Spectrum of the second variation

TL;DR

This paper investigates the spectral properties of the second variation operator in optimal control, deriving explicit formulas and classical identities, with potential for discovering new mathematical identities.

Contribution

It provides an explicit expression for the spectrum of the second variation operator and connects it to classical identities like Euler's sine product formula.

Findings

Derived asymptotics of the spectrum of the second variation operator.

Obtained an explicit formula for the determinant in terms of Jacobi equation solutions.

Revealed connections to classical identities such as Euler's sine product.

Abstract

Second variation of a smooth optimal control problem at a regular extremal is a symmetric Fredholm operator. We study 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 a classical Euler identity $\prod\limits_{n=1}^\infty\left(1-\frac{x^2}{(\pi n)^2}\right)=\frac{\sin x}{x}$. General case may serve as a rich source of new nice identities.