Bounds for Deterministic and Stochastic Dynamical Systems using Sum-of-Squares Optimization

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Title: Bounds for Deterministic and Stochastic Dynamical Systems using Sum-of-Squares Optimization
Authors: Fantuzzi, G
Goluskin, D
Huang, D
Chernyshenko, SI
Item Type: Journal Article
Abstract: We describe methods for proving upper and lower bounds on infinite-time averages in deterministic dynamical systems and on stationary expectations in stochastic systems. The dynamics and the quantities to be bounded are assumed to be polynomial functions of the state variables. The methods are computer-assisted, using sum-of-squares polynomials to formulate sufficient conditions that can be checked by semidefinite programming. In the deterministic case, we seek tight bounds that apply to particular local attractors. An obstacle to proving such bounds is that they do not hold globally; they are generally violated by trajectories starting outside the local basin of attraction. We describe two closely related ways past this obstacle: one that requires knowing a subset of the basin of attraction, and another that considers the zero-noise limit of the corresponding stochastic system. The bounding methods are illustrated using the van der Pol oscillator. We bound deterministic averages on the attracting limit cycle above and below to within 1%, which requires a lower bound that does not hold for the unstable fixed point at the origin. We obtain similarly tight upper and lower bounds on stochastic expectations for a range of noise amplitudes. Limitations of our methods for certain types of deterministic systems are discussed, along with prospects for improvement.
Issue Date: 26-Oct-2016
Date of Acceptance: 10-Aug-2016
ISSN: 1536-0040
Publisher: Society of Industrial and Applied Mathematics
Start Page: 1962
End Page: 1988
Journal / Book Title: SIAM Journal on Applied Dynamical Systems
Volume: 15
Issue: 4
Copyright Statement: © 2016, Society for Industrial and Applied Mathematics. Published by SIAM under the terms of the Creative Commons 4.0 license (
Sponsor/Funder: Engineering & Physical Science Research Council (EPSRC)
Funder's Grant Number: EP/J011126/1
Keywords: Science & Technology
Physical Sciences
Mathematics, Applied
Physics, Mathematical
sum-of-squares optimization
time averages
stochastic expectations
34C28, 34D05, 90C22, 93E03
0102 Applied Mathematics
Fluids & Plasmas
Publication Status: Published
Appears in Collections:Aeronautics

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