[Paper Review] Homogenization of a transmission problem with Hamilton-Jacobi equations and a two-scale interface. Effective transmission conditions

Yves Achdou, Nicoletta Tchou|arXiv (Cornell University)|Jul 6, 2017

Advanced Mathematical Modeling in Engineering29 references4 citations

TL;DR

This paper studies the homogenization of a transmission problem with Hamilton-Jacobi equations across a two-scale oscillatory interface, where the interface amplitude is $\varepsilon$ and the period is $\varepsilon^2$. As $\varepsilon \to 0$, the value function converges to a solution of Hamilton-Jacobi equations in two half-planes separated by a flat interface $\Gamma$, with an effective transmission condition on $\Gamma$ that encodes the memory of the vanishing oscillations, derived via viscosity solution techniques and asymptotic analysis of oscillatory Hamiltonians.

ABSTRACT

We consider a family of optimal control problems in the plane with dynamics and running costs possibly discontinuous across a two-scale oscillatory interface. Typically, the amplitude of the oscillations is of the order of $\\epsilon$ while the period is of the order of $\\epsilon$ 2. As $\\epsilon$ $\ ightarrow$ 0, the interfaces tend to a straight line $\\Gamma$. We study the asymptotic behavior of the value function as $\\epsilon$ $\ ightarrow$ 0. We prove that the value function tends to the solution of Hamilton-Jacobi equations in the two half-planes limited by $\\Gamma$, with an effective transmission condition on $\\Gamma$ keeping track of the oscillations.

Motivation & Objective

• To analyze the asymptotic behavior of the value function in a two-dimensional optimal control problem with discontinuous dynamics and running costs across a two-scale oscillatory interface.
• To characterize the limit problem as a system of Hamilton-Jacobi equations in two half-planes separated by a flat interface $\Gamma$, with an effective transmission condition on $\Gamma$.
• To derive the effective transmission condition that retains information from the vanishing oscillations of the interface, which are of amplitude $\varepsilon$ and period $\varepsilon^2$.
• To extend previous homogenization results in the case of equal-scale oscillations ($\varepsilon$ amplitude and period) to the singularly perturbed regime where the period is $\varepsilon^2$.
• To establish convergence of the value function to the viscosity solution of the effective problem using comparison principles and test-function techniques from PDE theory.

Proposed method

• Use of viscosity solution theory for Hamilton-Jacobi equations with transmission conditions on oscillatory interfaces.
• Application of a reduced set of test-functions, inspired by Imbert and Monneau (2018), to handle the transmission condition at the interface.
• Asymptotic analysis in the limit $\varepsilon \to 0$, where the interface tends to a flat line $\Gamma$, with the oscillations scaled via $\varepsilon$ and $\varepsilon^2$.
• Construction of correctors and use of ergodic-type arguments to derive the effective Hamiltonian at the interface, based on the behavior near the junction point.
• Employment of a single test-function at the interface, following techniques from [19, 18], to handle the non-mixing condition at the interface.
• Use of modulus of continuity estimates and contradiction arguments in the proof of convergence, relying on uniform bounds on gradients and Hamiltonian regularity.

Experimental results

Research questions

• RQ1 How does the value function of an optimal control problem behave when the interface has oscillations of amplitude $\varepsilon$ and period $\varepsilon^2$ as $\varepsilon \to 0$?
• RQ2 What is the effective transmission condition on the limiting flat interface $\Gamma$ that captures the influence of the vanishing two-scale oscillations?
• RQ3 Can the convergence of the value function to a solution of a Hamilton-Jacobi system in two half-planes be established when no mixing of dynamics is allowed at the interface?
• RQ4 How does the effective transmission condition depend on the dynamics, running costs, and the geometry of the oscillatory interface?
• RQ5 What techniques are necessary to handle the lack of a change of variables that flattens the interface, unlike in the case of equal-scale oscillations?

Key findings

• The value function converges locally uniformly to the viscosity solution of a Hamilton-Jacobi equation in each half-plane separated by the flat interface $\Gamma$.
• The effective transmission condition on $\Gamma$ is derived as a limit of oscillatory data and depends on the dynamics, running costs, and the two-scale structure of the interface.
• The effective transmission condition preserves memory of the oscillations through a homogenized Hamiltonian that emerges from the asymptotic behavior near the interface.
• The convergence proof relies on a comparison principle for the original problem and a contradiction argument using uniform continuity estimates on the Hamiltonian and gradient bounds.
• The effective transmission condition is characterized via the limit of ergodic constants associated with the oscillatory cell problem, analogous to junction problems in networks.
• The method extends previous results in [3] to the singular perturbation regime where the period is $\varepsilon^2$, requiring new analytical tools due to the inability to flatten the interface via change of variables.