First- and second-order necessary conditions with respect to components for discrete optimal control problems

Abstract

This paper is devoted to the study of discrete optimal control problems. We aim to obtain more constructive optimality conditions under weakened convexity assumptions. Based on a new approach introduced in this work, an optimality condition with respect to every component is obtained in the form of a global maximum principle. In addition, an optimality condition with respect to one of the components of a control in the form of the global maximum principle and with respect to another component of a control in the form of the linearized maximum principle are obtained. Furthermore, various second-order optimality conditions in terms of singular and quasi-singular controls with respect to the components are obtained on the fly.

Second‐order necessary optimality conditions for discrete‐time stochastic systems

“…Our work generalizes and strengthens the previously known optimality conditions of References 23,26,27,29. Simultaneously, it extends and promotes the classical results 19–22 to stochastic framework.…”

“…Hence, it is indispensable to adopt more subtle and characteristic techniques to investigate the discrete‐time SMP. The researchers have been devoted to establishing the necessary optimality conditions 18–22 under various types of weaker convexity assumptions.…”

“…Here, we introduce the concept of $$$ \gamma $$$‐convex set. Completely different from the methods in Reference 28, by virtue of improving the new techniques adopted in Mardanov, Melikov, Malik & Malikov 22 to discrete‐time deterministic systems, also see References 19–21, and using a special variation of the control as well as a discrete‐time backward stochastic equation, we establish a discrete‐time global SMP. Moreover, we also obtain the second‐order multipoint necessary optimality conditions of singular controls with the aid of a discrete‐time backward stochastic matrix equation.…”

Optimization of Chemotherapy Using Hybrid Optimal Control and Swarm Intelligence

“…The Pontryagin Maximum Principle is said to be the preceding methodology to solve continuous optimal problems [42]. In order to solve the optimization problem with PMP, it is necessary to find auxiliary steps like a formalization of the optimization problem and encoding the notation for PMP formalization [43].…”

A maximum principle for fully coupled controlled forward–backward stochastic difference systems of mean-field type

“…Xu et al [16] considered the solvability of fully coupled FBS Es, in which the BS E was given as the conditional expectation form and the coefficients in the backward equation were degenerate. Some representative works in this direction include [17][18][19][20]. Very recently, Ji and Liu [21] first discussed the SMP for FBS Es under the convex control domain, which made substantial progresses in discrete-time forward-backward systems.…”