arXiv subjects

A. Habbal

Publications and source records attributed to A. Habbal.

2 recordsLinked to original sources

arXiv2026

Backward problem for a degenerate viscous Hamilton-Jacobi equation: stability and numerical identification

This work is devoted to the analysis of the backward problem for a viscous Hamilton-Jacobi equation with degenerate diffusion and a general Hamiltonian that is not necessarily quadratic. First, we focus on linear degenerate parabolic equations in the nondivergence setting. We prove the conditional stability of the backward problem using Carleman estimates. Then, by a linearization technique, we prove similar results for the nonlinear viscous Hamilton-Jacobi equation. Regarding numerical identification, we first investigate the linear degenerate equation with noisy data using the adjoint state method, combined with a Conjugate Gradient algorithm, to solve the associated minimization problem. Finally, the numerical identification for the nonlinear viscous Hamilton-Jacobi equation is investigated by the Van Cittert iteration. Numerical tests are presented to show the performance of the proposed algorithms.

math.AP↗

arXiv2024

Stability of backward inverse problems for degenerate mean-field game systems

We investigate inverse backward-in-time problems for a class of second-order degenerate Mean-Field Game (MFG) systems. More precisely, given the final datum $(u(\cdot, T),m(\cdot, T))$ of a solution to the one-dimensional mean-field game system with a degenerate diffusion coefficient, we aim to determine the intermediate states $(u(\cdot,t_{0}),m(\cdot,t_{0}))$ for any $t_{0} \in [0, T)$, i.e., the value function and the mean distribution at intermediate times, respectively. We prove conditional stability estimates under suitable assumptions on the diffusion coefficient and the initial state $(u(\cdot,0),m(\cdot,0))$. The proofs are based on Carleman's estimates with a simple weight function. We first prove a Carleman estimate for the Hamilton-Jacobi-Bellman (HJB) equation. A second Carleman estimate will be derived for the Fokker-Planck (FP) equation. Then, by combining the two estimates, we obtain a Carleman estimate for the mean-field game system, leading to the stability of the backward problems.

math.AP↗