Please I have been trying to write the mathematical formulation of my nonlinear dynamical system for quite some time and I will appreciate any input.
** Problem Description**
Assuming, I am traveling from Point A to Point B and there are hundred different paths (1 to 100). I do not know the optimum path (path with the shortest distance).
Let's assume again that I have a controller input $u$ representing the different paths. i.e. if $ u = 1$ then the system (me) will travel along path 1.
Let's also assume that the state $x$ of the system is the distance traveled at a given time.
The question is how can one express this dynamical system? Knowing that the control input $u$ is not directly related to the state $x$.
Does it make sense to represent the system described like this:
$$\dot{x} = f(t,x) + g(t,x,u)$$
Thanks for your help.
From my current understanding of the problem, I would represent the system as a hybrid system of the form $$\dot{x}(t)=\left\{\begin{array}{rcl} v,&& \mathrm{if\ }x(t)\le f(u)\\ 0,&& \mathrm{if\ }x(t)>f(u)\\ \end{array}\right.$$
where $x(t)$, as required, represents the distance traveled from point A at time $t$ and $v>0$ is the traveling speed which I assume here to be constant.
The function $f:U\mapsto\mathbb{R}_{\ge0}$ is a function that maps the input $u$ to the total traveling distance from $A$ to $B$ using path $u\in U=\{1,\ldots\}$. I am also assuming here that once a path has been chosen, then we are not allowed to change; i.e. $u$ is constant.
Basically, what the system is describing is 1) traveling at constant speed $v$ until we reach the destination and 2) once B is reached, we just stop.
This answer is based on my current understanding of the problem and I will be happy to update it in case there is something that I misunderstood. I have other, more complex, solutions that can capture more complex behaviors but I wanted to provide the simplest one.
Edit. If the speed is time-varying and also path-dependent, then you can consider the following model $$\dot{x}(t)=\left\{\begin{array}{rcl} v_u(t),&& \mathrm{if\ }x(t)\le f(u)\\ 0,&& \mathrm{if\ }x(t)>f(u)\\ \end{array}\right.$$ where $v_u(t)$ is now the input/path-dependent time-varying velocity. Here again, path is chosen at the beginning and remains the same until reaching the final destination.