the above two cases that sparse patches occur in or around diseased hair follicles, gradually develop into diffuse or total hair loss and ultimately induce AA. [ABSTRACT FROM AUTHOR]

This paper deals with the modelling and stabilization of a flexible clamped beam controlled with a piezoelectric actuator in the self-sensing configuration. We derive the model starting from general principles, using the general laws of piezoelectricity. The obtained model is composed by a PDE, describing the flexible deformations dynamics, interconnected with an ODE describing the electric charge dynamics. Firstly, we show that the derived linear model is well-posed and the origin is globally asymptotically stable when a voltage control law, containing the terms estimated in the self-sensing configuration, is applied. Secondly, we make the more realistic assumption of the presence of hysteresis in the electrical domain. Applying a passive control law, we show the well-posedness and the origin's global asymptotic stability of the nonlinear closed-loop system. [ABSTRACT FROM AUTHOR]

Kidneys are the most commonly transplanted organs, and renal transplant is the best treatment for patients with advanced stages of renal disease. Immunosuppressive drugs are used after renal transplant to prevent the body from rejecting the transplanted kidney and ensure its proper kidney functioning. However, suppression of the immune system increases the risk of viral infections and other complications. Therefore, careful monitoring and management of immunosuppressive and antiviral drugs are essential for the success of the transplants. This article presents a hybrid fast non-singular integral terminal sliding mode control technique to adjust the efficacies of these drugs in renal transplant recipients, ensuring successful transplants and preventing viral infections. The proposed strategy tracks system trajectories to reference values and adjusts the treatment plan accordingly. The Lyapunov stability theorem is used to prove the asymptotic stability of the closed-loop system. Several simulation studies are conducted in MATLAB/Simulink environment to evaluate the performance of the proposed control technique in maintaining a balance between over-suppression and under-suppression. Genetic Algorithm is used to optimize the gain values to further improve the performance of the proposed control technique. Its performance is compared with two other variants of terminal sliding mode controllers to demonstrate its effectiveness against them. [Display omitted] • We have proposed an advanced nonlinear control for safe conduct of immunosuppressive treatment. • This controller is designed for both immunosuppressive and hybrid treatment modes. • The comparison with few other controllers depicts its superior performance. • We have optimized controller gains using Genetic algorithm, showcasing the benefits over a trial-and-error approach. • Simulation results highlighted fast convergence, negligible steady-state error, singularity handling, and reduced chattering. [ABSTRACT FROM AUTHOR]

This paper studies a dynamical system that models the free recall dynamics of working memory. This model is an attractor neural network with n modules, named hypercolumns, and each module consists of m minicolumns. Under mild conditions on the connection weights between minicolumns, the authors investigate the long-term evolution behavior of the model, namely the existence and stability of equilibria and limit cycles. The authors also give a critical value in which Hopf bifurcation happens. Finally, the authors give a sufficient condition under which this model has a globally asymptotically stable equilibrium consisting of synchronized minicolumn states in each hypercolumn, which implies that in this case recalling is impossible. Numerical simulations are provided to illustrate the proposed theoretical results. Furthermore, a numerical example the authors give suggests that patterns can be stored in not only equilibria and limit cycles, but also strange attractors (or chaos). [ABSTRACT FROM AUTHOR]

The study of orbits about the artificial equilibrium points of the solar sail problem (or satellite equipped with solar sail) is an important part for a mission design. This paper investigates about the motion of a solar sail in the presence of oblate primaries and Stokes drag effect. First, we have formulated solar sail problem and then determined the artificial equilibrium points (AEPs). It is found that due to Stokes drag, collinear AEPs do not exist but two non-collinear AEPs ( L ¯ 4 , 5) exist, which are asymptotically stable with respect to all values of oblateness (A 1 , 2) as well as dissipative constant (k), whereas relative to sail lightness number (β) , these are asymptotically stable for the range 0 ≤ β < 0.4102 . Again, the long as well as short periodic orbits are determined and impact of perturbing factors are observed by finding the amplitude, time period and phase of the respective orbits. Further, tadpole orbit of the solar sail near AEPs L ¯ 4 , 5 are computed and effect of perturbations are analysed on the basis of number of loops and its shape in the orbits. [ABSTRACT FROM AUTHOR]

