Why dynamics and control?
A dynamical system is a rule for time evolution on a state space, e.g. the
- locations and speeds of planets in a solar system described by Newton’s law „force = mass x acceleration“, or the
- motion of billiard balls on a billiard table.
Dynamical systems are generated e.g. by the solutions of ordinary deterministic or random differential equations, difference equations, iterated function systems, cellular automata, dynamic equations with time delay, differential algebraic equations and many more. Dynamical systems are semigroup actions, usually of the non-negative reals or integers, on a manifold which is often a subset of a Banach space.
Control theory studies systematically the controllability of the time evolution of states under constraints, e.g. the
- reachability of stable orbits of satellites with a prescribed amount of fuel, or the
- development of a heat control which stabilizes the room temperature.
Control systems are e.g. differential equations which depend on a control which can be chosen from a set of admissible functions. Laplace transforms for the analysis in the frequency domain and Lyapunov functions from stability theory are used to develop a mathematical control strategy which can then be implemented in a real system.