Optimal Control, Geometry, and Mechanics
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This essay highlights the contributions of geometric control theory to the calculus of variations. The Maximum Principle is recognized as a covariant necessary condition of optimality valid for variational problems defined on subsets C of the tangent bundle of the ambient manifold, rather than the entire tangent bundle, as was commonly assumed by the classical theory. Control-theoretic interpretations provide a uniform framework for variational problems with non-holonomic constraints and reveal the significance of controllability theory for such problems. The merits of this theory are then illustrated through solutions of variational problems on Lie groups having either left or right invariance. By focusing on problems fromm geometry and mechanics, the essay visibly demonstrates the innovative ideas that geometric control brings to the calculus of variations.
KeywordsVector Field Symmetric Space Heisenberg Group Elastic Problem Hamiltonian Vector Field
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