The State of Dynamical Input]Output Systems as an Operator

A causal input]output system operating for all time from the indefinite past to the indefinite future may be described by a function space for inputs, a function space for outputs, and a causal operator mapping the input space into the output space. The state of such a system at any instant is defined here as an operator from the space of possible future inputs to that of future outputs. This operator is called the natural state. The output space is taken to be a time-shift-invariant normed linear function space, and the input space is either also such a space or a time-shift-invariant subset thereof. There is flexibility allowed in the choice of these spaces. Both the input]output operator and the operator giving the natural state are themselves taken to be elements of normed linear spaces with one of a particular family of norms called N-power norms. The general development applies to nonlinear and time-varying systems. Continuity and boundedness of the natural state as an operator.and properties of the natural state and its trajectory as related to the input]output description of the system are investigated. Two examples are presented.