Optimal Response for Hyperbolic Systems by the fast adjoint response method

In a uniformly hyperbolic system, we consider the problem of finding the optimal infinitesimal perturbation to apply to the system, from a certain set $P$ of feasible ones, to maximally increase the expectation of a given observation function. We perturb the system both by composing with a diffeomorphism near the identity or by adding a deterministic perturbation to the dynamics. In both cases, using the fast adjoint response formula, we show that the linear response operator, which associates the response of the expectation to the perturbation on the dynamics, is bounded in terms of the $C^{1,\alpha}$ norm of the perturbation. Under the assumption that $P$ is a strictly convex, closed subset of a Hilbert space $\cH$ that can be continuously mapped in the space of $C^3$ vector fields on our phase space, we show that there is a unique optimal perturbation in $P$ that maximizes the increase of the given observation function. Furthermore since the response operator is represented by a certain element $v$ of $\cH$, when the feasible set $P$ is the unit ball of $\cH$, the optimal perturbation is $v/||v||_{\cH}$. We also show how to compute the Fourier expansion $v$ in different cases. Our approach can work even on high dimensional systems. We demonstrate our method on numerical examples in dimensions 2, 3, and 21.