8,235 research outputs found
Discrete-Time Fractional Variational Problems
We introduce a discrete-time fractional calculus of variations on the time
scale , . 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
tends to zero. Our Legendre type condition is useful to eliminate false
candidates identified via the Euler-Lagrange fractional equation.Comment: Submitted 24/Nov/2009; Revised 16/Mar/2010; Accepted 3/May/2010; for
publication in Signal Processing
Melnikov analysis in nonsmooth differential systems with nonlinear switching manifold
We study the family of piecewise linear differential systems in the plane
with two pieces separated by a cubic curve. Our main result is that 7 is a
lower bound for the Hilbert number of this family. In order to get our main
result, we develop the Melnikov functions for a class of nonsmooth differential
systems, which generalizes, up to order 2, some previous results in the
literature. Whereas the first order Melnikov function for the nonsmooth case
remains the same as for the smooth one (i.e. the first order averaged function)
the second order Melnikov function for the nonsmooth case is different from the
smooth one (i.e. the second order averaged function). We show that, in this
case, a new term depending on the jump of discontinuity and on the geometry of
the switching manifold is added to the second order averaged function
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