Exponential Mapping in Euler’s Elastic Problem
The Euler problem on optimal configurations of elastic rod in the plane with fixed endpoints and tangents at the endpoints is considered. The global structure of the exponential mapping that parameterises extremal trajectories is described. It is proved that open domains cut out by the Maxwell strata in the preimage and image of the exponential mapping are mapped diffeomorphically. As a consequence, computation of globally optimal elasticae with given boundary conditions is reduced to solving systems of algebraic equations having unique solutions in the open domains. For certain special boundary conditions, optimal elasticae are presented.
KeywordsEuler elastica Optimal control Exponential mapping
Mathematics Subject Classifications (2010)49J15 93B29 93C10 74B20 74K10 65D07
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