Fractional h-difference equations arising from the calculus of variations

Abstract

The recent theory of fractional h-difference equations introduced in [N. R. O. Bastos, R. A. C. Ferreira, D. F. M. Torres: Discrete-time fractional variational problems, Signal Process. 91 (2011), no. 3, 513-524], is enriched with useful tools for the explicit solution of discrete equations involving left and right fractional difference operators. New results for the right fractional h sum are proved. Illustrative examples show the effectiveness of the obtained results in solving fractional discrete Euler-Lagrange equations. 2000 Mathematics Subject Classification. Primary 39A12; Secondary 49J05, 49K05. Key words and phrases. Fractional discrete calculus, fractional difference calculus of variations, Euler-Lagrange equations, explicit solutions.

Remarks on Fractional Discrete Cone Control Systems with n-Orders and Their Stability

“…Let us draw our attention to the problem of controllability of system (8). Classically, controllability of the given system means that it is possible to transfer the considered system from a given initial state to a final state using controls from a certain set, see for example [16,17].…”

“…Classically, controllability of the given system means that it is possible to transfer the considered system from a given initial state to a final state using controls from a certain set, see for example [16,17]. Suppose that a set V is a subset of the state space of system (8).…”

“…Suppose that there exists some feedback law u(k) = κ(X(k)) so that X e is a local asymptotically stable state for P cone system (15). Then system (8) is asymptotically controlled to X e .…”

“…Basic information on fractional calculus concept, ideas, and applications of these operators can be found for example in [15,18,24]. In [8], there was adopted a more general fractional h-difference Riemman-Liouville operator, where on the one hand h represents sample step, on the other hand, for h tending to zero, the solutions of the fractional difference equation may be seen as approximations of the solutions of corresponding Riemann-Liouville fractional equations.…”

Generalized Fractional-Order Discrete-Time Integrator

“…As crucial realizations, those, which reveal such important dynamic properties as stability, controllability, and so forth, are considered. The first-order differential equations can be generalized to the fractional-order ones [3][4][5][6][7][8][9][10]. Hence, we get the fractional-order integrator, which can be used in a modelling of fractional-order dynamic systems.…”