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This paper defines a new class of hybrid systems called piecewise polynomial (PWP) systems in strict form and develops a backstepping controller synthesis methodology for these systems. The main contribution of the paper is to formulate controller design for a large class of PWP systems as a convex feasibility problem. The controller synthesis problem for PWP systems in strict feedback form is divided into two cases. The first case consists of the construction of a sum of squares (SOS) Lyapunovdoi:10.1109/acc.2009.5160371 dblp:conf/amcc/SamadiR09 fatcat:swojv5h32bglng6rqroze5t7za