Infinite horizon linear quadratic gene regulation in fluctuating environments

We study the evolution of gene regulation in response to environmental fluctuations. For gene dynamics modeled by a linear system, we compute the optimal gene regulator that minimizes the expected square difference between the current protein level and the level that is assumed to be optimal for the current environment plus the cost of protein production/decay, integrated over the life span of the cell. We show that such cost could be represented by a discounted infinite horizon LQR problem with switching equilibria. We also derive a necessary and sufficient condition for the existence of the optimal controller, which can be expressed in terms of a system of Linear Matrix Inequalities (LMI).