On Necessary Optimality Conditions for Discrete Control Systems

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

“…Consequently, this paper is the first that studies DOCPs with respect to the components, enabling us to obtain more constructive optimality conditions under a new type of weakened convexity assumptions, in contrast to the existing results, e.g., [13,15,18,20,22]. Our results have practical implications as they can be used in solving various problems.…”

“…In Theorem 3.2, we obtain an optimality condition with respect to one of the components of a control in the form of a global maximum principle and with respect to another component of a control in the form of the linearized maximum principle. Note that this theorem has its own application areas compared to Theorem 3.1, and the relevant results for the necessary optimality conditions are studied in [4,13,15].…”

“…Consequently, we obtain second-order necessary optimality conditions by components in the forms of (10), (11), and (12). Although these conditions have various areas of application, the applications of (10) and (11) are less constructive relative to those of (12).…”

“…Following [15], we introduce the vector functionψ(t;û), t ∈ I, as the solution of the linear discrete system…”

“…Soon after this work, extensive studies in this area were devoted to obtaining a number of first-and second-order optimality conditions in various forms. For example, the works [5][6][7][8][9][10][11][12] obtain optimality conditions in the form of a global maximum principle, while [3,4,[13][14][15][16][17][18] obtain similar conditions in the form of a local maximum principle, the linearized maximum principle or the Euler equation. Moreover, second-order optimality conditions in terms of singular (in the sense of the discrete maximum principle) as well as quasi-singular controls are obtained in [14][15][16][19][20][21][22].…”