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 provoke maximal responses of statistical properties of the system. We treat systems whose transfer operator has an L 2 kernel, and we consider the problems of finding (i) the infinitesi...

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Bibliographic Details
Published in:Journal of nonlinear science 2022-12, Vol.32 (6), Article 79
Main Authors: Antown, Fadi, Froyland, Gary, Galatolo, Stefano
Format: Article
Language:English
Subjects:
Analysis
Classical Mechanics
Discrete time systems
Dynamical systems
Economic Theory/Quantitative Economics/Mathematical Methods
Kernels
Mathematical and Computational Engineering
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Maximization
Operators (mathematics)
Optimal control
Optimization
Perturbation
Theoretical
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Summary: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 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 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.
ISSN:0938-8974
1432-1467
DOI:10.1007/s00332-022-09839-0