Gross Domestic Product (GDP) growth and national debt are like two faces of the same coin. The national debt is the major source of growth of GDP. GDP is completely paralyzed in the absence of national debt. The national debt in turn is hugely dependent on foreign funding. The GDP is growing faster as a result of these investments. It is believed that the external debt will never be entirely settled. It takes some time for the agreement to mature before external investments become available in response to demand. The primary topic of this study is the delay in foreign investment’s real arrival and how it affects the dynamics of GDP and national debt. We investigate this impact with a delay parameter τ. The stability analysis is done on the system and the nonzero equilibrium is computed. For a crucial delay parameter value, Hopf bifurcation is seen. The research plays a significant role in economic growth. [ABSTRACT FROM AUTHOR]

This paper is devoted to studying the initial-boundary value problem for the radiative full Euler equations, which are a fundamental system in the radiative hydrodynamics with many practical applications in astrophysical and nuclear phenomena, with the non-slip boundary condition on an impermeable wall. Due to the difficulty from the disappearance of the velocity on the impermeable boundary, quite few results for compressible Navier-Stokes equations and no result for the radiative Euler equations are available at this moment. So the asymptotic stability of the rarefaction wave proven in this paper is the first rigorous result on the global stability of solutions of the radiative Euler equations with the non-slip boundary condition. It also contributes to our systematical study on the asymptotic behaviors of the rarefaction wave with the radiative effect and different boundary conditions such as the inflow/outflow problem and the impermeable boundary problem in our series papers including [5,6]. [ABSTRACT FROM AUTHOR]

Under study is the stability of Persidskii systems with distributed delay. We assume that the sector-type functions on the right-hand sides of the system are essentially nonlinear. Also, we propose some original construction of the Lyapunov–Krasovskii functional of use in deriving new asymptotic stability conditions of the zero solution. The approach is applied to the stability analysis of the Lurie indirect control system and a mechanical system with essentially nonlinear positional forces. Using some development of the averaging method, we obtain the conditions that guarantee stability under nonstationary perturbations with zero mean values for the systems under study. [ABSTRACT FROM AUTHOR]

The system of equations describing motion of compressible viscoelastic fluids is considered in a one dimensional half space under the outflow boundary condition. We investigate the existence and stability of stationary solutions. It is shown that the stationary solution exists for large Mach number and small number of propagation speed of elastic wave. We next show that the stationary solution is asymptotically stable, provided that the initial perturbation is sufficiently small. [ABSTRACT FROM AUTHOR]

For the issue of tracking control of motor servo system with input dead-zone, a novel robust adaptive prescribed performance control is proposed. Firstly, a smooth dead-zone inverse model is introduced and parameterized, which can help compensate for dead-zone. Secondly, the prescribed performance function is used to constrain the convergence process of tracking error. Then, a robust adaptive controller is designed based on the estimation of the upper bound of disturbance to weaken the influence from disturbance. Comparative tracking verification under two position command cases is carried out and the simulation results show that the proposed controller can improve the tracking accuracy well. [ABSTRACT FROM AUTHOR]

This article considers the problem of output feedback stabilization for a class of stochastic high‐order nonlinear time‐delay systems with unknown output function. For stochastic high‐order nonlinear time‐delay systems, based on the Lyapunov stability theorem, by combining the addition of one power integrator and homogeneous domination method, the maximal open sector Δ$\Delta$ of output function is given. As long as output function belongs to any closed sector included in Δ$\Delta$, an output feedback controller can be developed to guarantee the closed‐loop system globally asymptotically stable in probability. [ABSTRACT FROM AUTHOR]

We provide the well-posedness for a partially dissipative viscous system of balance laws in smooth Sobolev spaces under the same assumptions as in the case of inviscid balance laws. A priori estimates for coupled hyperbolic-parabolic linear systems with coefficients having limited regularity are derived using Friedrichs regularization and Moser-type estimates. Local existence for nonlinear systems will be established using the results of the linear theory and a suitable iteration scheme. The local existence theory is then applied to the Kuznetsov–Westervelt equation with damping for nonlinear wave acoustic propagation. Existence of global solutions for small data and their asymptotic stability are established. [ABSTRACT FROM AUTHOR]

This paper is devoted to the asymptotic stability of diverging traveling waves for reaction–advection–diffusion equation u t - Δ u + α (t , y) u x = f (t , y , u) in cylinders. By the sliding method, we first establish a Liouville-type result. Then, using the Liouville-type result and truncation technique, we prove the asymptotic stability of the diverging traveling wave. [ABSTRACT FROM AUTHOR]

