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Extra resources for Algorithms and Theory in Filtering and Control: Proceedings of the Workshop on Numerical Techniques for Systems Engineering Problems, Part L

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1) is given by P ( t ) = Y ( t ) X ( t ) -1. Proof. 8) is still satisfied with Y exchanged for PX, where P is the solution of the Riccati equation. 3), may be written X ( t ) = [I - H ' R ( t ) - x H P ( t ) ] A ' X ( t + l). 2). Hence, by uniqueness, Y = PX. 1, so is X ( t + 1). Hence, since X(O) = I, X ( t ) is nonsingular for t = 0, 1. . . tl. 1. Let A be nonsingular. For t = 0, 1..... tr, let S ( ( t ) : = [X(t)'] -l. A. 15) Proof. 11). 4. 4), we shall take a closer look at the Hamiltonian function ~.

N - 1, f o r j - n. -~-) = f ( y . _~ j) 1)^ " n - l - j -- A. Bensoussan! _j). T h e r e f o r e we deduce, by addition when j ranges from 0 to n - 1, u,,(x)= J~,(~'") which c o m p l e t e s the proof of the lemma. Let us now interpret the decreasing scheme. We set T'" = {V I vi(xo .... (xo ..... 3. The decreasing scheme has the following interpretation u"(x) = Min Jx(V). 24) Proof. Let V @ ~". ~n-1f(YJ, ] u"(x) < EV'XL~=0 vj)a j + a " ~ ( y . ) . ) <---E v'x ~ aJ-"f(yj, vi). 1. 13). ), wM(yj, vi) = fM(yj, vj) + ~q~VJwM(yj, v,), since V E ~V", and j --- n.

Proceedings of the seventh Princeton symposium on information and systems science (Department of Electrical Engineering, Princeton University, Princeton, NJ) pp. 344-352. [9] A. Lindquist, "A new algorithm for optimal filtering of discrete-time stationary processes", SIAM Journal on Control 12 (1974) 736-746. [10] A. Lindquist, "Optimal filtering of continuous-time stationary processes by means of the backward innovation process", S l A M Journal on Control 12 (1974) 747-754. [ll] A. Lindquist, "On Fredholm integral equations, Toeplitz equations and Kalman-Bucy filtering", Applied Mathematics and Optimization l (1975) 355-373.

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