Analysis of the set of trajectories of fuzzy equations of perturbed motion (Q317629)

In this paper, the authors consider a fuzzy model of equations of a perturbed motion of a system with inexact values of the parameters, i.e., \[ \frac{dx}{dt}=f(t,x,\alpha),\,\, x(t_0)=x_0, \] where \(x\in E^n,\) \(f\in C(\mathbb{R}_+\times E^n\times J,E^n),\) \(\alpha\in J\) is a fuzziness parameter, and \(J\) is a compact set in \(\mathbb{R}^d\). As a result of regularization, the authors obtain a family of fuzzy equations \[ \frac{du}{dt}=f_{\chi}(t,u),\,\, u(t_0)=u_0, \] where \( f_{\chi}(t,u)= \chi f_{M }(t,u)+(1-\chi) f_{m }(t,u),\) \(f_M(t,\cdot)=\overline{co}\bigcup\limits_{\alpha\in J}f(t,\cdot,\alpha),\) \( f_m(t,\cdot)=\overline{co}\bigcap\limits_{\alpha\in J}f(t,\cdot,\alpha),\) and establish conditions for the existence of solutions of these equations, deduce an estimate for the distance between two families of solutions, indicate conditions for the existence of successive approximations, investigate the continuous dependence of a family of solutions on the initial data, establish conditions for the global existence of solutions and conditions for the existence of \(\varepsilon\)-aproximations to the solutions, and present a general theorem on the stability of the stationary solution of a family of regularized equations.