We prove the asymptotic stability of a finite sum of well-ordered solitary waves for the Zakharov–Kuznetsov equation in dimensions two and three. We also derive a qualitative version of the orbital stability result, which will be useful for studying the collision of two solitary waves in a forthcoming paper. The proof extends the ideas of Martel, Merle and Tsai for the sub-critical gKdV equation in dimension one to the higher-dimensional case. It relies on monotonicity properties on oblique half-spaces and rigidity properties around one solitary wave introduced by Côte, Muñoz, Pilod and Simpson in dimension two, and by Farah, Holmer, Roudenko and Yang in dimension three. [ABSTRACT FROM AUTHOR]

In this research, a new hybrid backstepping control strategy based on a neural network is proposed for tractor-trailer mobile manipulators in the presence of unknown wheel slippage and disturbances. To minimize the negative impacts of wheel slippage, the desired velocities of the tractor's wheels are computed with a proposed kinematic control model with an adaptive term. As the system's dynamical model contains unavoidable uncertainties, model-based backstepping control technique is unable to effectively manage these systems. Hence, the proposed controller blends a radial basis function neural network with the merits of a dynamical model-based backstepping approach. The neural networks are employed to approximate the non-linear unknown smooth function. To minimize the impact of external disturbances, and network reconstruction error an adaptive term is added to the control law. The Lyapunov theorem and Barbalat's lemma are employed to guarantee the stability of the control method. The tracking error is shown to be bounded and to rapidly converge to zero with the proposed method. To demonstrate the efficacy and validity of the control mechanism, comparison simulation results are presented. • A hybrid controller is proposed for tractor-trailer systems with wheel slippage. • By putting forward a kinematic control law, the desired velocities are calculated. • At the dynamical level, RBFNN is used to approximate the uncertainties of the system. • The stability of the system is analyzed with Lyapunov theory and Barbalat's Lemma. • The proposed control scheme is verified through simulation studies. [ABSTRACT FROM AUTHOR]

In this study, we investigate the stability and asymptotic stability properties of Caputo fractional time-dependent systems with delay by employing vector Lyapunov functions. Utilizing the Caputo fractional Dini derivative on Lyapunov-like functions, along with a new comparison theorem and differential inequalities, we derive and prove sufficient conditions for the stability and asymptotic stability of these complex systems. An example is included to showcase the method's practicality and to specifically illustrate its advantages over scalar Lyapunov functions. Our results improves, extends, and generalizes several existing findings in the literature. [ABSTRACT FROM AUTHOR]

In this article, we are concerned with the asymptotic stability of the rarefaction wave for the one-dimensional isentropic compressible Navier-Stokes-Korteweg equations under initial perturbations, which tend to space-periodic functions at infinities. It is shown that the solution of the one-dimensional isentropic compressible Navier-Stokes-Korteweg equations tends to background rarefaction wave as time tends to infinity, provided that the initial perturbation and strength of the rarefaction wave are suitably small. The proof is based on the delicate constructions of the proper ansatzes, which capture the interactions between the background waves and the periodic perturbations, and the energy method in Eulerian coordinates involving the effect of the Korteweg stress term. [ABSTRACT FROM AUTHOR]

As presented in this paper, we explore the control of a discrete-time two–dimensional (2-D) system using the Lyapunov approach. The Giovane–Roesser model (G–R) for 2-D systems was introduced, and we presented the asymptotic stability analysis for this class of systems while maintaining the strictly bounded real (SBR) property. In the next step, we solve the stability problem in the presence of uncertainties in the system while preserving the SBR condition. We design state feedback and output feedback controllers to control 2-D discrete-time systems with preceding uncertainties, introducing algorithms to design such controllers. In order to ensure the validity of our findings, we present the simulation results as numerical and practical examples. [ABSTRACT FROM AUTHOR]

The system of equations describing motion of compressible viscoelastic fluids is considered in a one dimensional half space under the outflow boundary condition. We investigate the existence and stability of stationary solutions. It is shown that the stationary solution exists for large Mach number and small number of propagation speed of elastic wave. We next show that the stationary solution is asymptotically stable, provided that the initial perturbation is sufficiently small.

Abstract This article considers the problem of output feedback stabilization for a class of stochastic high‐order nonlinear time‐delay systems with unknown output function. For stochastic high‐order nonlinear time‐delay systems, based on the Lyapunov stability theorem, by combining the addition of one power integrator and homogeneous domination method, the maximal open sector Δ of output function is given. As long as output function belongs to any closed sector included in Δ, an output feedback controller can be developed to guarantee the closed‐loop system globally asymptotically stable in probability.

