Barrier Functionals for the Analysis of Complex Systems: An Optimization-Based Approach

TL;DR

This paper introduces a novel optimization-based methodology using barrier functionals to analyze complex systems, enabling bounding of stochastic differential equations and verification of set avoidance in PDE systems without solution approximation.

Contribution

It develops a new barrier functional approach for complex system analysis that avoids solution approximation and leverages semi-definite programming for verification tasks.

Findings

Provides moments bounds for nonlinear SDEs

Formulates analysis problems as semi-definite programs

Does not require solution approximation or trajectory stability

Abstract

We propose a methodology to address two analysis problems concerning complex systems, namely bounding state functionals of stochastic differential equations (SDEs) and verifying set avoidance of systems described by partial differential equations (PDEs). The proposed method is based on barrier functionals, which are functionals of the states of the studied systems. The proposed method does not require the approximation of solutions nor the stability of trajectories. In the case of SDEs, the formulation relies on a generalized version of the Feynman-Kac formula and results in moments bounds for nonlinear SDEs. Furthermore, we show that the analysis problems can be cast as optimization problems and can be solved by semi-definite programming.