# A.A. Agrachev, L. Rizzi, P. Silveira, On conjugate times of LQ optimal control problems

Title: On conjugate times of LQ optimal control problems
Authors: A.A. Agrachev, L. Rizzi, P. Silveira
Journal title: Dynamical and Control Systems
Year: 2015
Issue: 4
Volume: 21
Pages: 625–641
Abstract:

Motivated by the study of linear quadratic optimal control problems, we consider a dynamical system with a constant, quadratic Hamiltonian, and we characterize the number of conjugate times in terms of the spectrum of the Hamiltonian vector field $\overrightarrow{H}$. We prove the following dichotomy: the number of conjugate times is identically zero or grows to infinity. The latter case occurs if and only if $\overrightarrow{H}$ has at least one Jordan block of odd dimension corresponding to a purely imaginary eigenvalue. As a byproduct, we obtain bounds from below on the number of conjugate times contained in an interval in terms of the spectrum of $\overrightarrow{H}$.

ArXiv ID (ENG): 1311.2009