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Generalizing the variational theory on time scales to include the delta indefinite integral. (English) Zbl 1221.49040
Summary: We prove necessary optimality conditions of Euler-Lagrange type for generalized problems of the calculus of variations on time scales with a Lagrangian depending not only on the independent variable, an unknown function and its delta derivative, but also on a delta indefinite integral that depends on the unknown function. Such kinds of variational problems were considered by Euler himself and have been recently investigated by {\it J. Gregory} [Methods Appl. Anal. 15, No. 4, 427--436 (2008; Zbl 1178.49024)]. Our results not only provide a generalization to previous results, but also give some other interesting optimality conditions as special cases.

49K21Optimal control problems involving relations other than differential equations
34A08Fractional differential equations
Full Text: DOI arXiv
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