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Multi-point Gaussian States, Quadratic–Exponential Cost Functionals, and Large Deviations Estimates for Linear Quantum Stochastic Systems

Multi-point Gaussian States, Quadratic–Exponential Cost Functionals, and Large Deviations... This paper is concerned with risk-sensitive performance analysis for linear quantum stochastic systems interacting with external bosonic fields. We consider a cost func- tional in the form of the exponential moment of the integral of a quadratic polynomial of the system variables over a bounded time interval. Such functionals are related to more conservative behaviour and robustness of systems with respect to statistical uncertainty, which makes the challenging problems of their computation and mini- mization practically important. To this end, we obtain an integro-differential equation for the time evolution of the quadratic–exponential functional, which is different from the original quantum risk-sensitive performance criterion employed previously for measurement-based quantum control and filtering problems. Using multi-point Gaus- sian quantum states for the past history of the system variables and their first four moments, we discuss a quartic approximation of the cost functional and its infinite- horizon asymptotic behaviour. The computation of the asymptotic growth rate of this approximation is reduced to solving two algebraic Lyapunov equations. Further approximations of the cost functional, based on higher-order cumulants and their growth rates, are applied to large deviations estimates in the form of upper bounds for tail distributions. We discuss an auxiliary classical Gaussian–Markov http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics and Optimization Springer Journals

Multi-point Gaussian States, Quadratic–Exponential Cost Functionals, and Large Deviations Estimates for Linear Quantum Stochastic Systems

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References (89)

Publisher
Springer Journals
Copyright
Copyright © 2018 by Springer Science+Business Media, LLC, part of Springer Nature
Subject
Mathematics; Calculus of Variations and Optimal Control; Optimization; Systems Theory, Control; Theoretical, Mathematical and Computational Physics; Mathematical Methods in Physics; Numerical and Computational Physics, Simulation
ISSN
0095-4616
eISSN
1432-0606
DOI
10.1007/s00245-018-9512-y
Publisher site
See Article on Publisher Site

Abstract

This paper is concerned with risk-sensitive performance analysis for linear quantum stochastic systems interacting with external bosonic fields. We consider a cost func- tional in the form of the exponential moment of the integral of a quadratic polynomial of the system variables over a bounded time interval. Such functionals are related to more conservative behaviour and robustness of systems with respect to statistical uncertainty, which makes the challenging problems of their computation and mini- mization practically important. To this end, we obtain an integro-differential equation for the time evolution of the quadratic–exponential functional, which is different from the original quantum risk-sensitive performance criterion employed previously for measurement-based quantum control and filtering problems. Using multi-point Gaus- sian quantum states for the past history of the system variables and their first four moments, we discuss a quartic approximation of the cost functional and its infinite- horizon asymptotic behaviour. The computation of the asymptotic growth rate of this approximation is reduced to solving two algebraic Lyapunov equations. Further approximations of the cost functional, based on higher-order cumulants and their growth rates, are applied to large deviations estimates in the form of upper bounds for tail distributions. We discuss an auxiliary classical Gaussian–Markov

Journal

Applied Mathematics and Optimization – Springer Journals

Published: Jul 24, 2018

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