Optimizing a Multivariate Nonlinear System

Question

I am trying to optimize a multivariate system. The equations of motion are: $$x^i_{n+1}-x^i_n=\sum_j g^{ij}(|x^l_n-x^m_n|)f^j(|x^l_n-x^m_n|)$$ I want after an arbitrary number of steps $N$: $$\sum_{i=0}^{M-1}(x^i_N)/M\rightarrow max$$ The lower index indicates the time step and upper index the particle index. $g$ is given tensor depending non-linearly on the relative positions and f is the sought for vector valued function also depending non-linearly on the relative positions. $l,m$ indicate that the vector valued function $f$ depends on all positions $l,m\in [0,M-1]$ can take, where $M$ is the total particle number. I think one could use calculus of variations. But I am not sure how, because in Calculus of variations one usually has an integral and one can use partial integration. But I am not sure how this can be done here. I hope someone can give me a hint.

Answer 1

I'm guessing $f$ has some kind of restriction. Otherwise you can just select $f^j = \infty$. I assume that $\sum_j |f^j| = 1$. Now, note that

$$x_N = x_0 + \sum_{n=0}^{N-1} g(x_n) f(x_n)$$

where $x_n^T := [x_n^1 ~ \dots ~ x_n^M]$. You want to maximize the following:

$$ \textbf{1}^T \sum_{n=0}^{N-1} g(x_n) f(x_n) $$

with respect to $f$ where $\textbf{1}^T := [1 ~ \dots ~ 1]$. Note that $\textbf{1}^T g(x_n)$ is the sum of columns of $g$. So, the best function you can use is the one with

$$f(x_n) = [0 \dots 1 \dots 0]^T$$ where the index of $1$ in $f$ is the index of the column of $g(x_n)$ with maximum sum.