A control system is a family of dynamical systems defined on the same state space.
The family is parameterized by some parameters and it is allowed to change the values of parameters (i.e. to switch from one dynamics of the family to another one) at any time moment. Control function is a way to choose parameters as a function of time.
Involved in the family dynamical systems usually do not commute. In other words, the state where the system arrives at prescribed time heavily depends on the order in which we activate different dynamics and on the number of switchings. The structure and behaviour of the control system is actually determined by the commutator relations between the flows generated by the involved dynamical systems. This simple observation links links Control Theory with transformations groups and, more generally, with the entire world of geometry
In this expository talk I am going to exploit this link in both directions: from control to geometry and back. The talk should be accessible for the third year undergraduate students.