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Discrete-time fractional variational problems. (English) Zbl 1203.94022
Summary: We introduce a discrete-time fractional calculus of variations on the time scale (h) a ,a,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.
MSC:
94A12Signal theory (characterization, reconstruction, filtering, etc.)
26A33Fractional derivatives and integrals (real functions)
62M10Time series, auto-correlation, regression, etc. (statistics)
49J99Existence theory for optimal solutions
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