Safety Verification for Distributed Parameter Systems Using Barrier Functionals

TL;DR

This paper introduces barrier functionals for safety verification of distributed parameter systems modeled by PDEs, enabling the checking of safety constraints without finite-dimensional approximations, using semi-definite programming for polynomial PDEs.

Contribution

It extends barrier certificates to infinite-dimensional PDE systems via barrier functionals and provides a semi-definite programming approach for polynomial data.

Findings

Successfully verifies safety for PDE systems using barrier functionals.

Avoids finite-dimensional approximations in safety verification.

Demonstrates effectiveness through illustrative examples.

Abstract

We study the safety verification problem for a class of distributed parameter systems described by partial differential equations (PDEs), i.e., the problem of checking whether the solutions of the PDE satisfy a set of constraints at a particular point in time. The proposed method is based on an extension of barrier certificates to infinite-dimensional systems. In this respect, we introduce barrier functionals, which are functionals of the dependent and independent variables. Given a set of initial conditions and an unsafe set, we demonstrate that if such a functional exists satisfying two (integral) inequalities, then the solutions of the system do not enter the unsafe set. Therefore, the proposed method does not require finite-dimensional approximations of the distributed parameter system. Furthermore, for PDEs with polynomial data, we solve the associated integral inequalities using semi-definite programming (SDP) based on a method that relies on a quadratic representation of the integrands of integral inequalities. The proposed method is illustrated through examples.