## Should the Control of a Dynamical System Depend on the State?

Hey, all.

I am working with a dynamical system of the form `[; begin{align*}dot{x}_i(t) &= u_i(t)\ &= d_isum_{jinmathcal N_i}w_{ij}(x_j-x_i)end{align*} ;]`, where `[; d_i ;]` are control parameters, and `[; w_{ij} ;]` are predetermined weights specified arbitrarily (randomly).

I want to find optimal control of the system, but I have not ever worked with a dynamical system where the control depends on the state itself. In particular, the dynamical system can be written in the form `[; dot{bf x} = -DL{bf x} ;]`, where `[; D ;]` is an invertible diagonal matrix containing the `[; d_i ;]`‘s created to stabilize the system by sending `[; dot {bf x} ;]` to 0, `[; L ;]` is a matrix containing the weights (Laplacian), and `[; {bf x} ;]` is the state vector.

How does setting up a performance index differ in this case? In particular, what I want to do is have `[; dot{bf x} approx {bf 0};]` in a given time `[; t_f ;]`, and the way I can check this is with determining if `[; |dot{bf x}|_2^2 approx 0 ;]`. Thus, how do I need to set up my performance index? I think I need to set it up like so:

`[; J = int_0^{t_f} lambda^top(left[|dot x|_2^2 - 0) + mu^top(-DL{bf x} - dot{bf x})right],dt ;]`

But I’m not sure if this is appropriate setup. I need to incorporate the dynamics of the system with a Lagrange multiplier, given here by `[; mu ;]`, right? Do I also need the Lagrange multiplier `[; lambda ;]` on `[; |dot{bf x}| ;]`?

I apologize for the nature of my questions. I just unfortunately haven’t seen a system like this before in control theory, and I’m not sure how it affects finding the Euler-Lagrange equations.

submitted by Nevin Manimala Nevin Manimala /u/Lafojwolf

## Need help finding a particular website

There was a webpage I used to visit a long time ago that listed every positive integer from 1 to 1000 (or it might have been 10,000), and beside each number it gave a mathematical fact that uniquely characterized that number (i.e. 137 is the first such and such to have the property such and such).

Does anyone remember this site? If so, I’d really like some help finding it. Thank you in advance.

Edit: Has been found

submitted by Nevin Manimala Nevin Manimala /u/American_Gambit