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Optimal and $H_\infty$ Control of Stochastic Reaction Networks

Corentin Briat, Mustafa Khammash

TL;DR

The continuous-time finite-horizon optimal control problem is formulated first and explicitly solved in the case of unimolecular reaction networks and the problems of the optimal sampled-data control, the continuous $H_\infty$ control, and the sampled- data control of such networks are addressed next.

Abstract

Stochastic reaction networks is a powerful class of models for the representation a wide variety of population models including biochemistry. The control of such networks has been recently considered due to their important implications for the control of biological systems. Their optimal control, however, has been relatively few studied until now. The continuous-time finite-horizon optimal control problem is formulated first and explicitly solved in the case of unimolecular reaction networks. The problems of the optimal sampled-data control, the continuous $H_\infty$ control, and the sampled-data $H_\infty$ control of such networks are addressed next. The results in the unimolecular case take the form of nonstandard Riccati differential equations or differential Lyapunov equations coupled with difference Riccati equations, which can all be solved numerically by backward-in-time integration.

Optimal and $H_\infty$ Control of Stochastic Reaction Networks

TL;DR

The continuous-time finite-horizon optimal control problem is formulated first and explicitly solved in the case of unimolecular reaction networks and the problems of the optimal sampled-data control, the continuous control, and the sampled- data control of such networks are addressed next.

Abstract

Stochastic reaction networks is a powerful class of models for the representation a wide variety of population models including biochemistry. The control of such networks has been recently considered due to their important implications for the control of biological systems. Their optimal control, however, has been relatively few studied until now. The continuous-time finite-horizon optimal control problem is formulated first and explicitly solved in the case of unimolecular reaction networks. The problems of the optimal sampled-data control, the continuous control, and the sampled-data control of such networks are addressed next. The results in the unimolecular case take the form of nonstandard Riccati differential equations or differential Lyapunov equations coupled with difference Riccati equations, which can all be solved numerically by backward-in-time integration.
Paper Structure (21 sections, 12 theorems, 165 equations)

This paper contains 21 sections, 12 theorems, 165 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Stochastic Reaction Networks
  4. Kolmogorov's equations
  5. Moment equations
  6. Continuous-Time Optimal Control of Stochastic Reaction Networks
  7. General stochastic reaction networks
  8. Unimolecular stochastic reaction networks
  9. Optimal Sampled-Data Control of Stochastic Reaction Networks
  10. Sampled-data stochastic reaction networks as stochastic hybrid systems
  11. General stochastic reaction networks
  12. Unimolecular stochastic reaction networks
  13. Continuous-Time $H_\infty$ Control of Stochastic Reaction Networks
  14. Preliminaries
  15. General stochastic reaction networks
  16. ...and 6 more sections

Key Result

Theorem 2

Let us consider a general reaction network $(\boldsymbol{X},\mathcal{R})\ $ satisfying Assumption hyp:CTLQ. Then, the optimal control law that minimizes the cost eq:costCTgeneral is given by where the function $V(t,x)$ is the solution to the Hamilton-Jacobi-Bellman equation given by for $(x,t)\in\mathbb{Z}_{\ge0}^d\times[0,T]$. Moreover, in such a case, the optimal cost $J_T(u^*)$ coincides with

Theorems & Definitions (17)

  • Theorem 2
  • Theorem 4
  • Remark 5
  • Theorem 6
  • Theorem 7
  • Theorem 8
  • Theorem 9
  • Definition 11
  • Definition 12
  • Theorem 14
  • ...and 7 more