State of circuit components in the behavioral framework

A general, rigorous but reasonably simple, and meaningful approach to the concept of state for circuit components is proposed in the behavioral framework. First, some basic notions of the behavioral setting (e.g., trajectory, event, universum of trajectories, admissible trajectory, and behavior) are recalled. Then, using the notion of trajectory concatenation, the forwards and backwards concatenation sets of an admissible trajectory are introduced, and the class of components with well-ordered behavior is defined. Finally, for these components, the concept of state is established as the parametrization that rules the time-dependent partition of the component behavior whose sets are closed under trajectory concatenation. To provide a clear and significant illustration of the theory, the state investigation of the class of the two-terminal components whose behavior is generated by means of an ordinary differential relation of any order, linear, and with constant coefficients, is carried out in all details. This class includes, in particular, all compound one-ports of the Classical Circuit Theory, that is, those made of customary resistors, capacitors, and inductors, coupled or not.