Continuous and Sampled-Data H∞ Control of Linear Stochastic Reaction Networks

Abstract

The effective control of cellular processes in living cells requires that models of such processes take into account the discrete and stochastic nature of chemical reaction networks that underlie the dynamics inside the cell. Stochastic biochemical reaction networks have emerged as a powerful paradigm for realistic mathematical models of cellular phenomena. While classical control algorithm like PI and PID controllers have been recently proposed as controllers in living cells, methods for designing more advanced optimal controllers remain elusive. Here we formulate the H <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</inf> control problem for linear stochastic reaction networks and provide an approach for its solution using Dynamic Programming—one that gives rise to a nonstandard Riccati differential equation. We next address the H <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</inf> sampled-data control problem for stochastic networks, and present a solution based on Hybrid Dynamic Programming that requires the solution of coupled Riccati differential and difference equations.