Optimal growth for linear processes with affine control

Vincent Calvez (1, 2) , Pierre Gabriel (3)

1 UMPA-ENSL - Unité de Mathématiques Pures et Appliquées
2 NUMED - Numerical Medicine
3 BEAGLE - Artificial Evolution and Computational Biology

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.

Mots clés: Optimal control; Infinite horizon; Hamilton-Jacobi equation; Ergodic problem; Non coercive hamiltonian

Domaines: Equations aux dérivées partielles [math.AP]; Optimisation et contrôle [math.OC]; Systèmes dynamiques [math.DS]

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