ABSTRACT
An important class of dynamical systems are modeled by differential equations. This chapter introduces differential equations with inputs for modeling continuous-time control systems. It also discusses how classical tools from control theory such as Lyapunov functions can be used to establish relatively simple solution properties such as stability, boundedness, invariance, safety, and reachability. Existence and uniqueness theorems in terms of Carathéodory solutions are provided. Lyapunov theorems are proved for continuous-time dynamical systems subject to disturbance. To characterize safety and stability with safety constraints, classical invariance conditions a closed set with respect to a continuous flow are formulated for continuous-time dynamical systems with disturbance. Lyapunov and barrier functions are unified to provide sufficient conditions for stability with safety requirements. A brief discussion for control Lyapunov and barrier functions is also included.
