A class of hybrid-state continuous-time dynamic systems

A class of continuous time systems with part continuous, part discrete state is described by differential equations combined with multistable elements. Transitions of these elements between their discrete states are triggered by the continuous part of the state and not directly by inputs. The dynamic behavior of such systems, in response to piecewise continuous inputs, is defined under suitable assumptions. A general Mayer-type optimization problem is formulated. Conditions are given for a solution to be well-behaved, so that variational methods can be applied. Necessary conditions for optimality are stated and the jump conditions are interpreted geometrically.