Multistage systems are systems characterized by n stages, where each stage has a precise parametric evolution law. We propose a procedure to reduce the optimal parameter synthesis on a multistage system to the optimal control design on an automaton with a DAG structure that we will call multistage automaton. We describe the procedure using a thawing system as a case study. We address the optimization problem as a reachability problem and we use backtrack strategy to obtain a feasible solution in the linear case.