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This work offers a real-world application for the study of fuzzy dynamic equations. First, we propose the novel concept of granular delta differentiability for fuzzy-valued functions defined on time scales with the help of the relative distance measure fuzzy arithmetic and horizontal membership functions. Then, fundamental foundations of fuzzy calculus on time scales are provided. Discussion on the Malthusian model defined on particular time scales to illustrate the proposed approach is presented. [ABSTRACT FROM AUTHOR]

The aim of this paper is to analyze the behavior of Hegselmann‐Krause–type models. General sufficient conditions for achieving consensus are provided. Model predictive control is applied to the Hegselmann‐Krause system in the context of time scales. Numerical simulations show that the predictive control mechanism has the ability to steer the system to attain a consensus. Moreover, faster consensus speed is observed when compared with the classical model. Finally, in the case of the discrete time scale and all agents being neighbors at the initial time, conditions guaranteeing an average consensus are given. [ABSTRACT FROM AUTHOR]

We study incommensurate fractional variational problems in terms of a generalized fractional integral with Lagrangians depending on classical derivatives and generalized fractional integrals and derivatives. We obtain necessary optimality conditions for the basic and isoperimetric problems, transversality conditions for free boundary value problems, and a generalized Noether type theorem. [ABSTRACT FROM AUTHOR]

Both inflation and unemployment inflict social losses. When a tradeoff exists between the two, what would be the best combination of inflation and unemployment? A well known approach in economics to address this question is writing the social loss as a function of the rate of inflation p and the rate of unemployment u, with different weights, and then, using known relations between p, u, and the expected rate of inflation π, to rewrite the social loss function as a function of π. The answer is achieved by applying the calculus of variations in order to find an optimal path π that minimizes total social loss over a given time interval. Economists dealing with this question use a continuous or a discrete variational problem. Here we propose to use a time-scale model, unifying the results available in the literature. Moreover, the new formalism allows for obtaining new insights into the classical models when applied to real data of inflation and unemployment. [ABSTRACT FROM AUTHOR]

We prove optimality conditions for generalized quantum variational problems with a Lagrangian depending on the free end-points. Problems of calculus of variations of this type cannot be solved using the classical theory. [ABSTRACT FROM AUTHOR]

We extend the second Noether theoremto variational problems on time scales. As corollarieswe obtain the classical second Noether theorem, the second Noether theorem for the h-calculus and the second Noether theorem for the q-calculus. [ABSTRACT FROM AUTHOR]

We introduce a fractional theory of the calculus of variations for multiple integrals. Our approach uses the recent notions of Riemann–Liouville fractional derivatives and integrals in the sense of Jumarie. The main results provide fractional versions of the theorems of Green and Gauss, fractional Euler–Lagrange equations, and fractional natural boundary conditions. As an application we discuss the fractional equation of motion of a vibrating string. [ABSTRACT FROM AUTHOR]