Quasi-extremality for Control Systems
نویسنده
چکیده
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
منابع مشابه
Metric Diophantine Approximation for Systems of Linear Forms via Dynamics
The goal of this paper is to generalize the main results of [KM1] and subsequent papers on metric Diophantine approximation with dependent quantities to the set-up of systems of linear forms. In particular, we establish ‘joint strong extremality’ of arbitrary finite collection of smooth nondegenerate submanifolds of R. The proofs are based on quantitative nondivergence estimates for quasi-polyn...
متن کاملOrder Reduction of Optimal Control Systems
The paper presents necessary and sufficient conditions for the order reduction of optimal control systems. Exploring the corresponding Hamiltonian system allows to solve the order reduction problem in terms of dynamical systems, observability and invariant differential forms. The approach is applicable to non-degenerate optimal control systems with smooth integral cost function. The cost functi...
متن کاملStationarity and Regularity Concepts for Set Systems
The paper investigates stationarity and regularity concepts for set systems in a normed space. Several primal and dual constants characterizing these properties are introduced and the relations between the constants are established. The equivalence between the regularity property and the strong metric inequality is established. The extended extremal principle is formulated. keywords: nonsmooth ...
متن کاملStationarity and Regularity of Set Systems
Extremality, stationarity and regularity notions for a system of closed sets in a normed linear space are investigated. The equivalence of different abstract “extremal” settings in terms of set systems and multifunctions is proved. The dual necessary and sufficient conditions of weak stationarity (the Extended extremal principle) are presented for the case of an Asplund space.
متن کاملEstimates of Amplitudes of Transient Regimes in Quasi–Controllable Discrete Systems
Families of regimes for discrete control systems are studied possessing a special quasi–control lability property that is similar to the Kalman control lability property. A new approach is proposed to estimate the amplitudes of transient regimes in quasi–controllable systems. Its essence is in obtaining of constructive a priori bounds for degree of overshooting in terms of the quasi– control la...
متن کامل