## q-bio workshop

# Frontiers in Hamilton Jacobi Equation and its Applications

The Hamilton-Jacobi-(Bellman) equation (HJ(B) Equation) is a partial differential equation that originates from variational methods and analytical mechanics and plays fundamental roles in various fields, such as optimal control. In recent years, this equation has been used in the field of biology as well, in relation to nonequilibrium thermodynamics of chemical reaction networks and biological optimal control problems. In this workshop, we will explore the cutting-edge topics of these HJB equations, along with their connections to life science-related issues.

# GENERAL INFORMATION

Date: 12/May/2023 (Fri): 13:00-18:00

Location: Institute of Industrial Science, University of Tokyo

Style: Hybrid (online + physical)

For online participation, please register here.

If you are interested in participating in person, please contact us (the meeting room is small with a limited capacity)

# Program

13:00-13:05：Opening Remark

Tetsuya J. KOBAYASHI

13:05-13:20：Interface of Hamilton-Jacobi equations and theoretical biology

Tetsuya J. KOBAYASHI, Institute of Industrial Science, University of Tokyo

13:30-14:15：On the study of Hamilton-Jacobi equations from view point of the PDE theory

Hiroyoshi MITAKE, Graduate School of Mathematical Science, University of Tokyo

14:30-15:30：Hamilton-Jacobi method for the large deviation principle and energy landscape of non-equilibrium reactions

Yuan GAO, Department of Mathematics, Purdue University

15:30-16:00：Break

16:00-16:45: Spatially discrete total variation flow map and the time discrete approximation (online)

Yoshikazu GIGA, Graduate School of Mathematical Science, University of Tokyo

17:00-17:30: Stochastic Optimal Estimation and Control under Memory Limitation

Takehiko TOTTORI, University of Tokyo

# Title & Abstract

## Interface of Hamilton-Jacobi equations & theoretical biology

Despite its long history in mathematics, Hamilton-Jacobi equation has rarely been used in the field of theoretical biology. However, HJ equation recently finds some interface with biology. We briefly outline how HJ equation appears in the problems of biology, covering optimal control problems of biological agents, nonequilibrium chemical reaction networks, and others.

## On the study of Hamilton-Jacobi equations from view point of the PDE theory

In this talk, I give several topics of Hamilton-Jacobi equations. In particular, I focus on recent development of asymptotic problems.

I would present them to non-experts audiences.

## Hamilton-Jacobi method for the large deviation principle and energy landscape of non-equilibrium reactions

Non-equilibrium chemical reactions can be modeled by random time changed Poisson process on countable states. The concentration of each species, defined as the molecular number over the size V of the container, can be then regarded as a continuous time Markov chain. The WKB reformulation for the backward equation is Varadhan's discrete nonlinear semigroup and is also a monotone scheme which approximates the limiting first order Hamiltonian-Jacobi equations(HJE). The discrete Hamiltonian is a m-accretive operator, which generates a nonlinear semigroup on countable grids and justifies the well posedness of chemical master equation. The convergence from the monotone schemes to the viscosity solution of HJE is proved via constructing barriers to overcome the polynomial growth coefficient in Hamiltonian. This convergence of Varadhan's discrete nonlinear semigroup to the continuous Lax-Oleinik semigroup yields the large deviation principle for the chemical reaction process at any single time. Consequently, the macroscopic mean-field limit reaction rate equation is recovered. Moreover, the LDP for invariant measures can be used to construct the global energy landscape for non-equilibrium reactions. It is also proved to be a selected unique weak KAM solution to the corresponding stationary HJE.

## Spatially discrete total variation flow map and ite time discrete approximation

We propose a new numerical scheme for a spatially discrete model of total variation flows whose values are constrained to a Riemannian manifold. The difficulty of this problem is that the underlying function space is not convex; hence it is hard to calculate a minimizer of the functional with the manifold constraint even if it exists. We overcome this difficulty by “localization technique” using the exponential map and prove a finite-time error estimate. Finally, we show a few numerical results when the target manifolds are a unit two-dimensional sphere and the group of rotations of the three dimensional Euclidean space.

## Stochastic Optimal Estimation and Control under Memory Limitation – Towards Applications to Biological Systems -

Biological systems survive by estimating and controlling uncertain environments. To understand their survival strategies, partially observable stochastic control (POSC) is expected as an effective approach, which is a combined theory of stochastic optimal estimation and control [1,2]. However, biological systems have severely limited memory, which hampers the applications of POSC. Furthermore, in POSC, Hamilton-Jacobi-Bellman (HJB) equation that needs to be solved becomes a functional differential equation, which is difficult to be solved even numerically. As a result, POSC is insufficient for the practical applications to biological systems.

In order to address these issues, we propose an alternative theoretical framework to POSC, memory-limited POSC (ML-POSC) [3]. ML-POSC can be implemented in memory-limited controllers such as biological systems because it explicitly formulates the memory limitations of the controllers. Furthermore, ML-POSC can reduce HJB equation from a functional differential equation to a partial differential equation by using a recent technique in the mean-field control theory. We demonstrate the effectiveness of ML-POSC by applying it to a linear-quadratic-Gaussian (LQG) problem and a non-LQG problem.

# Support

This workshop is supported by

Ajinomoto (Inc)

Center for Research Enginnering in Medicine and Biology (CREMeB), IIS, UTokyo

JST CREST Mathematical Information Platform

JSPS Project: Information Physics of Living Matters