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Localized Inverse Design in Conservation Laws and Hamilton-Jacobi Equations

Rinaldo M. Colombo, Vincent Perrollaz

Abstract

Consider the inverse design problem for a scalar conservation law, i.e., the problem of finding initial data evolving into a given profile at a given time. The solution we present below takes into account localizations both in the final interval where the profile is assigned and in the initial interval where the datum is sought, as well as additional a priori constraints on the datum's range provided by the model. These results are motivated and can be applied to data assimilation procedures in traffic modeling and accidents localization.

Localized Inverse Design in Conservation Laws and Hamilton-Jacobi Equations

Abstract

Consider the inverse design problem for a scalar conservation law, i.e., the problem of finding initial data evolving into a given profile at a given time. The solution we present below takes into account localizations both in the final interval where the profile is assigned and in the initial interval where the datum is sought, as well as additional a priori constraints on the datum's range provided by the model. These results are motivated and can be applied to data assimilation procedures in traffic modeling and accidents localization.
Paper Structure (15 sections, 17 theorems, 64 equations, 2 figures)

This paper contains 15 sections, 17 theorems, 64 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Analytic Results
  3. Constrained Inverse Design
  4. Inverse Design Localized in Space
  5. Application to Traffic
  6. Technical Details
  7. Proofs Related to § \ref{['subsec:inverse-design']}
  8. 1. $U_o^\flat \in \mathinner\mathbf{Lip} ({\mathbb{R}}; {\mathbb{R}})$ and for a.e. $x \in {\mathbb{R}}$, $\frac{\mathinner{\mathrm{d}{~}}}{\mathinner{\mathrm{d}{x}}}U_o^\flat (x) \in J$.
  9. 2. For all $x$, the set $\mathcal{M} (x)$ is not empty.
  10. 3. There exists $R>0$ such that if $x \in {\mathbb{R}}$ and $\xi \in \mathcal{M} (x)$, then ${\left|x-\xi\right|} \leq R$.
  11. 4. $\mathcal{M}$ is monotone increasing.
  12. 5. For all $x$, the set $\mathcal{M} (x)$ is a compact interval.
  13. 6. $\mathcal{M}$ is surjective, in the sense that $\bigcup_{x \in {\mathbb{R}}} \mathcal{M} (x) = {\mathbb{R}}$.
  14. 7. Conclusion.
  15. Proofs Related to § \ref{['subsec:time-localization']}

Key Result

Theorem 2.1

Let $f$ satisfy item:1, $J$ be a non empty closed interval and $T$ be positive. Fix $u_T \in {\mathbf{L}^\infty} ({\mathbb{R}}; J)$ satisfying Condition item:2 at $T$. Define, for a fixed $\widecheck x \in {\mathbb{R}}$, for all $x \in {\mathbb{R}}$ Then, $U_o^\flat \in I^{ HJ}_T (U_T; J)$ and $u_o^\flat \in I^{ CL}_T (u_T; J)$.

Figures (2)

  • Figure 1: Notations used in Lemma \ref{['lem:TBA']}.
  • Figure 2: Notation used in \ref{['eq:31']}.

Theorems & Definitions (21)

  • Theorem 2.1
  • Corollary 2.2
  • Proposition 2.3
  • Corollary 2.4
  • Remark 2.5
  • Proposition 2.6: MR3468916
  • Proposition 2.7
  • Proposition 2.8
  • Theorem 2.9
  • Definition 2.10
  • ...and 11 more