Differentiability With Respect to the Initial Condition for Hamilton--Jacobi Equations

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