Reference Request: investigation of higher order dynamical systems

Question

In dynamical system and control theory, people usually investigate into system of the type $$\dot x = f(x,u)$$

Is there any references to looks into the theory of higher order dynamical systems of the type $${\ddot x} = f(x,u)$$ $$\vdots$$ $${x}^n = f(x,u)$$

and so on?

Thanks!

Answer 1

Hint:

any equation of the form $$ x^{(n)}=f(t,x,x',x'',\cdots,x^{(n-1)}) $$ can be reduced to a system of first order: $$ \begin{cases} x'=x_1\\ x_1'=x_2\\ x_2'=x_3\\ \cdots\\ x_{n-2}'=x_{n-1}\\ x_{n-1}'=f(t,x,x_1,x_2,\cdots,x_{n-1}) \end{cases} $$