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Fractional Hamilton-Jacobi equation for the optimal control of nonrandom fractional dynamics with fractional cost function. (English) Zbl 1111.49014
Summary: 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 series $f\left(x+h\right)={E}_{\alpha }\left({h}^{\alpha }{D}^{\alpha }\right)f\left(x\right)$ where ${E}_{\alpha }\left(·\right)$ is the Mittag-Leffler function.
##### MSC:
 49K15 Optimal control problems with ODE (optimality conditions) 90C39 Dynamic programming 60G18 Self-similar processes 49L20 Dynamic programming method (infinite-dimensional problems) 49K20 Optimal control problems with PDE (optimality conditions)
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