Positivstellensätze for polynomial matrices with universal quantifiers

Abstract: This paper investigates Positivstellens\"atze for polynomial matrices subject to universally quantified polynomial matrix inequality constraints. We first establish a matrix-valued Positivstellensatz under the Archimedean condition, incorporating universal quantifiers. For scalar-valued polynomial objectives, we further develop a sparse Positivstellensatz that leverages correlative sparsity patterns within these quantified constraints. Moving beyond the Archimedean framework, we then derive a series of generalized Positivstellens\"atze under analogous settings. These results collectively unify and extend foundational theorems in three distinct contexts: classical polynomial Positivstellens\"atze, their universally quantified counterparts, and matrix polynomial formulations. Applications of the established Positivstellens\"atze to robust polynomial matrix optimization are also discussed.