The nonlocal behavior of numerous dynamical systems, such as in diffusion process milling processes, data networks, transportation, plasma physics, Levy processes, economics, flexible structures, continuum robots, etc., can be well-described using differential equations with nonlocal operators. An operator acting on a function is local if its output at any arbitrary point only depends on the function values in a small neighborhood of the point; otherwise, the operator is called nonlocal like discrete delays, distributed delays, convolution integrals, fractional derivatives, integro-differential operators, Volterra integral equations, stochastic processes, etc. The behavior of nonlocal dynamical systems is not intuitive, similar to the classic ones governed by ordinary differential equations. Including these operators to the governing equations extends the inner dimension of the system to infinity, which makes their modeling and control laborious.

In this research, we shall demonstrate the importance and advantages of employing nonlocal operators in practical problems along with developing new control schemes . An essential step towards to use nonlocal operators for practical applications is developing an efficient numerical method that can solve differential equations with anomalous forms. We have proposed new methods for solving differential equations with the nonlocal operators, where the obtained solution exhibits spectral convergence, and more importantly, it can be obtained in a closed form for linear cases. Then, we have developed new control theories for stabilizing dynamical systems with nonlocal operators by approximating their local state transition matrix using developed operational matrices for different types of nonlocal operators.

advantages of using the nonlocal operators along the proposed methods were shown in several practical applications. For instance, we showed that dynamical systems can be controlled by using the nonlocal operators more efficient than conventional controllers. Moreover, we demonstrated that the current framework is elegant for integrating the differential equations with nonstandard-forms since requires less amount of data and computation time. In another example, we showed that the proposed framework is elegant for the identification of dynamical systems since it needs minimum training data. For modeling impact problems, we addressed the issue of unrealistic impact forces for the conventional linear viscoelastic models by introducing nonlocal operators to them.

For this purpose, we developed operational matrices of differentiation and integration for fractional operators, which are entirely superior to existing ones. Our operational matrices do not use the gamma function in comparison to the current ones whose main burden is the overwhelming use of the gamma function, which makes them significantly limited in the size or the number of discretization points. The proposed operational matrices are constructed in a closed-form with minimal computation time by determining the mystery relationship between the well-known Chebyshev differentiation matrix and fractional-order derivatives. Consequently, we developed a computational toolbox for solving differential equations with nonlocal operators since there was none available. The advantages of using the nonlocal operators along the proposed methods were shown in several practical applications. For instance, we showed that dynamical systems can be controlled by using the nonlocal operators more efficient than conventional controllers. Moreover, we demonstrated that the current framework is elegant for integrating the differential equations with nonstandard-forms since requires less amount of data and computation time. In another example, we showed that the proposed framework is elegant for the identification of dynamical systems since it needs minimum training data. For modeling impact problems, we addressed the issue of unrealistic impact forces for the conventional linear viscoelastic models by introducing nonlocal operators to them.