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 , α norm of the perturbation. Under the assumption that P is a strictly convex closed subset of a Hilbert space H , and that P can be linearly and continuously mapped to the space of C3 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 H , when the feasible set P is the unit ball of H , the optimal perturbation is v / | | v | | H . We also show how to compute the Fourier expansion of 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. We also describe an approximation scheme for the computation of v in other cases where a Fourier basis of H is not easily accessible.