Is state space bounded in practical scenario?

Question

This is a follow up to one of my previous questions on manifolds and state space. As I read through the answer, another question popped up in me.

1)Can there be a dynamic system with state space bounded ? Such as the states evolve only in a subset of R(n) if initial conditions lie in the subset.

2)Typically when I look at state equations there is invariably a mention that the states span R(n) space. Assuming a nonlinear dynamic system for which someone could give a single analytical state equation will it typically apply for all of R(n)? What if there are state constraints to the system, should a seperate state equation be defined at boundaries?

Answer 1

In practice the state space is always bounded to a subset of $\mathbb{R}^n$.

1.): Just look at an object with mass $M$ (assuming it is a point mass) that is accelerated by a force $F$:

$$ \ddot{x} = \frac{1}{M}F \tag{1} $$

Assume you are interested in velocity $\dot{x}$ of that object. Define $x_1 = \dot{x} = y$ and $u = F$ so you get the transfer function

$$ Y(s) = \frac{1}{M s} U(s) $$

That is a scalar system, so if the state space would be whole $\mathbb{R}$ then $x_1$ could become larger than $c$, the speed of light. However, this is impossible as we know from physics.

Such practical limitations exist for every real system, so in practice, the state space will always be bounded. The example $(1)$ leaves out friction so in reality you would at some point be prevented to increase $x_1$ further.

2.) If you want to include state constraints to your system then you get a differential-algebraic system of equations (DAEs).

Answer 2

Since you already have an answer, perhaps this comes a bit late, but:

There are plenty dynamics in bounded spaces and plenty dynamics in unbounded spaces.

And in some cases we case use both for the same system. Take for example the pendulum equation $$x''+\sin x=0.$$ You can choose to describe it on $\mathbb R^2$ or on the infinite cylinder $\mathbb R\times S^1$, understanding $S^1$ as a circle obtained from $[0,2\pi]$ "identifying" the endpoints in some precise/rigorous manner. But depending on your interest you can also consider a specific subset of orbits that corresponds to a bounded subset of the cylinder $\mathbb R\times S^1$. The choice is basically yours in this respect, and may depend on whatever you want to study.

Whether a dynamics has full space $\mathbb R^n$ simply depends on whether the equations are defined on the whole space or not.

Concerning constraints, it is better to apply them a priori so that, a priori, we have already a "reduced" phase space. Hopefully these would be nice manifolds where you can consider a corresponding reduced dynamics. Sometimes this is not the case, and sometimes it is complicated to take the constraints into consideration a priori. It really depends on your specific system.