# Optimal Growth for Simple Polymerization Processes, and Application to Controlled Proliferation of Prion

Vincent Calvez^{*} and Pierre Gabriel

We analyze the following optimal control problem: $\dot x_\alpha(t) = (G + \alpha(t) F)x_\alpha(t)$, where $G$ and $F$ are $3\times 3$ matrices with some prescribed structure. This simple system mimicks a growth-fragmentation process modeling prion proliferation. Assuming constant control $\alpha(t)\equiv \alpha$, the system possesses an optimal Perron eigenvalue with respect to varying $\alpha$ under suitable hypotheses. We prove the existence of an eigenvalue (in the generalized sense) for the full optimal control problem under suitable hypotheses. Indeed the reward function $V(t,x)$ satisfies \[\exists \lambda\; \forall x \in \left(\mathbb{R}_+\setminus\{0\}\right)^3:\; \lim_{T\to +\infty}\dfrac{1}{T}\log V(T,x) = \lambda\, .\] Surprisingly enough, the two eigenvalues appear to be numerically the same.

Mathematics Subject Classification: 35Q92 35Q93

Keywords: optimal control; Hamilton-Jacobi-Bellman equation; ergodicity; prion proliferation

Minisymposion: Modelling and Optimization in Mathematical Biology