Barrier Functionals for Output Functional Estimation of PDEs

TL;DR

This paper introduces barrier functionals for PDEs to compute bounds on output functionals, utilizing SOS programming and SDPs for polynomial data, with an illustrative example.

Contribution

It presents a novel barrier functional approach for output estimation of PDEs, integrating SOS programming and SDPs for polynomial cases.

Findings

Successfully computes bounds on PDE output functionals.

Demonstrates the method with a practical example.

Provides a new tool for PDE output estimation.

Abstract

We propose a method for computing bounds on output functionals of a class of time-dependent PDEs. To this end, we introduce barrier functionals for PDE systems. By defining appropriate unsafe sets and optimization problems, we formulate an output functional bound estimation approach based on barrier functionals. In the case of polynomial data, sum of squares (SOS) programming is used to construct the barrier functionals and thus to compute bounds on the output functionals via semidefinite programs (SDPs). An example is given to illustrate the results.