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Fractional Euler-Lagrange differential equations via Caputo derivatives
Almeida, R.; Malinowska, A. B.; Torres, D. F. M.We review some recent results of the fractional variational calculus. Necessary optimality conditions of Euler–Lagrange type for functionals with a Lagrangian containing left and right Caputo derivatives are given. Several problems are considered: with fixed or free boundary conditions, and in presence of integral constraints that also depend on Caputo derivatives.
Fractional calculus of variations in terms of a generalized fractional integral...
Odzijewicz, T.; Malinowska, A.B.; Torres, D.F.M.We study fractional variational problems in terms of a generalized fractional integral with Lagrangians depending on classical derivatives, generalized fractional integrals and derivatives. We obtain necessary optimality conditions for the basic and isoperimetric problems, as well as natural boundary conditions for free-boundary value problems. The fractional action-like variational approach (FALVA) is extended...
A fractional calculus of variations for multiple integrals with application to ...
Almeida, R.; Malinowska, A.B.; Torres, D.F.M.We introduce a fractional theory of the calculus of variations for multiple integrals. Our approach uses the recent notions of Riemann-Liouville fractional derivatives and integrals in the sense of Jumarie. The main results provide fractional versions of the theorems of Green and Gauss, fractional Euler-Lagrange equations, and fractional natural boundary conditions. As an application we discuss the fractional e...
Strong minimizers of the calculus of variations on time scales and the Weierstr...
Malinowska, A.B.; Torres, D.F.M.We introduce the notion of strong local minimizer for the problems of the calculus of variations on time scales. Simple examples show that on a time scale a weak minimum is not necessarily a strong minimum. A time scale form of the Weierstrass necessary optimality condition is proved, which enables to include and generalize in the same result both continuous-time and discrete-time conditions.
Nonessential functionals in multiobjective optimal control problems
Malinowska, A.B.; Torres, D.F.M.We address the problem of obtaining well-defined criteria for multiple criteria optimal control problems. Necessary and sufficient conditions for an objective functional to be nonessential are proved. The results provide effective tools for determining nonessential objectives in multiobjective optimal control problems.
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