Inverse initial problem under Nash strategy for stochastic reaction-diffusion equations with dynamic boundary conditions

TL;DR

This paper addresses an inverse problem for stochastic reaction-diffusion equations with dynamic boundary conditions, employing Carleman estimates to establish stability and uniqueness results under Nash strategy constraints.

Contribution

It introduces new Carleman estimates and interpolation inequalities for coupled stochastic systems, advancing inverse problem analysis under Nash strategies.

Findings

Established backward uniqueness for the inverse problem

Derived conditional stability estimates for initial conditions

Developed new Carleman estimates for stochastic systems

Abstract

In this paper, we study a multi-objective inverse initial problem with a Nash strategy constraint for forward stochastic reaction-diffusion equations with dynamic boundary conditions, where both the volume and surface equations are influenced by randomness. The objective is twofold: first, we maintain the state close to prescribed targets in fixed regions using two controls; second, we determine the history of the solution from observations at the final time. To achieve this, we establish new Carleman estimates for forward and backward equations, which are used to prove an interpolation inequality for a coupled forward-backward stochastic system. Consequently, we obtain two results: backward uniqueness and a conditional stability estimate for the initial conditions.