Quasi-extremality for Control Systems

The main object of study will be the Hessian of the "input-output" map of a control system at a certain critical point (extremal of the system). Let us recall, therefore, the definition of the Hessian of a smooth map. Let r ~ § M be a smooth map of some smooth Banach manifold into a finite-dimensional manifold and let ~0 ~ ~. The differential of r at D0 is the linear map D$0~:TB0 ~ + T~(~0)M of the tangent spaces. If we fix local coordinates in the neighborhoods of ~0 and r we can also define the second differential (a symmetric bilinear map of a Banach space into a finite-dimensional space). However, this procedure does not yield a well-defined bilinear map of T~o~XT~o~ into Tr since the quadratic part of a smooth map depends essentially on the choice of local coordinates (for example, if Ds0# is a surjective linear map, then by the Implicit Function Theorem r will be represented by a linear map in certain local coordinates). But if we restrict the second differential to the kernel of the first differential and factorize its values modulo the image of the first differential, the result is a well-defined symmetric bilinear map