This work focuses on the asymptotic stability of nonlocal diffusion equations in N$$ N $$‐dimensional space with nonlocal time‐delayed response term. To begin with, we prove L2$$ {L}&#x0005E;2 $$ and L∞$$ {L}&#x0005E;{\infty } $$‐decay estimates for the fundamental solution of the linear time‐delayed equation by Fourier transform. For the considered nonlocal diffusion equation, we show that if l>p$$ l>\left&#x0007C;p\right&#x0007C; $$, then the solution u(t,x)$$ u\left(t,x\right) $$ converges globally to the trivial equilibrium time‐exponentially. If l=p$$ l&#x0003D;\left&#x0007C;p\right&#x0007C; $$, then the solution u(t,x)$$ u\left(t,x\right) $$ converges globally to the trivial equilibrium time‐algebraically. Furthermore, it can be proved that when r>q$$ r>\left&#x0007C;q\right&#x0007C; $$, the solution u(t,x)$$ u\left(t,x\right) $$ converges globally to the positive equilibrium time‐exponentially, and when r=q$$ r&#x0003D;\left&#x0007C;q\right&#x0007C; $$, the solution u(t,x)$$ u\left(t,x\right) $$ converges globally to the positive equilibrium time‐algebraically. Here, l,p,r$$ l,p,r $$, and q$$ q $$ are the coefficients of each term contained in the linear part of the nonlinear term f$$ f $$. All convergence rates above are L2$$ {L}&#x0005E;2 $$ and L∞$$ {L}&#x0005E;{\infty } $$‐decay estimates. The comparison principle and low‐frequency and high‐frequency analyses are significantly effective in proofs. Finally, our theoretical results are supported by numerical simulations in different situations. [ABSTRACT FROM AUTHOR]

In this paper, we consider the two-species chemotaxis-competition system with loop and nonlocal kinetics { u t = Δ u − χ 11 ∇ ⋅ (u ∇ v) − χ 12 ∇ ⋅ (u ∇ z) + f 1 (u , w) , x ∈ Ω , t > 0 , 0 = Δ v − v + u + w , x ∈ Ω , t > 0 , w t = Δ w − χ 21 ∇ ⋅ (w ∇ v) − χ 22 ∇ ⋅ (w ∇ z) + f 2 (u , w) , x ∈ Ω , t > 0 , 0 = Δ z − z + u + w , x ∈ Ω , t > 0 , subject to homogeneous Neumann boundary conditions in a smooth bounded domain Ω ⊂ R n (n ≥ 1) , where χ i j > 0 (i , j = 1 , 2) , f 1 (u , w) = u (a 0 − a 1 u − a 2 w − a 3 ∫ Ω u d x − a 4 ∫ Ω w d x) , f 2 (u , w) = w (b 0 − b 1 u − b 2 w − b 3 ∫ Ω u d x − b 4 ∫ Ω w d x) with a i , b i > 0 (i = 0 , 1 , 2) , a j , b j ∈ R (j = 3 , 4). It is shown that if the parameters satisfy certain conditions, then the corresponding initial boundary value problem admits a unique global-in-time classical solution in any spatial dimension, which is uniformly bounded. Moreover, based on the construction of suitable energy functionals, the globally asymptotic stabilization of coexistence and semi-coexistence steady states is considered. Our results generalize and improve some previous results in the literature. [ABSTRACT FROM AUTHOR]

This study focuses on formulating and analysing a COVID-19 transmission dynamics model using integer and fractional-order derivatives in the Caputo sense. The model considers two doses of vaccination, confinement, and treatment with limited resources. The control reproduction number is computed and the asymptotic stability analysis of the disease-free equilibrium is proved. We also prove the existence of at least one endemic equilibrium whenever $ \mathcal{R}_c>1 $. Using real data from Germany, we calibrate our models by performing parameter estimations. We find that the control reproduction number is approximately equal to 1.90, which implies that the disease remains endemic in Germany. We also perform global sensitivity analysis by computing partial rank correlation coefficients (PRCC) between $ \mathcal{R}_c $ (respectively compartments of infected individuals) and each model parameter. By fixing vaccine coverage at 70%, we observe that it might be more effective to increase the vaccine efficacy than increasing the numbers of vaccinated people. After that, we formulate the corresponding fractional model in the Caputo sense, proving positivity, boundedness, existence, and uniqueness of solutions. We calculate the control reproduction number of the fractional model, and prove the asymptotic stability of the DFE, and existence of at least one endemic equilibrium point. We find from numerical simulations, that for a long-term forecasting, it seems better to use a fractional derivative in the range $ \varphi\in (0,0.87] $ than using just ordinary derivatives. Indeed, for this range of the fractional-order parameter, daily detected cases are closer to those the classical model predicts. [ABSTRACT FROM AUTHOR]

