Positivstellens\"atze for polynomial matrices with universal quantifiers

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 two 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 inequality constrained optimization are also investigated.