Critical point theory for smooth functions on Hilbert manifolds with singularities and its application to some optimal control problems

Geometrical and topological methods in optimal control theory

“…)+~(oT!N(uo) C "~'-~o,V xo,Nk o) It ~ ag(T) is the coindex space of the Hessian H~ o (J) of the function J at the critical point Uo(-). As was already shown in [60,63], Propositions 6.5-6.8 imply Morse Inequalties. Let (6.2) be a finitely defined system of constant rank satisfying the continuation condition (6.3), and let fo be a smooth integrand of class C ~ for which conditions (6.6), (6.7), (6.15), (6.16), (6.17), (6.18) The author expresses his gratitude to his teachers, A.…”

“…Nevertheless, this loss-of-smoothness effect does not prevent the construction of a theory analogous to the Morse theory; it was already done by the author in [60] (for abstract version, see [63]). Another approach to the definition of an index of the Ct-functionals on a Hilbert manifold has been recently undertaken in [422].…”

“…Let (6.2) be a smooth control system of constant rank satisfying the finite-definiteness condition, continuation condition (6.3), and let fo be a smooth integrand of class C ~ for which conditions (6.6), (6.7), and conditions This result permits one at once to obtain the following version of the Lusternik-Schnirelman theory for the problem (P) (see [63] More precise bounds also hold in terms of numbers similar to the numbers in the formulation of the Lusternik-Schnirelman theorem in Subset. 6.1 (see [58,59,63]).…”

“…The proofs of these statements are based on the deformations defined by the so-called pseudogradient vector field (see [63,56,59]). It is necessary to point out that the latter notion is different from Palais' construction [1254] for the Finsler case and is motivated essentially by the geometry of a transversal convex subset with respect to which it is defined.…”

Corrigendum to “The Conley conjecture for Hamiltonian systems on the cotangent bundle and its analogue for Lagrangian systems” [J. Funct. Anal. 256 (9) (2009) 2967–3034]

“…As in the proof of Lemma 2 in [26, p. 201] (see also Lemma 5.2 of [27]) we can prove: As in the proof of Lemma 2 in [26, p. 201] (see also Lemma 5.2 of [27]) we can prove:…”