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• We develop a new finite-time method for controlling and synchronising fractional-order financial systems • A Deep Learning Recurrent Neural Network is integrated by terminal sliding mode control • The proposed technique is applied to a chaotic fractional-order financial model with market confidence under highly complex and time-varying uncertainties • It is shown comparatively that when the disturbances are high, the conventional TSMC methods fail Disturbances are inevitably found in almost every system and, if not rejected, they could jeopardize the effectiveness of control methods. Thereby, employing state-of-the-art observers could improve the reliability and performance of controllers dramatically. Motivated by this, we develop a new finite-time method for controlling and synchronising fractional-order systems. The deep learning recurrent neural network, which is a strong tool in handling highly complex and time-varying uncertainties, is integrated by terminal sliding mode control. On the basis of Lyapunov stability theorem, the finite-time convergence and stability of the closed-loop system are proven. Then, the proposed technique is applied to a chaotic fractional-order financial model with market confidence. In simulations, it is supposed that the system is operating in the presence of complex disturbances. The results of the proposed technique are compared with terminal sliding mode control. Numerical results illustratively confirm the theoretical claims about the robust performance of the deep learning control technique. Also, it is shown that when the disturbances are high, the conventional methods fail. [ABSTRACT FROM AUTHOR]

This paper investigates the stability properties for a class of impulsive Caputo fractional-order hybrid systems with impulse effects at fixed moments. By utilizing the variational Lyapunov method, a fractional variational comparison principle is established. Some stability and instability criteria in terms of two measures are obtained. These results generalize the known ones, extending the corresponding theory of impulsive fractional differential systems. An example is given to demonstrate their effectiveness. [ABSTRACT FROM AUTHOR]

Fractional calculus is an efficient tool, which has the potential to improve the performance of gradient methods. However, when the first order gradient direction is generalized by fractional order gradient one, the corresponding algorithm converges to the fractional extreme point of the target function which is not equal to the real extreme point. This drawback critically hampers the application of this method. To solve such a convergence problem, the current paper analyzes the specific reasons and proposes three possible solutions. Considering the long memory characteristics of fractional derivative, short memory principle is a prior choice. Apart from the truncation of memory length, two new methods are developed to reach the convergence. The former is the truncation of the infinite series, and the latter is the modification of the constant fractional order. Finally, six illustrative examples are performed to illustrate the effectiveness and practicability of proposed methods. [ABSTRACT FROM AUTHOR]

Highlights • A new nabla Taylor formula expanded at b ≥ n is derived for a discrete function. • Two kinds of series representations of are derived for nabla discrete fractional sum/difference. • Some essential properties of nabla discrete fractional calculus are investigated. • Besides fixed initial instant case, nabla fractional calculus with fixed memory step are introduced. Abstract This paper addresses the description and analysis problems of nabla discrete fractional calculus. The series representation framework is developed first, including two Taylor series expanded at the initial instant and the current time, respectively. Under this framework, several essential properties of fractional sum/difference are presented and investigated. Notably, the short memory principle is introduced for nabla discrete fractional calculus, along with which two corresponding series are studied. [ABSTRACT FROM AUTHOR]

This article focuses on the indirect model reference adaptive control problem for fractional order systems. A constrained gradient estimation method was established firstly, since parameter estimation is part and parcel of the whole control problem. Then a novel adaptive control law is designed, from which the two problems, i.e., parameter estimation and reference tracking, can be unified perfectly. On these basis, an effective control scheme is established. The stability of the resulting closed-loop system is analyzed rigorously via indirect Lyapunov method and frequency distributed model. Finally, a careful simulation study is reported to illustrate the effectiveness of the proposed scheme. [ABSTRACT FROM AUTHOR]