Simulation Relations and Controllability Properties of Linear and Nonlinear Control Systems

We examine to what extent nonlinear input-disturbance systems that are connected by a simulation relation share certain controllability properties. We derive several results that fit within the paradigm that if there is a simulation relation of a system $A$ by a system $B$, and if system $A$ has a specified controllability property, then system $B$ has that same property. As expected, one can only turn the paradigm into actual theorems by imposing appropriate assumptions on the systems and/or the simulation relation. We prove three such results. The first result we obtain deals with the property of complete controllability, where we impose minimal assumptions on the input-disturbance systems but require that the simulation relation be the graph of a smooth surjection between the systems' state spaces that satisfies a certain compactness condition. The second result deals with a somewhat weaker notion of controllability modulo the kernel of a linear mapping, where it is assumed that system $B$ is “almost l...