## A. Agrachev - L. Rizzi - P. Silveira

# On conjugate times of LQ optimal control problems

created by rizzi1 on 12 Nov 2013

modified on 07 Oct 2014

[BibTeX]

*Published Paper*

**Inserted:** 12 nov 2013

**Last Updated:** 7 oct 2014

**Journal:** Journal of Dynamical and Control Systems

**Year:** 2013

**Doi:** 10.1007/s10883-014-9251-6

**Links:**
Arxiv preprint, Journal link

**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 $\vec{H}$. 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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