Variational Methods and Optimal Control III
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Many problems of optimisation and control in the sciences and engineering seek to find the shape of a curve or surface satisfying certain conditions so as to maximise or minimise some quantity. For example, shape a yacht hull so as to minimise fluid drag. Variational methods involve an extension of calculus techniques to handle such problems. This course develops an appropriate methodology, illustrated by a variety of physical and engineering problems.
Topics covered are: Classical Calculus of Variations problems such as calculation of the shape of geodesics, the Cantenary, and the Brachystochrone; the derivation and use of the simpler Euler-Lagrange equations for second-order (the Euler-Poisson equation), multiple dependent variables (Hamilton's equations), and multiple independent variables (minimal surfaces); constrained problems and problems with non-integral constraints; Euler's finite differences, Ritz's method and Kantorich's method; conservation laws and Noether's theorem; classification of extremals using second variation; optimal control via the Pontryagin Maximum Principle, and its applications to space-flight calculations.
This course is not recorded as prequisite for other courses.