In this study, we investigate the stability and asymptotic stability properties of Caputo fractional time-dependent systems with delay by employing vector Lyapunov functions. Utilizing the Caputo fractional Dini derivative on Lyapunov-like functions, along with a new comparison theorem and differential inequalities, we derive and prove sufficient conditions for the stability and asymptotic stability of these complex systems. An example is included to showcase the method's practicality and to specifically illustrate its advantages over scalar Lyapunov functions. Our results improves, extends, and generalizes several existing findings in the literature.

The paper below is a written version of the 17th Andrzej Lasota Lecture presented on January 12th, 2024 in Katowice. During the lecture we tried to show the impact of Andrzej Lasota’s results on the author’s research concerning various fields of mathematics, including chaos and ergodicity of dynamical systems, Markov operators and semigroups and partial differential equations.

Abstract This paper presents various bifurcations of the McMillan map under perturbations of its coefficients, such as period-doubling, pitchfork, and hysteresis bifurcation. The associated existence regions are located. Using the quasi-Lyapunov function method, the existence of asymptotically stable fixed point is also demonstrated.

We study the long time behavior of the generalized solution of the flow and thermal explosion model of the reactive real micropolar gas. The dynamics of the chemical reaction involved and the usual laws of conservation of mass, momentum, angular momentum, and energy generate a complex governing system of partial differential equations. The fluid is nonideal and non‐Newtonian. In this work, we prove that the problem can be solved in an infinite time domain and establish the asymptotic properties of the solution. Namely, we conclude that for certain parameter values, the solution stabilizes exponentially to a steady‐state solution, while for others the stabilization occurs but at power decay rate. At the end, we conducted a few numerical tests whereby we experimentally confirmed theoretical findings about long‐term behavior of the solution. [ABSTRACT FROM AUTHOR]

On the basis of the known equations of motion of the system of coupled gyrostats by P.V. Kharlamov and the functions of state by S.L. Sobolev, the equations of rotation in a medium with resistance of a free system of two elastically connected rigid bodies with cavities completely filled with an ideal incompressible fluid were derived. Rigid bodies are connected by an elastic restoring spherical joint. Assuming that the center of mass of the rigid bodies is located on the third main axis of inertia and the fluid is ideal, the equation of disturbed motion of the considered mechanical system is obtained in the form of a countable system of ordinary differential equations. In the case of two Lagrangian gyroscopes with arbitrary axisymmetric cavities filled with an ideal fluid, a transcendental characteristic equation has been derived. Taking into account the fundamental tone of liquid oscillations, a characteristic equation of the sixth order was obtained, and on the basis of the Lenard–Schipar criterion, it was written in the innor form, and the conditions for the asymptotic stability of uniform rotation of Lagrange gyroscopes with a liquid were written out in the form of a system of five inequalities. These inequalities are presented in the form of the first, third, sixth, and eighth powers with respect to the coefficient of the spherical joint elasticity. It was proved that if the first tones of liquid oscillations in two cavities are greater than one and do not coincide, then this is sufficient for the higher inequality coefficients to be positive. It was shown that if the first oscillation tones coincide, only the degree of the last inequality decreases, while the higher inequality coefficients remain positive; therefore, internal resonance is impossible. Thus, when the first tones of fluid oscillations are greater than one, the asymptotic stability will always be possible with the increase in the elasticity coefficient. For ellipsoidal cavities, this means that they must be oblate along the axis of rotation. It was shown that in the absence of the spherical joint elasticity, the characteristic equation has a zero root, and the conditions of stability are already presented in the form of a system of four inequalities, which are only necessary. [ABSTRACT FROM AUTHOR]

