We introduce contracts for linear dynamical systems with inputs and outputs. Contracts are used to express formal specifications on the dynamic behaviour of such systems through two aspects: assumptions and guarantees. The assumptions are a linear system that captures the available knowledge about the dynamic behaviour of the environment in which the system is supposed to operate. The guarantees are a linear system that captures the required dynamic behaviour of the system when interconnected with its environment. In addition to contracts, we also define and characterize notions of contract refinement and contract conjunction. Contract refinement allows one to determine if a contract expresses a stricter specifications than another contract. On the other hand, contract conjunction allows one to combine multiple contracts into a single contract that fuses the specifications they express.

Motivated by the growing requirements on the operation of complex engineering systems, we present contracts as specifications for continuous-time linear dynamical systems with inputs and outputs. A contract is defined as a pair of assumptions and guarantees, both characterized in a behavioural framework. The assumptions encapsulate the available information about the dynamic behaviour of the environment in which the system is supposed to operate, while the guarantees express the desired dynamic behaviour of the system when interconnected with relevant environments. In addition to defining contracts, we characterize contract implementation, and we find necessary conditions for the existence of an implementation. We also characterize contract refinement, which is used to characterize contract conjunction in two special cases. These concepts are then illustrated by an example of a vehicle following system.