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J. E. Marsden, (1978), Foundations of Mechanics, 2nd Ed. Benjamin / Cummings Pub. Com. Inc. P. Novikov, Symplectic geometry, (1980) in Dynamical Systems IV, Encyclopedia if Mathematical Sciences, Vol. , (1986)Introduction to Differentiable Manifolds and Riemannian Geometry, 2nd ed. Academic Press, INC. [5] Brockett R. , (1977) Control theory and analytic mechanics, in Geometric Control Theory, C. Martin and R. Sci. al, (1998), On generalized Hamiltonian systems, Proc. , Z. Xi, Y. Hong, and H.

12 Stabilization and Hx control for dissipative Hamiltonian systems The following two propositions have been proved for standard Hamiltonian systems. ). Using parallel arguments, one can see that they remain true for the controlled dissipative Hamiltonian systems. 1) is passive. 42 D. 1) is stabilizable. 3 LetR bean nxn symmetric matrix, and g is an nxm matrix. Then there exists an mxm symmetric matrix P such that R + gPg1 >0 if and only if Span{R} + Span{g} = R" Proof. (Sufficiency) Choose a coordinate frame such that R= 0 0 where k = rank(R).

Choosing CC = \IE it is easy to see that aG — R>0. 8) 46 D. Cheng It is easy to see that there exists a constant a such that c = min{a,a2/2} . a2 SoaR + — Gj - G 2 >c(R + Gi)-G2 ^ 0 . 7) is satisfied. 1) again. It is easy to see that the equilibrium of the system is _ 1 . e. —— ^ (xd-xd)V}Td — <1. and it is easy to know that 0<5q

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