The present paper proposes the use of fractional derivatives in the definition of sliding function, giving a new mode control applied to induction motor drives in electric vehicle (EV) applications. The proposed Fractional-Order Sliding Mode Direct Torque Control-Space Vector Modulation (FOSM-DTC-SVM) strategy aims to address the limitations of conventional control techniques and mitigate torque and flux ripples in induction motor systems. The paper first introduces the motivation for using fractional-order control methods to handle the nonlinear and fractional characteristics inherent in induction motor systems. The core describes the proposed FOSM-DTC-SVM control strategy, which leverages a fractional sliding function and the associated Lyapunov stability analysis. The efficiency of the proposed strategy is validated via three scenarios. (i) The first scenario, where the acceleration of the desired speed is defined by pulses, leading to Dirac impulses in its second derivative, demonstrates the advantage of the proposed control approach in tracking the desired speed while minimizing flux ripples and generating pulses in the rotor pulsation. (ii) The second scenario demonstrates the effectiveness of filtering the desired speed to eliminate Dirac impulses, resulting in smoother rotor pulsation variations and a slightly slower speed response while maintaining similar flux ripples and stator current characteristics. (iii) The third scenario consists of eliminating the fractional derivatives of the pulses existing in the expression of the control, leading to the elimination of Dirac impulses. These results demonstrate the potential of the FOSM-DTC-SVM to revolutionize the performance and efficiency of EVs. By incorporating fractional control in the control scheme for PV-powered EVs, the paper showcases a promising avenue for sustainable transportation. [ABSTRACT FROM AUTHOR]

The Jungle Game is used in population dynamics to describe cyclic competition among species that interact via a food chain. The dynamics of the Jungle Game supports a heteroclinic network whose cycles represent coexisting species. The stability of all heteroclinic cycles in the network for the Jungle Game with four species determines that only three species coexist in the long-run, interacting under cyclic dominance as a Rock–Paper–Scissors Game. This is in stark contrast with other interactions involving four species, such as cyclic interaction and intraguild predation. We use the Jungle Game with four species to determine the success of a fourth species invading a population of Rock–Paper–Scissors players. [ABSTRACT FROM AUTHOR]

In this manuscript, it is investigated that the stability property of a discrete-time switched system consisting of all unstable subsystems. Under a switching signal with certain conditions, the sufficient constraints for asymptotic stability of a discrete-time switched system composed of all unstable subsystems are obtained via Lyapunov functions and the defined divergence time. Furthermore, based on this result, linear matrix inequalities are obtained for asymptotic stability of a linear discrete-time switched system composed of all unstable subsystems. The efficiency of the acquired theorems is exhibited by three numerical examples. [ABSTRACT FROM AUTHOR]

The relationship is studied here between the 3D incompressible Brinkman–Forchheimer problem with delay and its generalized steady state. First, with some restrictive condition on the delay term, the global well-posedness of 3D Brinkman–Forchheimer problem and its steady state problem are obtained by compactness method and Brouwer fixed point method respectively. Then the global L p (2 ≤ p < ∞) decay estimates are established for weak solution of non-autonomous Brinkman–Forchheimer equations with delay by using a retarded integral inequality. The global decay estimates can be proved for strong solution as well. Finally, the exponential stability property is investigated for weak solution of the 3D non-autonomous Brinkman–Forchheimer problem by a direct approach and also for the autonomous system by using a retarded integral inequality. Furthermore, the Razumikhin approach is utilized to achieve the asymptotic stability for strong solution of autonomous system under a relaxed restriction. [ABSTRACT FROM AUTHOR]

This article explores the existence and asymptotic stability in the p-th moment of mild solutions to a class of Hilfer fractional neutral stochastic differential equations with infinite delay in Hilbert spaces. To prove our main results, we use fractional calculus, stochastic analysis, semigroup theory, and the Krasnoselskii-Schaefer type fixed point theorem. Moreover, a set of novel sufficient conditions is derived for achieving the required result. Following that, we extend the given system to the Sobolev type and provided the existence results of the considered system. After that, we provided an example to illustrate the validity of our results. [ABSTRACT FROM AUTHOR]

We consider the problem of stability of nonzero integral manifolds of a nonlinear finitedimensional system of ordinary differential equations whose right-hand side is a periodic vector-valued function of the independent variable containing a parameter. We assume that the system has a trivial integral manifold for all values of the parameter and the corresponding linear subsystem does not possess the property of exponential dichotomy. The aim of this work is to find sufficient conditions for stability, instability, and asymptotic stability of a local nonzero integral manifold. For this purpose, we use the method of Lyapunov functions modified to the problem considered and singularities of the right-hand sides of the system. [ABSTRACT FROM AUTHOR]

The stability of systems of linear autonomous functional differential equations of neutral type is studied. The study is based on the well-known representation of the solution in the form of an integral operator, the kernel of which is the Cauchy function of the equation under study. The definitions of Lyapunov, asymptotic, and exponential stability are formulated in terms of the corresponding properties of the Cauchy function, which allows us to clarify a number of traditional concepts without loss of generality. Along with the concept of asymptotic stability, a new concept of strong asymptotic stability is introduced. The main results are related to the stability with respect to the initial function from the spaces of summable functions. In particular, it is established that strong asymptotic stability with initial data from the space L1 is equivalent to the exponential estimate of the Cauchy function and, moreover, exponential stability with respect to initial data from the spaces Lp for any p ≥ 1. [ABSTRACT FROM AUTHOR]

