By Daniel Liberzon
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Extra resources for Calculus of Variations and Optimal Control Theory: A Concise Introduction (Free preliminary copy)
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. 3 while the subset A is smaller because it reflects additional constraints (to be specified below).
26) must then also vanish, which gives us an additional necessary condition for optimality: Lz (b, y(b), y (b))η(b) = 0 or, since η(b) is arbitrary, Lz (b, y(b), y (b)) = 0. 27) as replacing the boundary condition y(b) = y 1 . Recall that we want to have two boundary conditions to uniquely specify an extremal. 27). 2, with Lagrangian L(x, y, z) = √ 1 + z 2 . In other words, we are looking for a shortest path from a given point to a vertical line. 27) amounts to y (b) = 0. This means that the optimal path must have a horizontal tangent at the final point.
16). , it is still a necessary condition for optimality. 26) is 0. But this means that the entire integral is 0, for all admissible η (not just those vanishing at x = b). 26) must then also vanish, which gives us an additional necessary condition for optimality: Lz (b, y(b), y (b))η(b) = 0 or, since η(b) is arbitrary, Lz (b, y(b), y (b)) = 0. 27) as replacing the boundary condition y(b) = y 1 . Recall that we want to have two boundary conditions to uniquely specify an extremal. 27). 2, with Lagrangian L(x, y, z) = √ 1 + z 2 .
Calculus of Variations and Optimal Control Theory: A Concise Introduction (Free preliminary copy) by Daniel Liberzon