Mixed Monotonicity for Reachability and Safety in Dynamical Systems

A dynamical system is mixed monotone if its vector field or update-map is decomposable into an increasing component and a decreasing component. In this tutorial paper, we study both continuous-time and discrete-time mixed monotonicity and consider systems subject to an input that accommodates, e.g., unknown parameters, an unknown disturbance input, or an exogenous control input. We first define mixed monotonicity with respect to a decomposition function, and we recall sufficient conditions for mixed monotonicity based on sign properties of the state and input Jacobian matrices for the system dynamics. The decomposition function allows for constructing an embedding system that lifts the dynamics to another dynamical system with twice as many states but where the dynamics are monotone with respect to a particular southeast order. This enables applying the powerful theory of monotone systems to the embedding system in order to conclude properties of the original system. In particular, a single trajectory of the embedding system provides hyperrectangular over-approximations of reachable sets for the original dynamics. In this way, mixed monotonicity enables efficient reachable set approximation for applications such as optimization-based control and abstraction-based formal methods in control systems.