  We consider a compositional construction of approximate abstractions of
interconnected control systems. In our framework, an abstraction acts as a
substitute in the controller design process and is itself a continuous control
system. The abstraction is related to the concrete control system via a
so-called simulation function: a Lyapunov-like function, which is used to
establish a quantitative bound between the behavior of the approximate
abstraction and the concrete system. In the first part of the paper, we provide
a small gain type condition that facilitates the compositional construction of
an abstraction of an interconnected control system together with a simulation
function from the abstractions and simulation functions of the individual
subsystems. In the second part of the paper, we restrict our attention to
linear control system and characterize simulation functions in terms of
controlled invariant, externally stabilizable subspaces. Based on those
characterizations, we propose a particular scheme to construct abstractions for
linear control systems. We illustrate the compositional construction of an
abstraction on an interconnected system consisting of four linear subsystems.
We use the abstraction as a substitute to synthesize a controller to enforce a
certain linear temporal logic specification.
