فصلنامه مدل سازی پیشرفته ریاضی
سال سیزدهم شماره 2 (تابستان 1402)

In this paper, we propose a fuzzy sliding mode controller (FSMC) for Multi-agent Systems (MAS) and investigate the leader-follower consensus problem for the network of nonlinear dynamic agents with an undirected graph topology. A new sliding mode controller is suggested to address the consensus problem in MASs. The proposed sliding mode controller is based on a separating hyperplane. In addition, a fuzzy controller is designed to eliminate the chattering phenomenon. According to the communication graph topology and the Lyapunov stability condition, the proposed FSMC satisfies the consensus condition. As an advantage of the control presented in this paper, the states of the system reach the sliding surface very quickly and remain on the surface. The advantage of the proposed approach is also illustrated by simulation results.

In this paper, we investigate the estimation of fixed point errors for two important fixed point theorems of quasi-contractive mappings in fuzzy norm spaces. For this purpose, we define 􀀀generalized contraction and ("; )􀀀uniformly generalized contraction functions on fuzzy metric spaces and show that these definition are generalization of contractive functions defined by Ćirić on classical metric space. Also, when Picard’s iteration is used to approximate fixed points in fuzzy norm spaces, we obtain complete expressions for Ćirić’s fixed point theorems, including estimates of the fuzzy error.

‎In this paper‎, ‎we consider a diffusive predator-prey model‎, ‎in which the prey population lives in groups and has a social behavior‎. ‎We show that Hopf bifurcation and the existence of a center manifold may occur‎. ‎The linear stability analysis shows that a Hopf bifurcation occurs in the corresponding homogeneous system‎. ‎Next‎, ‎we study the effect of diffusion parameters on homogeneous dynamics‎. ‎By choosing a proper bifurcation parameter‎, ‎we prove that a Hopf bifurcation occurs in the nonhomogeneous system‎. ‎We compute the normal form of this bifurcation up to the third order and obtain the direction of the Hopf bifurcation‎. ‎Finally‎, ‎we provide numerical simulations to illustrate our analytical findings‎.

In this short note, the following new characterization of contractibility is stated and proved: A Banach algebra is contractible if and only if for any pair of Banach bimodules over that algebra, the closed linear subspace of all continuous bimodule morphisms between them, in the space of all bounded linear operators between them, is naturally topological complemented. Here, the phrase ``natural'' has been used with its meaning in Category Theory.

In this paper, we study the existence of periodic solutions a certain autonomous non-linear ordinary differential equations (ODE's) of‎ ‎fifth‎ order. Our method is based on the evaluation of Brouwer's degree theory and making use of the homotopy invariance property of the topological degree. Finally, we present some numerical results obtained by using the analytical methods.

‎In this ‎paper, ‎we ‎investigate ‎the ‎structure ‎of ‎Jordan ‎centrali‎zer ‎maps‎ from a unital ‎alg‎ebra ‎$‎A‎$ ‎into a ‎unital‎ ‎‎$‎A‎$‎-bimodule ‎$‎M‎$‎.‎ We applied our results to triangular algebras. In particular, we prove that every ‎ Jordan centralizer map on a triangular algebra is a centralizer map. 

Let $R$ be a commutative Noetherian ring, and let $\mathfrak a$ be a proper ideal of $R$. Let $M$ be a non-zero finitely generated $R$-module with the finite projective dimension $p$. Also, let $N$ be a non-zero finitely generated $R$-module with $N\neq\mathfrak{a} N$, and assume that $c$ is the greatest non-negative integer with the property that $\operatorname{H}^i_{\mathfrak a}(N)$, the $i$-th local cohomology module of $N$ with respect to $\fa$, is non-zero. $\operatorname{H}^i_{\mathfrak a}(M, N)$, the $i$-th generalized local cohomology module of $M$ and $N$ with respect to $\mathfrak a$, is zero for all $i$ with $i>p+c$. In this paper, we obtain the coassociated prime ideals of $\operatorname{H}^{p+c}_{\mathfrak a}(M, N)$. Using this, in the case when $R$ is a local ring and $c$ is equal to the dimension of $N$, we obtain a necessary and sufficient condition for the vanishing of $\operatorname{H}^{p+c}_{\mathfrak a}(M, N)$ which extends the Lichtenbaum-Hartshorne vanishing theorem for generalized local cohomology modules.

In this paper, a numerical method for solving second kind nonlinear Volterra integral equations is presented. The method is based upon the extension of unknown function by single term Walsh series. Indeed, the interval [0, 1) is divided to m equal subinterval and in each interval, the unknown function is extended by the first term of Walsh series functions. By using a collocation method the coefficients of these extensions are computed and a block-pulse approximation of the unknown function is obtained. By the block-pulse approximation both continuous and pointwise approximations can be obtained. A convergence analysis for continuous approximations are investigated. The numerical examples confirm the ability and accuracy of the method. The method is computationally attractive and can easily be generalized for the systems of nonlinear volterra equations.

In this paper, necessary optimality conditions for a class of optimal control problems containing time-varying delays in control and state variables are discussed. There is an important aspect of these problems in that time-varying delays are applied to both state and control variables. Also, the cost functional of problems is influenced by the time-varying delays in state and control. We prove necessary optimality conditions in this study. A key aspect of the proof is calculating the variations of control and state variables when there are time-dependent delays. we make use of appropriate changing variables to derive these variations. In order to illustrate the use of these conditions, several examples are solved and numerical results are presented. At the end, some conclusions are drawn.

‎In this paper, we propose a three-component mathematical model, including Suspected-Infected-Recovered individuals (SIR), under the control of maple vaccination, for infectious diseases. In such, infectious disease can be transmitted through contact with an infected person (horizontal transmission). Vaccination of suspected population will reduce the horizontal transmission of patients in the community. The mathematical model has two disease-free and endemic equilibrium points. The basic reproduction rate of the model, the existence and local asymptotic stability of these two equilibrium points are investigated. By using Pontriagin's minimum principle, we have investigated the conditions of reducing the suspected and infected population and increasing the recovered population due to the use of vaccination in the community. Numerical simulations to the optimal control problem show that control measures can lead to a decrease in the number of suspected population and an increase in recovered population ‎a‎nd it prevents the spread of the disease and becoming into an epidemic.

‎In this paper, ‎numerical solution of a system of fractional differential equations which is the model of an epidemic disease is considered‎. ‎To this aim‎, ‎first‎, ‎third degree hat functions and their properties are introduced‎. After that‎, ‎using the expansion of the existing functions in the system in terms of the basis functions‎, ‎the system under consideration is transformed to a system of algebraic equations that can be solved using iterative methods‎. Then‎, ‎by solving the problem with a given initial data and comparing the results with the reported real data‎, ‎efficiency of the method is shown‎.

For a locally compact group $G$, $L^1(G)$ is its group algebra and $L^\infty(G)$ is the dual of $L^1(G)$. We consider on $L^\infty(G)$ the $\tau$-topology, i.e. the weak topology under all right multipliers induced by measures in $L^1(G)$. For such an arbitrary $G$ the $\tau$-topology is not weaker than the weak$^*$-topology and not stronger than the norm topology on $L^\infty(G)$. Among the other results we mention that except for discrete $G$ the $\tau$-topology is always different from the norm-topology. The properties of $\tau$ are then studied further and we pay attention to the $\tau$-almost periodic elements of $L^\infty(G)$.