Vol. 11, No. 1 (2022): Regular Issue

The paper presents a new scheme for sliding mode control (SMC) using single neuron PID (SNPID) to treat the shuttering signal output of SMC. This study develops a modified technique based on the combination of the SNPID, as a main controller and SMC, as an adaptation technique, to design an optimized self-tuned for SNPID controller that may overcome difficulties faced when a change in system operating points occurs. The purpose of the proposed controller is to track the reference speed of the electric vehicle (EV). A steady MATLAB/Simulink model was established and validated. It was then used to estimate the system performance. The optimal parameters of the proposed controller were obtained using the harmony search optimization based on an effective cost function. The simulation assumes a DC permanent magnet motor, ideal mechanical transmission. Two tests were executed, the first test was implemented at fixed reference speed while the second test was subjected at several commands of reference speed. The SNPID-SMC has been compared to the PID controller to ensure robustness. The obtained results can be summarized as follows. In the first test, The SNPID-SM controller reaches the steady-state speed at 8.8378 seconds while the PID controller stabilizes at 15.4530 seconds. Also, the SNPID-SM controller has a 0.06 % steady-state error however, the PID controller has a 4% steady-state error. Moreover, the settling time of SNPID-SMC is 14.4699 while the PID controller is 28.6823. Besides, the second test shows that the SNPID-SM controller can minimize the mean square by a percentage of 28.12 % compared to the PID controller. Lastly, the proposed SNPID-SM controller can enhance the dynamic response of EV significantly.

We introduce an ecological improvement of an eco-epidemiological model by improving the predation principles through predator switching between susceptible and infected prey. The model is analyzed for stability around the uninfected and coexisting equilibria to evaluate the thresholds that control the extinction and coexistence of the species. Next, we investigate the improved model to interpret the effect of changing the functional responses through predator switching. Applying the Arzel\`{a}-Ascoli theorem, we analyze the dynamics of the system around the origin. Numerical simulations are performed to validate the analytical findings. Finally, we conclude some eco-epidemiological comments made through mathematical and numerical observations.

The Halda, a 98-kilometre long major tributary of Karnaphuli River in the Chattogram Hill Tracts, is the only source of naturally fertilized eggs of carp fishes in South Asia and a great contributor to Bangladesh fisheries sector. Waste from large factories, Hathazari Peaking Power Plant and a housing estate are polluting the water body of Halda river to such an extent that the indigenous sweetwater brood fishes are facing death and the quantity of their release of carp spawn is decreasing. The present paper examines a predatorprey fishery system by taking into account the toxin waste which can lead to polluted system. Both fish species obey the logistic population growth with their respective environmental carrying capacities. The equilibria existed in the model are investigated together with the local and global stability. Bifurcation diagrams are studied to examine the dynamical behaviors of the system. Bionomic equilibria, optimal harvesting policy and Optimal Control Theory are applied to reduce the external toxic substance. Finally, a numerical simulation of the model has been discussed to illustrate the effect of toxicity and their control upon both the predator and the prey species.

In this paper we establish the existence of solutions of fractional integrodifferential equations of Sobolev type with deviating arguments. The solution representation is given by the resolvent operators and the existence is proved using the fixed point theorem. An example is provided to illustrate the theory.

We construct $2N$ real parameter solutions to the Burgers' equation in terms of determinant of order $N$ and we call these solutions, $N$ order solutions. We deduce general expressions of these solutions in terms of exponentials and study the patterns of these solutions in functions of the parameters for $N=1$ until $N=4$.

In this paper, we consider two-step iteration methods to solve a variational inclusion problem over the set of common fixed points of an infinite family of nonexpansive mappings on real reflexive and strictly convex Banach space with a uniformly Gateaux differentiable norm.

We study periodic ‘cyclic’ solutions of a population model of the cell cycle. In this model, clusters of cells in one fixed phase of the cell cycle may exert a negative influence on the progress of clusters in another phase. Previous results showed that stability of cyclic solutions is determined by the values of model parameters $s$ and $r$, and by which of two possible orderings of certain events the cyclic solution follows. The parameter triangle $\Delta = \{(s,r): 0 \le s \le r \le 1\}$ is subdivided into sub-triangles on which the stability of all cyclic solutions are the same. The stability for sub-triangles on the boundary of $\Delta$ was fully characterized in terms of number theoretic relationships between the number of clusters $k$ and certain indices of the sub-triangles. Interior sub-triangles with the order of events called sr1, were shown to have unstable solutions. In the present work, we focus on interior sub-triangles with the other order of events rs1. We show that when $k$ is prime, then the cyclic solutions are unstable for all interior sub-triangles. When $k$ is even, we show that there always exist a small number of sub-triangles on which cyclic solutions are at least neutrally stable. For $k$ odd and composite, we show that there are stable sub-triangles when $k = 9$ and $k = 15$ and no others.

Centre Manifold analysis of a $3-D$ nonlinear system with general second order nonlinearities have been worked out. The system is shown to possess two fixed points on the reduced $2-D$ centre manifold. By introducing a $2-D$ centre manifold one can show how an oscillatory dynamics may be generated in the system. We also state and prove a theorem to find the stability of the resultant centre manifold equation apriori from the parity of the nonlinear terms in the original equations. For a $2-D$ nonlinear model with the example picked up from biochemistry, the protein molecules in assembly, kinetic stability analysis is provided for the chosen example and establish herewith the validity of the theorem for our chosen example.

