  Control systems are usually modeled by differential equations describing how
physical phenomena can be influenced by certain control parameters or inputs.
Although these models are very powerful when dealing with physical phenomena,
they are less suitable to describe software and hardware interfacing the
physical world. For this reason there is a growing interest in describing
control systems through symbolic models that are abstract descriptions of the
continuous dynamics, where each "symbol" corresponds to an "aggregate" of
states in the continuous model. Since these symbolic models are of the same
nature of the models used in computer science to describe software and
hardware, they provide a unified language to study problems of control in which
software and hardware interact with the physical world. Furthermore the use of
symbolic models enables one to leverage techniques from supervisory control and
algorithms from game theory for controller synthesis purposes. In this paper we
show that every incrementally globally asymptotically stable nonlinear control
system is approximately equivalent (bisimilar) to a symbolic model. The
approximation error is a design parameter in the construction of the symbolic
model and can be rendered as small as desired. Furthermore if the state space
of the control system is bounded the obtained symbolic model is finite. For
digital control systems, and under the stronger assumption of incremental
input-to-state stability, symbolic models can be constructed through a suitable
quantization of the inputs.
