Differentiability With Respect To The Initial Condition For Hamilton-jacobi Equations

Esteve-Yague C, Zuazua Iriondo E (2022)


Publication Type: Journal article

Publication year: 2022

Journal

Book Volume: 54

Pages Range: 5388-5423

Journal Issue: 5

DOI: 10.1137/22M1469353

Abstract

We prove that the viscosity solution to a Hamilton-Jacobi equation with a smooth convex Hamiltonian of the form H(x, p) is differentiable with respect to the initial condition. More-over, the directional Gateaux derivatives can be explicitly computed almost everywhere in R-N by means of the optimality system of the associated optimal control problem. We also prove that, in the one-dimensional case in space and in the quadratic case in any space dimension, these directional Gateaux 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 T > 0 and a target function uT, construct an initial condition such that the corresponding viscosity solution at time T minimizes the L-2-distance to u(T). 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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APA:

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

MLA:

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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