© 2021 American Institute of Physics Inc.. All rights reserved.The present paper studies a Bolza problem of optimal control theory with third-order polyhedral delay-differential inclusions and state constraints. We aim to establish well verifiable sufficient conditions of optimality for the polyhedral thirdorder delay-differential inclusions. Discrete-approximate inclusions are investigated using the method of discretization to ensure the transition to a continuous problem. The idea for obtaining sufficient conditions of the problem is based on passing the formal limit on the optimality conditions of the discrete-approximation problem. Thus, the sufficient conditions are formulated by using polyhedral Euler-Lagrange inclusions and the distinctive "transversality" conditions.