By B. Jakubczyk, Witold Respondek

This paintings gathers vital and promising info ends up in subfields of nonlinear keep an eye on idea, formerly on hand in journals. It provides the state-of-the-art of geometric tools, their functions optimum keep watch over, and suggestions alterations. It goals to teach how geometric keep watch over concept attracts from different mathematical fields to create its personal robust instruments.

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**Example text**

_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).

It is easy to show that in this case X, is a trajectory of the Hamiltonian system defined by the Hamiltonian H . The following assertion is a geometric formulation of the classical method of characteristics for solution of differential equations in partial derivatives of first order. Proposition 2. Let H : T*M -+ R be a smooth function and assume that CO C T * M is a smooth Lagrangian submanifold. Suppose that H i I TACO# 0 VX E CO n H-l(O). I+ p(t,X) Be a trajectory of theHamiltoniansystemwithHamiltonian H andthe initialcondition p ( 0 , X) = X E COn H-'(O).

It is easily seen that the bilinear form (211, W Z ) I+ (wF:wl)wz is symmetric, hence it i s restored by the quadratic form wF,h. The following test for being a Morse mapping is established by a straightforward computation. Proposition 2. A smooth mappingâ€™Fis a Morse mapping iff the linear mapping wF$ is injective at every Lagrangian point ( W , U ) . If the Hessian wF,h is a nondegenerate quadratic form then wF; is injective. The opposite is not always true. For example, Symplectic Methodsfor Optimizationand Control - 31 for the following Morse mapping (ti) U1 (U1uz+(ul)S ) U' E R, i = 1,2, for which the origin is a cusp point, the Hessian is equal to zero for (0;l), 211 = 212 = 0.