By B. Jakubczyk, Witold Respondek
This paintings gathers very important and promising info ends up in subfields of nonlinear keep watch over concept, formerly to be had in journals. It provides the cutting-edge of geometric tools, their functions optimum regulate, and suggestions differences. It goals to teach how geometric keep watch over thought attracts from different mathematical fields to create its personal robust instruments.
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Additional info for Geometry of Feedback and Optimal Control
Since the dimension of the total space of the bundle F* (T*N ) is equal to dim U +dim N and the codimension of trivial section in T*U is equal to dim U we obtain dim CF = dim N . From Thorn's transversality theorem now easily follows U Proposition 1. For arbitrary manifolds U , N ,a generic mapping F is a Morse mapping. ; +N Let dim N = n and suppose the mapping F : U + N is a Morse mapping. ), t E R\O> of the bundle F * ( T * N ) . The projection ( W , U ) c) U maps the manifold CF onto the set of critical points of F .
I . . ,...... ,. _l . ,. . ,. _. . . , I . . Agrachev and Gamkrelidze 46 The last representationimplies the skew-symmetry of the Maslov index in all three variables: p(Ao, h , Aa) = - p ( h , Ao, ha) = -p(Ao, h , AI). Somewhat more difficult is,to prove the following identity rule, cf. [U]: - the chain p(Ao,A1,Az) - p ( A o , A l , A 3 ) + p ( A o , A z I A ~ -) p ( A 1 , A a 1 A 3 ) = O f (10) VAi E L(C), i = 011,2,3. Except the Maslov index there are the following trivial invariants of the triple Ao, A I , Az: (n a dim(Ai n Ai), 0 5i < j 5 2, dim Ai).
F ( s , a ( s ) ) . + :S Introduction 16 The author shows a correspondence of the QBC-problem with the Ashby relation whose application allows to construct a stabilizing feedback (a branching feedback, in general). In particular, he studies in detail the stabilization problem in the plane. In the final part the author considers the optimal control problem in terms of the Ashby relation. Instead system in the case of the stable dynamical system i = ~$(z)’added to the of the stabilization problem the adjoint equation is added in the case of optimal control problem.