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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Acknowledgments
The first author has been supported by the grant of the Russian Federation for the state support of research, Agreement No. 14 B25 31 0029. The second author has been supported by the European Research Council, ERC StG 2009 GeCoMethods, contract number 239748, by INdAM (GDRE CONEDP) and by the Institut Henri Poincaré, Paris, where part of this research has been carried out.
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Agrachev, A., Rizzi, L. & Silveira, P. On Conjugate Times of LQ Optimal Control Problems. J Dyn Control Syst 21, 625–641 (2015). https://doi.org/10.1007/s10883-014-9251-6
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DOI: https://doi.org/10.1007/s10883-014-9251-6