An existence theorem in the calculus of variations
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In this paper, a nonparametric variational problem is considered in the setting of the theory of generalized curves. It is assumed that the integrand of the problem does not grow at infinity faster than the norm\(|\dot x|\) of the variable\(\dot x\), for all values of the other variablest andx (which take their values in a compact product set). It is shown that a generalized curve exists such that the minimum of the functional over an appropriate set is achieved. This generalized curve does not in general have compact support.
Key WordsCalculus of variations existence theorems generalized control theory discontinuous solutions extremal points
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