As a continuity to the study by M. M. Al-Gharabli et al. in [1], we consider a one-dimensional porous thermoelastic system with second sound, distributed delay term and nonlinear feedback. We show the well-posedness, using the semigroup theory, and establish an explicit and general decay rate result, using some properties of convex functions and the multiplier method. Our result is obtained under suitable assumption on delay without imposing any restrictive growth assumption on the damping term.

In this work, we investigate an initial boundary value problem related to the nonlinear hyperbolic equation $u_{tt} + u_{xxxx} + \alpha u_{xxxxt} = f(u_x)_x$, for $f(s) = |s|^\rho + |s|^\sigma$, $1 < \rho, \sigma, \alpha > 0$. Under suitable conditions, we prove the existence of global solutions and the exponential decay of energy. The nonlinearity $f(s)$ introduces some obstacles in the process of obtaining a priori estimates and we overcome this difficulty by employing an argument due to Tartar (1978). The exponential decay is obtained via an integral inequality introduced by Komornik (1994).

A detailed study on the effect of refuge on the phytoplankton-zooplankton ecosystem is explored. At first, the coexistence and stability conditions of different equilibria of the plankton system are analyzed. Our observations established that refuges have a strong impact on plankton dynamics. When the strength of phytoplankton crosses a certain critical value, the coexistence equilibrium loses its stability and enters into Hopf bifurcation that leads to oscillations of all species. The direction of the Hopf bifurcation is also established. Next, we used Pontryagins maximum principle to study a path of optimal harvesting policy. Also, we observed that the bottom-up and top-down effects like constant nutrient input, rate of zooplankton decay due to the toxic effect of phytoplankton play important roles for switching from one steady state to another that relates to the transcritical bifurcation. We derived the bifurcation scenarios when two different parameters vary together at the same time. At last, numerical simulations are implemented to support our results.

In this work, we consider a one-dimensional Timoshenko system of thermoelasticity of type III with infinite memory damped by weakly nonlinear feedbacks. Under suitable conditions, we establish the well-posedness of the problem using semigroups theory, and a general stability estimates using the multiplier method with no growth assumption on $f$ at the origin and without assuming equal or nonequal speeds of propagation of waves which is mentioned in numerous works (e.g. \cite{ayadi,chen,Fareh,jh,hao,masap}). Our results show that the damping effect leads to general decay rate for the energy function and also remove the necessity of the assumption on equal speeds which has been imposed in the prior literature.

This paper has the goal to broaden the incidence rate of an SEIRS epidemic model to a wide range of monotonic, concave incidence rates and some non-monotonic or concave cases. These incidence functions could reflect media education or psychological effect or mass action. The model takes into account relapse, recovery and immunity rates but without disease-induced death one. Applying the novel geometric approach we establish the global stability of the SEIRS model. Our analytical results reveal that the basic reproduction number completely determines the global stability of equilibria. Our conclusions are applied to two special incidence functions reflecting media and mass action.

Kinesiology is defined as the scientific study of human movement. The relation between various musculoskeletal movements can be diversified as physical activities, exercises, postures for health lifestyle. These can be partitioned mutually exclusively into many different ways. Different muscular movements are an asset of physical activities which are planned, structured and sometimes repetitive. The nonlinear differential model determines change in concentration for oxygen during musculoskeletal physical movements based on two major components heart and energy utilized by the Adenosine triphosphate (ATP) molecules using the compartment model of breath function. In this study, model utilizes non-linear model equalities which are with respect to time at constant rate of metabolism. Lyapunov Characteristic Exponents (LCE) measures the relative stability of the system of the equations. Lyapunov exponents are the most direct indicators and quantifiers of deterministic chaos. The mathematical model for three important components: Heart, Lungs and Cells/Tissues in the body is proposed. The model helps to study the impact of musculoskeletal movements on these factors simultaneously with time and also to study how the change in one component influences the changes in other with respect to time. Body consumes oxygen which is proportional to metabolic rate. It is observed that keeping breath function R constant at 20s and varying Q (amount of oxygen in the body) from 8(L/min) to 70 (L/min), the system experiences regular to chaotic behaviour. Further, it is observed that keeping Q as constant at 70 (L/min), chaotic situation can be controlled and the system be transformed to normal state by enhancing breath function. Thus, it is recommended that for extensive musculoskeletal movements of the body and to avoid collapsing, primarily the breath function should be boosted.

The purpose of this article is to analytically prove the exis\-tence and linear stability of three $\hat{T}$-periodic solutions (two strictly positive and one strictly negative) for a comb-drive actuator where the moveable finger is initially shifted a small distance $u>0$ from the center of the two fixed ones. Here we assume a damping force that is proportional to the velocity and an $AC$-$DC$ driving voltage $\hat{V}(\tau)>0$ with period $\hat{T}>0$. Under appropriate conditions over $\hat{V}_{\min}:=\min \hat{V}(\tau)$ and $\hat{V}_{\max}:=\max V(\tau)$, one of these solutions will be elliptic and the other two are hyperbolic. The basic tools for proving our results are the Lower and Upper Solution Method, Degree Theory and the Lyapunov-Zukovskii stability criterion for Hill's equation. Some numerical examples are provided to illustrate the results.