# Optimal growth for linear processes with affine control

2 NUMED - Numerical Medicine
UMPA-ENSL - Unité de Mathématiques Pures et Appliquées, Inria Grenoble - Rhône-Alpes
3 BEAGLE - Artificial Evolution and Computational Biology
LIRIS - Laboratoire d'InfoRmatique en Image et Systèmes d'information, Inria Grenoble - Rhône-Alpes, LBBE - Laboratoire de Biométrie et Biologie Evolutive - UMR 5558
Abstract : We analyse an optimal control with the following features: the dynamical system is linear, and the dependence upon the control parameter is affine. More precisely we consider $\dot x_\alpha(t) = (G + \alpha(t) F)x_\alpha(t)$, where $G$ and $F$ are $3\times 3$ matrices with some prescribed structure. In the case of constant control $\alpha(t)\equiv \alpha$, we show the existence of an optimal Perron eigenvalue with respect to varying $\alpha$ under some assumptions. Next we investigate the Floquet eigenvalue problem associated to time-periodic controls $\alpha(t)$. Finally we prove the existence of an eigenvalue (in the generalized sense) for the optimal control problem. The proof is based on the results by [Arisawa 1998, Ann. Institut Henri Poincaré] concerning the ergodic problem for Hamilton-Jacobi equations. We discuss the relations between the three eigenvalues. Surprisingly enough, the three eigenvalues appear to be numerically the same.
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Cited literature [26 references]

https://hal.archives-ouvertes.fr/hal-00681920
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Submitted on : Thursday, March 22, 2012 - 6:53:10 PM
Last modification on : Friday, September 30, 2022 - 11:34:15 AM
Long-term archiving on: : Wednesday, December 14, 2016 - 5:07:56 PM

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• HAL Id : hal-00681920, version 1
• ARXIV : 1203.5189

### Citation

Vincent Calvez, Pierre Gabriel. Optimal growth for linear processes with affine control. 2012. ⟨hal-00681920⟩

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