Imperial College London


Faculty of EngineeringDepartment of Aeronautics

Chair in Aerodynamics



+44 (0)20 7594 5548s.chernyshenko Website




211aCity and Guilds BuildingSouth Kensington Campus






BibTex format

author = {Fantuzzi, G and Goluskin, D and Huang, D and Chernyshenko, SI},
doi = {10.1137/15M1053347},
journal = {SIAM Journal on Applied Dynamical Systems},
pages = {1962--1988},
title = {Bounds for Deterministic and Stochastic Dynamical Systems using Sum-of-Squares Optimization},
url = {},
volume = {15},
year = {2016}

RIS format (EndNote, RefMan)

AB - We describe methods for proving upper and lower bounds on infinite-time averages in deterministicdynamical systems and on stationary expectations in stochastic systems. The dynamics and thequantities to be bounded are assumed to be polynomial functions of the state variables. The methodsare computer-assisted, using sum-of-squares polynomials to formulate sufficient conditions that canbe checked by semidefinite programming. In the deterministic case, we seek tight bounds that applyto 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 describetwo closely related ways past this obstacle: one that requires knowing a subset of the basin ofattraction, 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 deterministicaverages on the attracting limit cycle above and below to within 1%, which requires a lower boundthat does not hold for the unstable fixed point at the origin. We obtain similarly tight upper andlower bounds on stochastic expectations for a range of noise amplitudes. Limitations of our methodsfor certain types of deterministic systems are discussed, along with prospects for improvement.
AU - Fantuzzi,G
AU - Goluskin,D
AU - Huang,D
AU - Chernyshenko,SI
DO - 10.1137/15M1053347
EP - 1988
PY - 2016///
SN - 1536-0040
SP - 1962
TI - Bounds for Deterministic and Stochastic Dynamical Systems using Sum-of-Squares Optimization
T2 - SIAM Journal on Applied Dynamical Systems
UR -
UR -
UR -
VL - 15
ER -