Categorical Systems Theory introduces the following very general approach to dynamical systems:

The idea is that

1. The objects of \mathcal {C} are "spaces" which the dynamics can happen in---we could have \mathcal {C} = \mathsf {Top}, \mathcal {C} = \mathsf {Man}, or even more simply, \mathcal {C} = \mathsf {Set}
2. \mathcal {A}(X \in \mathcal {C}) assigns to each space a category of "bundles over X", families of objects indexed over X in a suitable way. For example, if the spaces are manifolds, we could have \mathcal {A}(X) be the category of bundles over X, or just the vector bundles and linear maps. If \mathcal {C} is the category of sets, we could have \mathcal {A}(X) = \mathsf {Set}_{/X}.
3. The section takes each space to its "tangent bundle"---a collection of "possible changes" in X.

In this situation, a dynamical system is an object S \in \mathcal {C} and a morphism TS \leftrightarrows A in the fiberwise opposite \int \mathcal {A}(-)^\mathrm {op}. In the case where \mathcal {C} is the category of smooth manifolds and smooth maps, and \mathcal {A}(X) is the category of bundles over X, and T is the tangent bundle functor, this means that A = A \to B is some bundle, and a dynamical system consists of maps f: S \to B and f^\#: f^*A \to TS. The idea is that A \to B is the "interface" of the dynamical system, f computes the output associated to each state, A_b is the space of possible inputs for a certain output, and f^\# computes the tangent vector given an input---hence, this is a form of parametrized smooth dynamical system

The central idea of this paper is

1. A costate c: A \to I gives a controller for dynamical systems with interface A
2. Let \alpha : TS \to A be a system with interface A, \beta : TS' \to B one with interface B, \lambda : A \leftrightarrows B a lens, c: B \to I be a controller, and f: \lambda _*\alpha \to \beta be a morphism. Then f implements a bisimulation between the systems \alpha \star \lambda ^*c and \beta \star c
3. Hence the problem of controlling \alpha reduces to the problem of controlling \beta , at least as long as we only care about controlling properties observed by \lambda

For example, in the doctrine of stochastic discrete-time dynamical systems, controlling certain systems amounts to solving a Markov Decision Problem, which, while not necessarily trivial, is a well-studied problem. Hence, we can control more general systems by finding \lambda , f, \beta as above. We think of this as finding an abstraction of the system