Existence and optimality in discrete and differential inclusions for the Bolza problem

This study examines the existence theorems and optimality conditions within the framework of the Bolza problem involving discrete and differential inclusions. Initially, an existence theorem is established to rigorously confirm the presence of solutions under specified constraints, thereby providing a foundation for further analysis. Subsequently, the investigation focuses on formulating optimality conditions for discrete-time and differential inclusions. Optimality conditions and solution properties of discrete-time dynamical systems are formulated, offering insights into the fundamental principles underlying discrete-time inclusions. Furthermore, using discrete-time inclusions to approximate differential inclusions provides a critical bridge between continuous and discrete formulations. The derivation of optimality conditions further enhances the analytical framework, distinguishing optimal solutions from the broader feasible set. Leveraging the geometric framework of normal cones, this study provides a precise characterization of optimality within the constraints of the given differential inclusion. Consequently, both necessary and sufficient conditions for optimality are established, highlighting the theoretical and practical significance of the proposed approach. Finally, by presenting a concrete example, we demonstrate how the theoretical components of the framework become more accessible and practically interpretable.