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Barrier certificates, which serve as differential invariants that witnesssystem safety, play a crucial role in the verification of cyber-physicalsystems (CPS). Prevailing computational methods for synthesizing barriercertificates are based on semidefinite programming (SDP) by exploiting PutinarPositivstellensatz. Consequently, these approaches are limited by Archimedeancondition, which requires all variables to be bounded, i.e., systems aredefined over bounded domains. For the unbounded case, unfortunately, thesemethods become conservative and even fail to identify potential barriercertificates.  In this paper, we address this limitation by presenting a new computationalmethod. The main technique we use is the homogenization approach, which wasproposed in optimization community recently, to transform an unboundedoptimization problem to a bounded one. Our method can be applied to variousdefinitions of barrier certificates, thus expanding the scope of barriercertificate synthesis in the general sense. Experimental results demonstratethat our approach is more effective while maintaining a comparable level ofefficiency.

Generalizing SDP-Based Barrier Certificate Synthesis to Unbounded Domains by Dropping Archimedean Condition