For several species of microorganisms, a competition model described by a system of nonlinear differential equations with an infinite distributed delay is considered. The asymptotic stability of the equilibrium point corresponding to the survival of only one species and extinction of the others is studied. Conditions on the initial species population sizes and the initial nutrient concentration are indicated under which the system reaches the equilibrium. Additionally, the stabilization rate is estimated. The results are obtained using a Lyapunov–Krasovskii functional. [ABSTRACT FROM AUTHOR]

We address the stability of stationary solutions to a class of 2D non-newtonian fluid equations, when the external force contains hereditary characteristics involving unbounded delays. Firstly, when the unbounded variable delay is driven by a continuously differential function, we establish the stability of nontrivial weak stationary solutions and the asymptotic stability of trivial stationary solution. Then when the general unbounded delay is continuous with respect to time, the stability of nontrivial strong stationary solutions is also obtained. Eventually, when the proportional delay is considered, the polynomial stability of trivial stationary solution is verified. [ABSTRACT FROM AUTHOR]

This paper investigates the Cauchy problem for time‐space fractional Keller‐Segel‐Navier‐Stokes model in ℝd(d≥2)$$ {\mathrm{\mathbb{R}}}^d\kern0.1em \left(d\ge 2\right) $$, which can describe the memory effect and anomalous diffusion of the considered system. The local and global existence and uniqueness in weak Lp$$ {L}^p $$ space are obtained by means of abstract fixed point theorem. Moreover, we explore the asymptotic stability of solutions as time goes to infinity. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a discrete-time prey-predator model. The model is formulated by using the piecewise constant argument method for differential equations and taking into account Holling type III. The existence and local behavior of equilibria are studied. We established that the system experienced both Neimark-Sacker and perioddoubling bifurcations analytically by using bifurcation theory and the center manifold theorem. In order to control chaos and bifurcations, the state feedback method is implemented. Numerical simulations are also provided for the theoretical discussion. [ABSTRACT FROM AUTHOR]

This present work develops a nonlinear SIRS fractional-order model with a system of four equations in the Caputo sense. This study examines the impact of positive and negative attitudes towards vaccination, as well as the role of government actions, social behavior and public reaction on the spread of infectious diseases. The local stability of the equilibrium points is analyzed. Sensitivity analysis is conducted to calculate and discuss the sensitivity index of various parameters. It has been established that the illness would spread across this system when the basic reproduction number is larger than 1, the system becomes infection-free when the reproduction number lies below its threshold value of 1. Numerical figures depict the effects of positive and negative attitudes towards vaccination to make the system disease-free sooner. A comprehensive study regarding various values of the order of fractional derivatives together with integer-order derivatives has been discussed in the numerical section to obtain some useful insights into the intricate dynamics of the proposed system. The Pontryagin principle is used in the formulation and subsequent discussion of an optimum control issue. The study also reveals the significant role of government actions in controlling the epidemic. A numerical analysis has been conducted to compare the system's behavior under optimal control and without optimal control, aiming to discern their differences. The policies implemented by the government are regarded as the most adequate control strategy, and it is determined that the execution of control mechanisms considerably diminishes the ailment burden. [ABSTRACT FROM AUTHOR]

In this paper, a viscous shock wave under space-periodic perturbation of 1-D isentropic Navier-Stokes system in the half space is investigated. It is shown that if the initial periodic perturbation around the viscous shock wave is small, then the solution time asymptotically tends to a viscous shock wave with a shift partially determined by the periodic oscillations. Moreover the strength of shock wave could be arbitrarily large. This result essentially improves the previous work Matsumura and Mei (1999) [14] where the strength of shock satisfies some restrictions and the initial periodic oscillations vanish. [ABSTRACT FROM AUTHOR]

This paper presents various bifurcations of the McMillan map under perturbations of its coefficients, such as period-doubling, pitchfork, and hysteresis bifurcation. The associated existence regions are located. Using the quasi-Lyapunov function method, the existence of asymptotically stable fixed point is also demonstrated. [ABSTRACT FROM AUTHOR]

