Methods of solving of the optimal stabilization problem for stationary smooth control systems. Part II Ending

In this article some ideas of Hamilton mechanics and differential-algebraic Geometry are used to exact definition of the potential function (Bellman-Lyapunov function) in the optimal stabilization problem of smooth finite-dimensional systems 3 Geometrical methods. Continuation In the given section the greater attention is given to invariant description of the potential function and its Lagrange manifold. 3.4 Isomorphisms of algebras of smooth functions with marked derivation Let’s say, that the vector field X = ξi(x) ∂ partialxi is in the general position in relation to function V (x), if (n − 1)-th Lie derivative L (n−1) X (V ) = X (n−1)(V ) 6= F (V ). Vector fields X, being in the general position to the function V , will form an open dense set in space of jets of smooth cuts Jp(Rn, TRn), p = 0, 1, 2, . . . ,∞ of tangent stratification TRn in topology of pointwise convergence. Therefore choice of an appropriate vector field is rather free. However the set of vector field of the general position in relation to the function V (x) is not transitive concerning automorphisms TRn→̃TRn, saving function V (x). System of attributes defining orbits of an appropriate operation, c ©1999 by G.Kondrat’ev, A.Balabanov