Discrete-time fractional variational problems

We introduce a discrete-time fractional calculus of variations on the time scale (hZ)a,a∈R,h>0. First and second order necessary optimality conditions are established. Examples illustrating the use of the new Euler–Lagrange and Legendre type conditions are given. They show that solutions to the considered fractional problems become the classical discrete-time solutions when the fractional order of the discrete-derivatives are integer values, and that they converge to the fractional continuous-time solutions when h tends to zero. Our Legendre type condition is useful to eliminate false candidates identified via the Euler–Lagrange fractional equation.