formatique et d'Automatiquein France, where he worked on the time delays), linear time-varying systems, linear two-dimensional sysmathematical theory of infinite-dimensional systems. At Georgia Tech, tems, and bilinear systems.he has been the principal investigator for several research grants dealing Dr. Kamen is a member of Sigma Xi, Eta Kappa Nu, Tau Beta Pi; with the theory of infinite-dimensional systems (including systems with and Phi Kappa Phi.Abstract--'Ihe pmpose of this paper is to develop a theory of smoothing for linite dimensional linear stochastic systems in the context of storhastic realization theory. "le basic idea is to embed the given stochastic system ina~ofsimllarsystemsallhavlagthesameootputprocesfandthe same Kalman-Bucy filter. 'Ibis class bas a lafflce shucture with a smallest and a largest element; these two elements completely determine the smoothing estimates.approach enables rn to obtain stochastic interpretations of many important smoothing formulas and to explain the relattomhip between t hem.the representation S is minimal in the sense that there is recommended by A. Ephremides, Chairman of the Estimation Com-Manuscript received April 2, 1979; revised August 17, 1979. Paper mittee. This work was supported by the Air Force Office of Scientific 78-3519. Research, Air Force Systems Command, USAF, under Grant AFOSR-( Q > 0) means that the symmetric matrix Q is positive (nonnegative) definite. Lemma 2.2: Let P be the state covariance function of the linear stochastic system S defined in Section I . litren, for any e > 0, P -' exists and is analytic on the interval [ E , r]. If ll> 0, the same holds for the compIete interval [0, TI. Proof: Integrating (1.2) yields P( t ) = @( t, O)II@( t, 0)' + ~r~( t y 7 ) B (~) B (~) '~( t ,~) ' (2.6) 'Some of these shortcomings have been pointed out in a recent thesis which is positive definite if I I > O ; hence, since A and B by Wall [&I, brought to OUT attention after the submission of this paper. are analytic an [o, TI, SO is ?' -'. NOW assume that n $0. 'For example, the Moore-Penrase pseudo-inverse can be used, Since S is minimal , ( A , B ) must be completely controlla-
