Optimization of polyhedral differential inclusions and state constraints with multiple delays

Abstract

This paper presents a constructive approach to a class of optimal control problems governed by polyhedral discrete and differential inclusions subject to multiple time delays and state constraints. While the theoretical exposition initially focuses on systems with two distinct delays to maintain notational clarity, the proposed framework provides a rigorous basis for generalizing the results to systems governed by multiple delays. By exploiting the geometric structure of polyhedral set-valued mappings, we rigorously employ the method of discrete approximations, utilizing polyhedral Euler–Lagrange type inclusions to bridge the gap between discrete and continuous-time formulations. Within a conjugate analysis framework, we establish that each delay parameter is associated with a distinct absolutely continuous function in the adjoint inclusion. Specifically, we derive necessary and sufficient optimality conditions for the discrete-approximate problem. Subsequently, by formally passing to the limit, we establish sufficient optimality criteria for the continuous-time problem. To demonstrate the computational applicability of the proposed method, numerical experiments are provided for both discrete and continuous settings. These examples explicitly verify the derived conditions and illustrate the behavior of conjugate variables under active state constraints, confirming the effectiveness of the discrete approximation scheme in solving complex constrained control problems.