Operator Algebra for Differential Systems

This paper presents the development of an operator algebra for differential systems which is useful in that it allows the transmittance methods commonly applied to linear stationary systems to be extended, almost without modification, to nonstationary systems as well. This operator algebra for differential systems has two operations, addition and multiplication, and two special elements, the null and the identity elements. The null element is defined as any differential system operator that maps every input into a null output. The identity element is defined as any differential system operator that maps every input into an identical output. In a similar fashion the negative and inverse of a particular differential system are defined. Also included is a treatment of reducibility of nonstationary systems and its relation to controllability. Matrix-vector notation is used exclusively. This has the advantage of yielding explicit expressions for sum and product differential systems. Examples illustrate the application of the differential system operator algebra.