The vanishing discount problem for monotone systems of Hamilton-Jacobi equations: a counterexample to the full convergence

Hitoshi Ishii; Hitoshi Ishii

doi:10.3934/mine.2023072

Mathematics in Engineering

2023, Volume 5, Issue 4: 1-10. doi: 10.3934/mine.2023072

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The vanishing discount problem for monotone systems of Hamilton-Jacobi equations: a counterexample to the full convergence

Hitoshi Ishii ^,

Institute for Mathematics and Computer Science, Tsuda University, 2-1-1 Tsuda, Kodaira, Tokyo, 187-8577, Japan

Received: 03 March 2022 Revised: 14 December 2022 Accepted: 04 January 2023 Published: 16 January 2023

In recent years there has been intense interest in the vanishing discount problem for Hamilton-Jacobi equations. In the case of the scalar equation, B. Ziliotto has recently given an example of the Hamilton-Jacobi equation having non-convex Hamiltonian in the gradient variable, for which the full convergence of the solutions does not hold as the discount factor tends to zero. We give here an explicit example of nonlinear monotone systems of Hamilton-Jacobi equations having convex Hamiltonians in the gradient variable, for which the full convergence of the solutions fails as the discount factor goes to zero.
- systems of Hamilton-Jacobi equations,
- vanishing discount,
- full convergence
Citation: Hitoshi Ishii. The vanishing discount problem for monotone systems of Hamilton-Jacobi equations: a counterexample to the full convergence[J]. Mathematics in Engineering, 2023, 5(4): 1-10. doi: 10.3934/mine.2023072

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Abstract

In recent years there has been intense interest in the vanishing discount problem for Hamilton-Jacobi equations. In the case of the scalar equation, B. Ziliotto has recently given an example of the Hamilton-Jacobi equation having non-convex Hamiltonian in the gradient variable, for which the full convergence of the solutions does not hold as the discount factor tends to zero. We give here an explicit example of nonlinear monotone systems of Hamilton-Jacobi equations having convex Hamiltonians in the gradient variable, for which the full convergence of the solutions fails as the discount factor goes to zero.

References

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