Keskeiset käsitteet
The authors propose a new framework for optimal control of stochastic reaction networks using a control cost function based on Kullback-Leibler divergence, which allows for efficient computation of optimal solutions by linearizing the Hamilton-Jacobi-Bellman equation.
Tiivistelmä
The authors present a new framework for optimal control of stochastic reaction networks, which are inherently nonlinear and involve a discrete state space. They formulate the optimal control problem using a control cost function based on the Kullback-Leibler (KL) divergence, which naturally accounts for population-specific factors and simplifies the complex nonlinear Hamilton-Jacobi-Bellman (HJB) equation into a linear form.
The key highlights of the paper are:
- The KL control cost function allows for the linearization of the HJB equation through the Cole-Hopf transformation, facilitating efficient computation of optimal solutions.
- The authors demonstrate the effectiveness of their approach by applying it to the control of interacting random walkers, Moran processes, and SIR models, and observe the emergence of mode-switching phenomena in the control strategies.
- For the interacting random walker problem, the authors derive analytical solutions for the optimal control due to the linearization.
- In the Moran process and SIR model, the authors identify a critical parameter value at which the optimal control strategy undergoes a transition between strong control and no control, leading to mode-switching behavior.
- The authors discuss the potential extensions of their framework, such as incorporating risk-sensitivity, addressing large-scale models, and exploring more flexible control cost functions.
Overall, the authors' approach provides new opportunities for applying control theory to a wide range of biological problems involving stochastic reaction networks.
Tilastot
The authors provide analytical expressions for the value function and optimal control strategies in the following cases:
Minimum exit time problem for a single random walker:
V(x, β) = -γ(β)|x*-x|
k†(x, x+1) = κeγ(β) if x < x*, κe-γ(β) if x > x*
k†(x, x-1) = κe-γ(β) if x < x*, κeγ(β) if x > x*
Minimum exit time problem for two interacting random walkers:
V(x1, x2, β) ≤ min{V(x1, β), V(x2, β)} = -γ(β)max{|x*-x1|, |x*-x2|}
Maximum exit time problem for the Moran process:
V(n, β) = log(eK0(n,β) + eKN(n,β))
where K0(n, β) and KN(n, β) have explicit expressions using the eigenvalues of the transition rate matrix.