Geodesic
Spectrum of the Second Variation
Informatics and Automation, Optimal control and differential equations, Tome 304 (2019), pp. 32-48.

Voir la notice de l'article provenant de la source Math-Net.Ru

Second variation of a smooth optimal control problem at a regular extremal is a symmetric Fredholm operator. We study the asymptotics of the spectrum of this operator and give an explicit expression for its determinant in terms of solutions of the Jacobi equation. In the case of the least action principle for the harmonic oscillator, we obtain the classical Euler identity $\prod _{n=1}^\infty (1-x^2/(\pi n)^2)= \sin x/x$. The general case may serve as a rich source of new nice identities. 
@article{TRSPY_2019_304_a2,
     author = {A. A. Agrachev},
     title = {Spectrum of the {Second} {Variation}},
     journal = {Informatics and Automation},
     pages = {32--48},
     publisher = {mathdoc},
     volume = {304},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TRSPY_2019_304_a2/}
}
TY  - JOUR
AU  - A. A. Agrachev
TI  - Spectrum of the Second Variation
JO  - Informatics and Automation
PY  - 2019
SP  - 32
EP  - 48
VL  - 304
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/TRSPY_2019_304_a2/
LA  - ru
ID  - TRSPY_2019_304_a2
ER  - 
%0 Journal Article
%A A. A. Agrachev
%T Spectrum of the Second Variation
%J Informatics and Automation
%D 2019
%P 32-48
%V 304
%I mathdoc
%U http://geodesic.mathdoc.fr/item/TRSPY_2019_304_a2/
%G ru
%F TRSPY_2019_304_a2
A. A. Agrachev. Spectrum of the Second Variation. Informatics and Automation, Optimal control and differential equations, Tome 304 (2019), pp. 32-48. http://geodesic.mathdoc.fr/item/TRSPY_2019_304_a2/

[1] A. A. Agrachev and Yu. L. Sachkov, Control Theory from the Geometric Viewpoint, Encycl. Math. Sci., 87, Springer, Berlin, 2004 | MR | Zbl

[2] S. V. Bolotin and D. V. Treschev, “Hill's formula”, Russ. Math. Surv., 65:2 (2010), 191–257 | DOI | DOI | MR | Zbl

[3] Conway J.B., Functions of one complex variable, Grad. Texts Math., 11, Springer, New York, 1973 | DOI | MR | Zbl

[4] Courant R., Hilbert D., Methods of mathematical physics, v. 1, Interscience, New York, 1953 | MR

[5] Kato T., Perturbation theory for linear operators, Grundl. Math. Wiss., 132, Springer, Berlin, 1980 | MR | Zbl