Realization theory in Hilbert space

A representation theorem for infinite-dimensional, linear control systems is proved in the context of strongly continuous semigroups in Hilbert spaces. The result allows for unbounded input and output operators and is used to derive necessary and sufficient conditions for the realizability in a Hilbert space of a time-invariant, causal input-output operator ℐ. The relation between input-output stability and stability of the realization is discussed. In the case of finite-dimensional input and output spaces the boundedness of the output operator is related to the existence of a convolution kernel representing the operator ℐ. This research has been supported by the Nuffields Foundation.