Differentiability with respect to the initial condition for Hamilton-Jacobi equations

Esteve-Yague C, Zuazua Iriondo E (2022)

Publication Language: English

Publication Status: Submitted

Publication Type: Journal article, Original article

Future Publication Type: Journal article

Publication year: 2022


Book Volume: 54

Pages Range: 5388-5423

Journal Issue: 5

DOI: 10.1137/22M1469353

Open Access Link: https://doi.org/10.1137/22M1469353


We prove that the viscosity solution to a Hamilton-Jacobi equation with a smooth convex Hamiltonian of the form [katex]H(x,p)[/katex] is differentiable with respect to the initial condition. Moreover, the directional Gâteaux derivatives can be explicitly computed almost everywhere in [katex]R^N[/katex] by means of the optimality system of the associated optimal control problem. We also prove that these directional Gâteaux derivatives actually correspond to the unique duality solution to the linear transport equation with discontinuous coefficient, resulting from the linearization of the Hamilton-Jacobi equation. The motivation behind these differentiability results arises from the following optimal inverse-design problem: given a time horizon [katex]T>0[/katex] and a target function [katex]u_T[/katex], construct an initial condition such that the corresponding viscosity solution at time [katex]T[/katex] minimizes the [katex]L^2[/katex]-distance to [katex]u_T[/katex]. Our differentiability results allow us to derive a necessary first-order optimality condition for this optimization problem, and the implementation of gradient-based methods to numerically approximate the optimal inverse design.

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Esteve-Yague, C., & Zuazua Iriondo, E. (2022). Differentiability with respect to the initial condition for Hamilton-Jacobi equations. SIAM Journal on Mathematical Analysis, 54(5), 5388-5423. https://doi.org/10.1137/22M1469353


Esteve-Yague, Carlos, and Enrique Zuazua Iriondo. "Differentiability with respect to the initial condition for Hamilton-Jacobi equations." SIAM Journal on Mathematical Analysis 54.5 (2022): 5388-5423.

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