0
votes
0answers
42 views

fundamental optimal-trajectory result known?

It's well-known and obvious that if you have a spaceship and your sole constraint is an upper bound on magnitude of acceleration/deceleration, the fastest way to get to a distant star (a fixed ...
2
votes
0answers
60 views

Approximating solutions to minima of the discrete Lagrangian

I have been stuck on this problem for a week and I'm not sure whether or not it is hard or I'm just missing something obvious. General gist of the problem I have a variational problem on a ...
4
votes
1answer
219 views

Minimizing action squared versus action

I have a very basic question in the calculus of variations: Suppose I want to minimize the functional $$A[r, r'] = \int_\Omega L(r, r') dx $$ When is it possible to say that extremals of $A$ agree ...
5
votes
1answer
216 views

A Lagrangian problem with a countable family of local extrema ?

Dear MO contributors, let $r > 0, L > 0$. I am interested in maximizing the integral: $$ \int_0^{2\pi} \frac{f(\alpha)^2 f'(\alpha)^2}{\sqrt{f(\alpha)^2 + f'(\alpha)^2}} \ \mathrm{d} \alpha $$ ...
2
votes
2answers
204 views

Equitable division of a contiguous resource

I have come across the following result regarding equitable division of a resource, which is a simple and immediate consequence of linear programming complementarity (in the infinite-dimensional ...
2
votes
1answer
132 views

Proving that a constructed curve solves an optimization problem

Caveat up-front: I'm not a mathematician, so please excuse any stupidity/ignorance that follows. First, let me explain what I'm trying to do: I want to choose a function that maximizes the following ...
1
vote
1answer
184 views

Proving a variational problem has no solutions

Consider the following integral $ \int_{0}^{\frac{\pi}{2}} \left( \sqrt{y(x)^2 + y'(x)^2} \left( \ln \left( \frac{\sin(x)}{1 -\cos(x)} \right) + \frac{\pi}{2} \right) + \frac{\pi}{2} y'(x) + 1 ...
1
vote
1answer
158 views

Conditions ensuring extrema are twice continuously differentiable?

For a functional $J[y]=\int_{a}^{b}F(t,y,y')dt$, are there any conditions that ensure extrema over the class of piecewise continuously differentiable functions are all in $C^2[a,b]$?
3
votes
3answers
437 views

An optimization problem

The following problem optimization problem arose in a project I am working on with a student. I would like to minimize the quantity: $$ M=\frac{1}{12} + \int_0^\frac{1}{2} \left( \tfrac{1}{2}-x ...
0
votes
1answer
122 views

optimize with respect to domain shape

Let $\Gamma$ be the set of all closed $C^2$ curves in the plane which enclose unit area and let $\Omega$ be the set of all subsets of $\mathbb{R}^2$ that are enclosed by some curve in $\Gamma$. Now ...
12
votes
1answer
682 views

What braking strategy is most fuel-efficient?

You notice a stop-light ahead of you and it is currently red. You can't run the red light, so you will have to brake, but braking wastes energy and you want to be as fuel efficient as possible. What ...
4
votes
2answers
949 views

Minimizing a function containing an integral

I am trying to optimize a function of the following form: $L = \int_{t=0}^{T}(AR-x)dt$, where A is a system parameter i.e. I am trying to find the optimum x(t) that minimizes L over all admissible ...