# On conjugate times of LQ optimal control problems

2 GECO - Geometric Control Design
Inria Saclay - Ile de France, X - École polytechnique, CNRS - Centre National de la Recherche Scientifique : UMR7641
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 $\vec{H}$.
Document type :
Journal articles
Domain :

https://hal.archives-ouvertes.fr/hal-01096715
Contributor : Luca Rizzi <>
Submitted on : Thursday, December 18, 2014 - 9:37:05 AM
Last modification on : Wednesday, March 27, 2019 - 4:08:31 PM

### Identifiers

• HAL Id : hal-01096715, version 1
• ARXIV : 1311.2009

### Citation

Andrei Agrachev, Luca Rizzi, Pavel Silveira. On conjugate times of LQ optimal control problems. Journal of Dynamical and Control Systems, Springer Verlag, 2014, pp.14. ⟨hal-01096715⟩

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