# Optimal linear response for Markov Hilbert-Schmidt integral operators and stochastic dynamical systems

arXiv: https://arxiv.org/abs/2101.09411 · HTML: https://mlsanity.com/paper/2101.09411/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Jan 2021.

## Built on by (most cited first, 1 of 1)

- [Invariant Measures in Time-Delay Coordinates for Unique Dynamical System Identification](https://mlsanity.com/paper/2412.00589/) (2024)

## Abstract (arXiv)

We consider optimal control problems for discrete-time random dynamical systems, finding unique perturbations that provoke maximal responses of statistical properties of the system. We treat systems whose transfer operator has an L2 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 in which 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.