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Inverse boundary problem for a mean field game system with probability density constraint

Hongyu Liu, Shen Zhang

TL;DR

An inverse boundary problem for the mean field game system where a probability density constraint is enforced on the game agents and hence the population distribution of the game agents should be treated as a probability measure which preserves both positivity and the total population is considered.

Abstract

By following the study in [24], we consider an inverse boundary problem for the mean field game system where a probability density constraint is enforced on the game agents. That is, we consider the case that reflective boundary conditions are enforced and hence the population distribution of the game agents should be treated as a probability measure which preserves both positivity and the total population. This poses significant challenges for the corresponding inverse problems in constructing suitable ``probing modes" which should fulfill such a probability density constraint. We develop an effective scheme in tackling such a case which is new to the literature.

Inverse boundary problem for a mean field game system with probability density constraint

TL;DR

An inverse boundary problem for the mean field game system where a probability density constraint is enforced on the game agents and hence the population distribution of the game agents should be treated as a probability measure which preserves both positivity and the total population is considered.

Abstract

By following the study in [24], we consider an inverse boundary problem for the mean field game system where a probability density constraint is enforced on the game agents. That is, we consider the case that reflective boundary conditions are enforced and hence the population distribution of the game agents should be treated as a probability measure which preserves both positivity and the total population. This poses significant challenges for the corresponding inverse problems in constructing suitable ``probing modes" which should fulfill such a probability density constraint. We develop an effective scheme in tackling such a case which is new to the literature.
Paper Structure (9 sections, 6 theorems, 56 equations)

This paper contains 9 sections, 6 theorems, 56 equations.

Table of Contents

  1. Introduction
  2. Problem setup and background
  3. Preliminaries and statement of main results
  4. Main unique identifiability result
  5. Auxiliary results on the forward problem
  6. A-priori estimates and analysis of the linearized systems
  7. Higher-order linearization
  8. Construction of probing modes
  9. Proof of Theorem \ref{['der F']}

Key Result

Theorem 2.1

Assume that $F_j \in\mathcal{A}$ ($j=1,2$) . Let $\mathcal{N}_{F_j}$ be the measurement map associated to the following system: If for all $m_0\in C^{2+\alpha}(\Omega) \cap \mathcal{O}$, where $\mathcal{O}$ is defined in eq:distr1, then it holds that

Theorems & Definitions (12)

  • Definition 2.1
  • Remark 2.2
  • Theorem 2.1
  • Lemma 3.1
  • Theorem 3.1
  • Theorem 4.1
  • proof
  • Theorem 4.2
  • proof
  • Lemma 5.1
  • ...and 2 more