ing yet further from reality, we might proscribe the simultaneous effect of two or more methods on an object's state; doing so, we impose a monoid structure on the fixed set of methods proper to an object class. Applying methods one after the other corresponds to multiplication in the monoid, and applying no methods corresponds to the identity of the monoid. A monoid is a set M with an associative binary operation ffl M : M \ThetaM ! M , usually referred to as `multiplication', which has an identity element e M 2 M . If M = (M; ffl M ; e M ) is a monoid, we often write just M for M, and e for e M ; moreover for m;m 0 2 M , we usually write mm 0 instead of m ffl M m 0 . For example, A , the set of lists containing elements of A, together with concatenation ++ : A \ThetaA ! A and the empty list [ ] 2 A , is a monoid. This example is especially important for the material in later sections. A monoid homomorphism is a structure preserving map between the carriers of ...

Abstract. In this paper, we introduce and study quotients of fully nonlinear control systems. Our definition is inspired by categorical definitions of quotients as well as recent work on abstractions of affine control systems. We show that quotients exist under mild regularity assumptions and characterize the structure of the quotient state/input space. This allows one to understand how states and inputs of the quotient system are related to states and inputs of the original system. We also introduce a notion of projectability which turns out to be equivalent to controlled invariance. This allows one to regard previous work on symmetries, partial symmetries, and controlled invariance as leading to special types of quotients. We also show the existence of quotients that are not induced by symmetries or controlled invariance. Such decompositions have a potential use in a theory of hierarchical control based on quotients.

Abstract. Abstraction is a natural way to hierarchically decompose the analysis and design of hybrid systems. Given a hybrid control system and some desired properties, one extracts an abstracted system while preserving the properties of interest. Abstractions of purely discrete systems is a mature area, whereas abstractions of continuous systems is a recent activity. In this paper we present a framework for abstraction that applies to discrete, continuous, and hybrid systems. We introduce a composition operator that allows to build complex hybrid systems from simpler ones and show compatibility between abstractions and this compositional operator. Besides unifying the existing methodologies we also propose constructions to obtain abstractions of hybrid control systems.

In this paper we provide a bridge between the infinite state models used in control theory to describe the evolution of continuous physical processes and the finite state models used in computer science to describe software. We identify classes of control systems for which it is possible to construct equivalent (bisimilar) finite state models. These constructions are based on finite, but otherwise arbitrary, partitions of the set of inputs or outputs of a control system.

There is a growing interest in models for probabilistic systems. This fact is motivated by engineering applications, namely in problems concerning the evaluation of the performance of systems. It is of

In this sequel to [HM02], we explore some of the global aspects of the category of paracategories. We establish its (co)completeness and cartesian closure. From the closed structure we derive the relevant notion of transformation for paracategories. We set-up the relevant notion of adjunction between paracategories and apply it to define (co)completeness and cartesian closure, exemplified by the paracategory of bivariant functors and dinatural transformations. We introduce partial multicategories to account for partial tensor products. We also consider fibrations for paracategories and their indexed-paracategory version. Finally, we instantiate all these concepts in the context of probabilistic automata.