Calculus of Variations and Geometric Measure Theory
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A. Agrachev - L. Rizzi - P. Silveira

On conjugate times of LQ optimal control problems

created by rizzi1 on 12 Nov 2013
modified on 15 May 2017


Published Paper

Inserted: 12 nov 2013
Last Updated: 15 may 2017

Journal: Journal of Dynamical and Control Systems
Year: 2013
Doi: 10.1007/s10883-014-9251-6

ArXiv: 1311.2009v2 PDF
Links: Journal link


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 $\vec{H}$. 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 $\vec{H}$ 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 $\vec{H}$.


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