Asynchronous Control Systems and Differential Inclusions

In discrete or continuous-time models of distributed control systems, each of the equations usually describes a change in the state of one of the constituent blocks (or nodes) of the modeled object or process. In real processes it is possible that at the next clock cycle not all blocks can be triggered simultaneously (synchronous case), but only one or several (asynchronous case). One of the classes of models that take into account possible asynchrony of processes are Petri nets. The transition from a synchronous model to an asynchronous one (“asynchronization”) can change the reachability domains for control system states. In the continuous case, asynchronization of a conventional dynamic system (without control) can be described as a transition to differential inclusion of a special type. In this article we study the influence of asynchronization on the reachability domains in the case of some simple linear dynamic systems on the plane. The topics discussed in the article will be useful in the educational process when studying courses in control theory and differential equations.