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

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. Moreover, the directional Gâteaux derivatives can be explicitly computed almost everywhere in $\mathbb{R}^N$ 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 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 $T>0$ and a target function $u_T$, 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.