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Local Hölder stabilities for inverse problems of first-order hyperbolic equations

Giuseppe Floridia, Hiroshi Takase

TL;DR

This work establishes local Hölder stability results for inverse problems of a first-order hyperbolic equation with time-dependent coefficients, using a Carleman estimate. By constructing a weight function and exploiting dissipativity, the authors derive Hölder-stable reconstructions of both an inverse source term and time-independent coefficients from partial lateral boundary data, without imposing extra conditions on the inaccessible boundary. The key contributions are the local stability theorems (ISP and ICP) and their proofs via a detailed Carleman framework, including a quantitative bound that relates interior sources or coefficients to boundary/initial data. These results extend inverse problems for first-order hyperbolic systems by accommodating space-time dependent coefficients and providing explicit stability exponents and data norms, which have potential implications for controllability and identification in hyperbolic PDE models.

Abstract

In this paper, we consider a Cauchy problem for a first-order hyperbolic equation with time-dependent coefficients. Cauchy data are given on a lateral subboundary and we obtain local Hölder stabilities for inverse source and coefficient problems via a Carleman estimate.

Local Hölder stabilities for inverse problems of first-order hyperbolic equations

TL;DR

This work establishes local Hölder stability results for inverse problems of a first-order hyperbolic equation with time-dependent coefficients, using a Carleman estimate. By constructing a weight function and exploiting dissipativity, the authors derive Hölder-stable reconstructions of both an inverse source term and time-independent coefficients from partial lateral boundary data, without imposing extra conditions on the inaccessible boundary. The key contributions are the local stability theorems (ISP and ICP) and their proofs via a detailed Carleman framework, including a quantitative bound that relates interior sources or coefficients to boundary/initial data. These results extend inverse problems for first-order hyperbolic systems by accommodating space-time dependent coefficients and providing explicit stability exponents and data norms, which have potential implications for controllability and identification in hyperbolic PDE models.

Abstract

In this paper, we consider a Cauchy problem for a first-order hyperbolic equation with time-dependent coefficients. Cauchy data are given on a lateral subboundary and we obtain local Hölder stabilities for inverse source and coefficient problems via a Carleman estimate.
Paper Structure (6 sections, 3 theorems, 64 equations, 1 figure)

This paper contains 6 sections, 3 theorems, 64 equations, 1 figure.

Table of Contents

  1. Introduction and main result
  2. Inverse source problem
  3. Inverse coefficient problem
  4. Carleman estimate
  5. Proof of Theorem \ref{['ISP']}
  6. Proof of Theorem \ref{['ICP']}

Key Result

Theorem 1.5

Let $g\in H^1(0,T;H^\frac{1}{2}(\partial\Omega))$, $A^0\in C^1(\overline{Q})\cap L^\infty(\Omega\times(0,\infty))$ satisfying $\min_{(x,t)\in\overline{Q}}A^0(x,t)>0$, and $A\in C^2(\overline{Q};\mathbb{R}^d)$ satisfying positivity, finiteness, and spd. Assume given and Let $\Sigma\subset\Sigma_+$ be an open subset satisfying Then, there exist $\varepsilon^*>0$ such that for any $\varepsilon\in(\

Figures (1)

  • Figure 1: On the assumption \ref{['geometry']}.

Theorems & Definitions (10)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Remark 1.4
  • Theorem 1.5
  • Example 1.6
  • Theorem 1.7
  • Proposition 1.8
  • proof : Proof of Theorem \ref{['ISP']}
  • proof : Proof of Theorem \ref{['ICP']}