Optimal Linear Response for Markov Hilbert-Schmidt Integral Operators and Stochastic Dynamical Systems

We consider optimal control problems for discrete-time random dynamical systems, finding unique perturbations that provokemaximal responses of statistical properties of the system. We treat systemswhose transfer operator has an L-2 kernel, and we consider the problems of finding (i) the infinitesimal perturbation maximising the expectation of a given observable and (ii) the infinitesimal perturbation maximising the spectral gap, and hence the exponential mixing rate of the system. Our perturbations are either (a) perturbations of the kernel or (b) perturbations of a deterministic map subjected to additive noise. We develop a general setting inwhich these optimisation problems have a unique solution and construct explicit formulae for the unique optimal perturbations. We apply our results to a Pomeau-Manneville map and an interval exchange map, both subjected to additive noise, to explicitly compute the perturbations provoking maximal responses.

Automated summary

This paper considers optimal control problems for discrete-time random dynamical systems, aiming to find unique perturbations that can cause maximal responses in statistical properties of the system. The authors focus on systems with an L-2 kernel in their transfer operator and solve two problems: finding the infinitesimal perturbation that maximizes the expectation of a given observable, and finding the infinitesimal perturbation that maximizes the spectral gap and exponential mixing rate of the system. They develop a general framework to ensure the uniqueness of the solution to these optimization problems and provide explicit formulas for the optimal perturbations. The authors apply their results to specific examples, such as the Pomeau-Manneville map and interval exchange map subjected to additive noise, to compute the perturbations that provoke maximal responses.