Several types of linear and nonlinear singularly perturbed time-delay differential systems are considered. Asymptotic stability of the linear systems and asymptotic stability of the trivial solution of the nonlinear systems, valid for any sufficiently small value of the parameter of singular perturbation, are analyzed. For the stability analysis in the linear case, a partial exact slow–fast decomposition of the original system and an application of the Symmetric Matrix Riccati Equation method are proposed. Such an analysis yields parameter-free conditions, providing the asymptotic stability of the considered linear singularly perturbed time-delay differential systems for any sufficiently small value of the parameter of singular perturbation. Using the asymptotic stability results for the considered linear systems and the method of asymptotic stability in the first approximation, parameter-free conditions, guaranteeing the asymptotic stability of the trivial solution to the considered nonlinear systems for any sufficiently small value of the parameter of singular perturbation, are derived. Illustrative examples are presented. [ABSTRACT FROM AUTHOR]

We extend the detailed study of the linearized dynamics obtained for cnoidal waves of the Korteweg–de Vries equation by Rodrigues [J. Funct. Anal. 274 (2018), pp. 2553–2605] to small-amplitude periodic traveling waves of the generalized Korteweg–de Vries equations that are not subject to Benjamin–Feir instability. With the adapted notion of stability, this provides for such waves, global-in-time bounded stability in any Sobolev space, and asymptotic stability of dispersive type. When doing so, we actually prove that such results also hold for waves of arbitrary amplitude satisfying a form of spectral stability designated here as dispersive spectral stability. [ABSTRACT FROM AUTHOR]

We consider a system of autonomous nonlinear ordinary differential equations for which the existence conditions for an invariant manifold are satisfied in the case where this manifold is central. It is well known that the theorem on the existence of a central invariant manifold cannot be supplemented with the statement of its uniqueness. We obtain sufficient conditions that guarantee the uniqueness of the central invariant manifold. [ABSTRACT FROM AUTHOR]

To solve the control problem of time-varying state-scale nonlinear systems whose initial state is not affected by settling time, fixed-time convergence algorithms are proposed for first-order systems and higher-order systems in this paper. First, a scalar model is used to illustrate how the time-varying feedback parameter can guarantee that the system achieves asymptotic stability while achieving finite-time convergence, and it is proved that the settling time obtained in this paper is only related to the prescribed boundary. This allows us to design the settling time with an appropriate parameter based on the prescribed boundary. To exhibit the effectiveness and extensibility of the proposed algorithm for first-order scalar systems, the results are subsequently extended to general higher-order systems based on the backstepping method. By introducing numerical simulation results, this paper verifies that the proposed algorithm will make the system achieve asymptotic stability and its output can converge to a given boundary, regardless of the system's initial states. [ABSTRACT FROM AUTHOR]

This paper proposes a novel stable economic model predictive control (EMPC) strategy for constrained nonlinear systems with changing economic criteria. The traditional EMPC may lead to infeasibility and even instability of the closed-loop system when the criterion has been changed. Firstly, a generalised terminal constraint is introduced to ensure the recursive feasibility of the economic optimisation problem, which enforces the terminal states to converge to an arbitrary equilibrium state rather than a predetermined fixed one at the initial time. Then the stability constraint is constructed by solving an auxiliary optimisation problem online, which is the key to guaranteeing the asymptotic stability of the closed-loop system. Finally, sufficient conditions for feasibility and asymptotic stability are derived in the context of changing economic criteria. The main feature of the proposed strategy is that it does not need to know the changes of economic criterion in advance, and its properties and effectiveness are exemplified by simulations on a nonlinear chemical process example. [ABSTRACT FROM AUTHOR]

This paper proposes a tube-based model predictive control strategy for linear systems with bounded disturbances and input delay to ensure input-to-state stability. Firstly, the actual disturbed system is decomposed into a nominal system without disturbances and an error system. For the nominal system, solving an optimization problem, where the delayed control input is set as an optimization variable, yields a nominal control law that enables the nominal state signal to approach to zero. Then, for the error system, the Razumikhin approach is used to identify a robust control invariant set. Using the set invariance theorem, an ancillary control law is developed to confine the error state signal in the invariant set. Combining the two results, we obtain a control law that enables the state signal to remain within a robustly invariant tube. Finally, the effectiveness of the developed control strategy is validated by simulations. • An ancillary control is used to ensure the state errors stay in the invariant set. • A constraint on the delayed input is added to ensure feasibility of the algorithm. • By introducing a state error term, a novel Lyapunov function is proposed. [ABSTRACT FROM AUTHOR]