Mixed Cost Function and State Constrains Optimal Control Problems - Hugo Leiva; Guido Tapia-Riera; Jhoana P. Romero-Leiton; Cosme Duque (AppliedMath, 2025)
In this paper, we analyze an optimal control problem with a mixed cost function, which combines a terminal cost at the final state and an integral term involving the state and control variables. The problem includes both state and control constraints, which adds complexity to the analysis. We establish a necessary optimality condition in the form of the maximum principle, where the adjoint equation is an integral equation involving the Riemann and Stieltjes integrals with respect to a Borel measure. Our approach is based on the Dubovitskii–Milyutin theory, which employs conic approximations to efficiently manage state constraints. To illustrate the applicability of our results, we consider two examples related to epidemiological models, specifically the SIR model. These examples demonstrate how the developed framework can inform optimal control strategies to mitigate disease spread. Furthermore, we explore the implications of our findings in broader contexts, emphasizing how mixed cost functions manifest in various applied settings. Incorporating state constraints requires advanced mathematical techniques, and our approach provides a structured way to address them. The integral nature of the adjoint equation highlights the role of measure-theoretic tools in optimal control. Through our examples, we demonstrate practical applications of the proposed methodology, reinforcing its usefulness in real-life situations. By extending the Dubovitskii–Milyutin framework, we contribute to a deeper understanding of constrained control problems and their solutions.