Principal bundles and connections modelled by Lie group bundles

In this work, generalized principal bundles modelled by Lie group bundle actions are investigated. particular, the definition of equivariant connections in these bundles, associated to connections, is provided, together with analysis their existence and main properties. The final part gives some examples. since research was initially originated problems on geometric reduction gauge field theories, we revisit classical Utiyama Theorem from perspective investigated article.