THE STATE OF DYNAMICAL INPUT-OUTPUT SYSTEMS AS AN OPERATOR

A causal input-output system operating for all time from the indefinit e past to the indefinite future may be described by a function space f or 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 n atural state. The output space is taken to be a time-shift-invariant n ormed 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 e lements of normed linear spaces with one of a particular family of nor ms called N-power norms. The general development applies to nonlinear and time-varying systems. Continuity and boundedness of the natural st ate (as an operator) and properties of the natural state and its traje ctory as related to the input-output description of the system are inv estigated. Two examples are presented. (C) 1998 Academic Press.