Using the recent formulation of Noether’s theorem for the problems of the calculus of variations with fractional derivatives, the Lagrange multiplier technique, and the fractional Euler-Lagrange equations, we prove a Noetherlike theorem to the more general context of the fractional optimal control. As a corollary, it follows that in the fractional case the autonomous Hamiltonian does not define...

Abstract Fractional calculus is a branch of classical mathematics, which deals with the generalization of operations of differentiation and integration to fractional order. Such a generalization is not merely a mathematical curiosity but has found applications in various fields of physical sciences. In this paper, we review the definitions and properties of fractional derivatives and integrals,...

We study the boundary value problem for a kind N-dimension nonlinear fractional differential system with the nonlinear terms involved in the fractional derivative explicitly. The fractional differential operator here is the standard Riemann-Liouville differentiation. By means of fixed point theorems, the existence and multiplicity results of positive solutions are received. Furthermore, two exa...

A unified approach is presented for the synthesis of continuous-time fractional orthogonal bases including Laguerre-like, Kautz-like and the GeneralizedOrthogonal-Basis-like (GOB-like) bases. They extend the definitions of their rational counterpart to fractional differentiation orders. Modes can either be chosen to be real or by pairs complex conjugate. Completeness of fractional Laguerre-like...

In this paper, completeness problem for a class of fractional rational basis functions with incommensurable differentiation orders is studied. Using the Müntz-Szász theory, it is established that the completeness problem for this class of the basis functions is equivalent to another completeness problem for a particular set of uncountably many basis functions. This equivalence allows one to dra...

The paper is devoted to the theoretical and experimental analysis of an electric circuit consisting two elements that are described by fractional derivatives different orders. These designed performed as RC ladders with properly selected values resistances capacitances. Different orders differentiation lead state-space system model, in which each state variable has a order derivative. Solutions...

We propose the model, which allows us to approximate fractional Levy noise and fractional Levy motion. Our model is based (i) on the Gnedenko limit theorem for an attraction basin of stable probability law, and (ii) on regarding fractional noise as the result of fractional integration/differentiation of a white Levy noise. We investigate self-affine properties of the approximation and conclude ...

Fractional order sliding mode control has been applied for speed of PMSM. However, in many previous works, the effects controller's parameters have not studied. This paper investigates fractional on performance FOSMC In this work, order, α and β FOSMS-PID were varied, their performances compared. The simulation experimental results show that variation integration, differentiation, can affect FO...