Object-oriented hybrid systems of coalgebras plus monoid actions

Abstract Hybrid systems combine discrete and continuous dynamics. We introduce a semantics for such systems consisting of a coalgebra together with a monoid action. The coalgebra captures the (discrete) operations on a state space that can be used by a client (like in the semantics of ordinary (non-temporal) object-oriented systems). The monoid action captures the influence of time on the state space, where the monoids that we consider are the natural numbers monoid ( N ,0,+) of discrete time, and the positive reals monoid ( R ⩾0 ,0,+) of real time. Based on this semantics we develop a hybrid specification formalism with timed method applications: it involves expressions like s . meth @α , with the following meaning: in state s let the state evolve for α units of time (according to the monoid action), and then apply the (coalgebraic) method meth . In this formalism we specify various (elementary) hybrid systems, investigate their correctness, and display their behaviour in simulations. We further define a suitable notion of homomorphism between our hybrid models (of coalgebras plus monoid actions), in such a way that minimal realizations (of the specified behaviour) appear as terminal models. We identify the terminal models of our example specifications, and give general constructions. This leads to an investigation of various topics related to terminality: bisimilarity, invariance, refinement and behaviour-realization adjunctions. In a final section we briefly discuss non-homogeneous hybrid systems (with continuous inputs).