By Jean-Paul Gauthier

This paintings provides a normal thought in addition to optimistic method with a purpose to remedy "observation problems," specifically, these difficulties that pertain to reconstructing the entire information regarding a dynamical procedure at the foundation of partial saw information. A basic method to manage procedures at the foundation of the observations can be constructed. Illustrative yet sensible purposes within the chemical and petroleum industries are proven.

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One should be con- scious of the fact that this is not a restriction: Any C ∞ manifold possesses a compatible C ω structure (see [26, p. 66]). Again, in this chapter, U = I du , where I is a closed bounded interval. Because we make an extensive use of subanalytic sets and their properties, this compactness assumption cannot be relaxed. 1. Definitions and Notations The systems under consideration are of the form ( ) dx = f x, u (0) ; y = h x, u (0) dt (33) dx = f x, u (0) ; y = h(x), dt (34) or ( ) in order to take into account the more practical cases in which the output function h does not depend on u: The proofs of the genericity results in that case are not different from the proofs in the general case, where h depends on u, but these results do not follow from the results in the general case.

The case f (y, u (0) ) = 0 is similar. Let (x, y, u (0) ) ∈ (X × X \ X ) × U. The typical fiber Bˆ 5 (k, x, y, u (0) ) of Bˆ 5 (k) in J k S(x, u (0) ) × J k S(y, u (0) ) × R (k−1)du is characterized by the following properties: (i) f (x, u (0) ) = 0 and (ii) k (x, u (0) , u ) − k (y, u (0) , u ) = 0. 4. P1: FBH CB385-Book CB385-Gauthier June 21, 2001 11:11 Char Count= 0 The Case d y > du 50 Let G be the subset of J k S(x, u (0) ) × J k S(y, u (0) ) × R (k−1)du of all tuples (x, u (0) ), j k (y, u (0) ), u ), such that f (x, u (0) ) = 0 and let χ : G → R kd y be the mapping: χ( j k (x, u (0) ), j k (y, u (0) ), u ) = k (x, u (0) , u ) − (0) (0) −1 ˆ k (y, u , u ).

Ti ). This shows (i). ˆ x ). To prove Let Y be any open subset of X. Set c = supx∈Y Codim(D(u) (ii), it is sufficient to prove that c = n. ,r open. Now, consider an open subset W of Z , such that D(u) Ker(ω(x, ti )) for some t1 , . . , tr . , ti ) = d X Fi, where Fi : W → R d y is the function ti Fi (x) = 0 ˆ x), u(τ ˆ ))dτ. h(ϕτ (u, ˆ Hence, the restriction D(u)|W is an integrable distribution, the leaves of which are the connected components of the level manifold of the mappings F1 , . . , Fr .