Calculus of Variations and Optimal Control Theory: A Concise by Daniel Liberzon

By Daniel Liberzon

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These are familiar expressions for the angular momentum. , time shift, translation, rotation) implies the existence of a conserved quantity. 3 we showed that the Euler-Lagrange equation is a necessary condition for optimality in the context of the Basic Calculus of Variations Problem, where the boundary points are fixed but the curves are otherwise unconstrained. In this section we generalize that result to situations where equality constraints are imposed on the admissible curves. 2 devoted to constrained optimality and the method of Lagrange multipliers for finite-dimensional problems.

38) can indeed be solved for y . This issue did not arise earlier when we were working with the Hamiltonian H = H(x, y, y , p). The above approach has another, more important drawback. 31) that H has a stationary point as a function of y along an optimal curve. 30), it will lead us to the maximum principle. But it only makes sense when we treat y as an independent variable in the definition of H. 38). This is probably why it was not until the late 1950s that the maximum principle was discovered.

However, we will see that it does not quite provide the right point of view for our future developments, and it is included here mainly for historical reasons. 10 is a possible graph of f ). For simplicity we are considering the scalar case, but the extension to f : R n → R is straightforward. The Legendre transform of f will be a new function, f ∗ , of a new variable, p ∈ R. Let p be given. Draw a line through the origin with slope p. Take a point ξ = ξ(p) at which the (directed) vertical distance from the graph of f to this line is maximized: ξ(p) := arg max{pξ − f (ξ)}.

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