Controlled Invariance in Fully Actuated Max-plus Linear Systems with Precedence Semimodules

Given a max-plus linear system and a semimodule, the problem of computing the maximal controlled invariant subsemimodule is still open to this day. In this paper, we consider this problem for the specific class of fully actuated systems and constraints in the form of precedence semimodules. The assumption of full actuation corresponds to the existence of an input for each component of the system state. A precedence semimodule is the set of solutions of inequalities typically used to represent time-window constraints. We prove that, in this setting, it is possible to (i) compute the maximal controlled invariant subsemimodule and (ii) decide the convergence of a fixed-point algorithm introduced by R.D. Katz in strongly polynomial time.