Optimal growth for linear processes with affine control

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 $\dot x_α(t) = (G + α(t) F)x_α(t)$, where $G$ and $F$ are $3\times 3$ matrices with some prescribed structure. In the case of constant control $α(t)\equiv α$, we show the existence of an optimal Perron eigenvalue with respect to varying $α$ under some assumptions. Next we investigate the Floquet eigenvalue problem associated to time-periodic controls $α(t)$. 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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