Operators Arising as Second Variation of Optimal Control Problems and Their Spectral Asymptotics
Tóm tắt
We compute the asymptotic for the eigenvalues of a particular class of compact operators deeply linked with the second variation of optimal control problems. We characterize this family in terms of a set of finite dimensional data and we apply this results to a particular class of singular extremal to get a nice description of the spectrum of the second variation.
Tài liệu tham khảo
Agrachev A, Stefani G, Zezza P. An invariant second variation in optimal control. Internat J Control 1998;71(5):689–715.
Agrachëv AA. Quadratic mappings in geometric control theory. Problems in geometry, Vol. 20 (Russian). Itogi Nauki i Tekhniki. Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1998. Translated in J. Soviet Math.; 1988. p. 111–205. 51 (1990), no. 6, 2667–2734.
Agrachev AA. Spectrum of the second variation. Tr Mat Inst Steklova 2019;304(Optimal noe Upravlenie i Differentsial nye Uravneniya):32–48.
Agrachev A, Barilari D, Boscain U, vol. 181. A comprehensive introduction to sub-Riemannian geometry, Cambridge studies in advanced mathematics. Cambridge: Cambridge University Press; 2020. From the Hamiltonian viewpoint, With an appendix by Igor Zelenko.
Agrachev A, Beschastnyi I. Jacobi fields in optimal control: one-dimensional variations. J Dyn Control Syst 2020;26(4):685–732.
Agrachev A, Beschastnyi I. 2022. Jacobi fields in optimal control: Morse and Maslov indices. Nonlinear Anal, 214. Paper No. 112608, 47.
Agrachev AA, Sachkov YL. Control theory from the geometric viewpoint. Encyclopaedia of mathematical sciences. Berlin: Springer; 2004. Control Theory and Optimization, II.
Agrachev AA, Beschastnyi IY. Symplectic geometry of constrained optimization. Regul Chaotic Dyn 2017;22(6):750–770.
Chui CK. Concerning rates of convergence of Riemann sums. J Approximation Theory 1971;4:279–287.
Jean F. Control of nonholonomic systems: from sub-Riemannian geometry to motion planning, SpringerBriefs in Mathematics. Cham: Springer; 2014.
Kato T. Perturbation theory for linear operators, Classics in mathematics. Berlin: Springer; 1995. Reprint of the 1980 edition.
Rudin W. Functional analysis, 2nd ed., International series in pure and applied mathematics. New York: McGraw-Hill, Inc.; 1991.