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After a brief survey of the literature about sufficient conditions, we give
different sufficient conditions of optimality for infinite-horizon calculus
of variations problems in the general (non concave) case. Some sufficient
conditions are obtained by extending to the infinite-horizon setting the
techniques of extremal fields. Others are obtained in a special
qcase of reduction to finite horizon. The last result uses auxiliary
functions. We treat five notions of optimality. Our problems are essentially
motivated by macroeconomic optimal growth models.
We prove the Legendre Necessary Condition of the Calculus of Variations in Mean an arbitrary finite dimension. When the Lagrangian is convex, we establish that if the Euler-Lagrange equation possesses an almost periodic solution then it possesses periodic and constant solutions. We deduce from this fact various consequences on the structure of the set of almost periodic solutions.
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