By Danny C. Sorensen, Roger J. B. Wets

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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**

**Sample text**

W h e r e A ( P ) = F P F ' - P - ( F P H ' + G ) ( H P H ' + S ) - ~ ( F P H ' + G ) ' + Q. 2) H e r e F, G, H, S a n d Q are c o n s t a n t m a t r i c e s of d i m e n s i o n s n x n, n x m, m x n, m • m a n d n x n, r e s p e c t i v e l y ; S, Q a n d P0 are s y m m e t r i c . ( P r i m e d e n o t e s t r a n s p o s e . ) A l t h o u t h it is a c t u a l l y sufficient to a s s u m e that S is i n v e r t i b l e , for simplicity we shall here take S to be either p o s i t i v e or n e g a t i v e definite.

Daniel D. A. Y. A. Received 15 October 1980 In [1] Brent has given an elegant and concise derivation of a fast, O(n log22n), algorithm for computing Pad6 fractions on downward sloping 'stair-cases' in the Pad6 table. Use of the Gohberg-Semenecul formula, a matrix interpretation of the Christoffel-Darboux formula, then permits the fast solution of related Hankel or Toeplitz systems of linear equations. Unfortunately, Brent's 'proof of correctness' of his algorithm is not complete. In this note we give a similar, and complete, result which leads to an algorithm of more general applicability.

In our approach, we use ideas introduced by Bensoussan and Lions [1] to study impulse control problems in continuous time. They are also related to the theory of fixed points for non decreasing maps, for which the reader is referred to [5]. 1. 1. Assumptions and notations Let X be a metric space and ~ be the Borel ~r-algebra on X. We call X the state space. Let U be a closed subset of a metric space, which will be called the set of controls. 2) is a probability law on ~ for any fixed x, v. Let B be the space of Borel bounded functions on X and C be the subspace of functions which are uniformly continuous and bounded.