I am interested in variational problems of the following form: $$ \max J(y) \quad\text{such that}\quad y\in C, $$
where $J(y)=\int_0^b f(x,y,y')\,dx$ is a functional and $C$ is a specified family of smooth curves. As an example, $C$ could be the family of all straight lines: $C=\{y\in C^1[0,b] : y(x) = xu \,\exists\, u\in\mathbb{R}^d\}$. More generally, $C$ could specify bounds on the curvature of $y$, the length of $y$, etc.
"Traditional" calculus of variations can be used to solve problems in which $C$ is the family of all curves satisfying certain boundary conditions (e.g. $y(0)=x_1$, $y(b)=x_2$), but I am not familiar with a more general theory for general families $C$. (One well-known extension is the variable endpoint problem, but I am interested in even more general families as above.)
So, I have two questions:
- Is there a name for these kind of constraints?
- Are there any good references for these types of problems?
The family of all straight lines is finite dimensional, and your problem reduces to a finite dimensional calculus problem for them.
Anyway, for more generality, here are some things you may want to search for:
If you fix the length, you have an "isoperimetric constraint", usually solved by Lagrange multipliers. If you only specify bounds, you will get some type of "variational inequality" or "obstacle problem".