Fractional Hamilton-Jacobi equation for the optimal control of nonrandom fractional dynamics with fractional cost function

By using the variational calculus of fractional order, one derives a Hamilton-Jacobi equation and a Lagrangian variational approach to the optimal control of one-dimensional fractional dynamics with fractional cost function. It is shown that these two methods are equivalent, as a result of the Lagrange’s characteristics method (a new approach) for solving nonlinear fractional partial differential equations. The key of this results is the fractional Taylor’s seriesf(x + h) = E _α(h^αD^α)f(x) whereE _α(·) is the Mittag-Leffler function.