Abstract
The object of research is the linear optimal control problem described by discrete two-parameter systems under the assumption that the controlled process is stepwise.The work is aimed at deriving the necessary first-order optimality conditions in the case of a non-smooth quality function. And also to establish the necessary conditions of second-order optimality in stepwise control problems for discrete two-parameter systems. The paper investigates one linear two-parameter discrete optimal control problem with a non-smooth quality criterion. A special increment of the quality functional is calculated. Cases under the condition of a convex set are considered. The concept of special control in the problem under study is given. A number of necessary optimality conditions for the first and second orders are established. And also the necessary second-order optimality conditions are obtained in terms of directional derivatives. In the case of a linear quality criterion, the necessary and sufficient optimality condition is proved using the increment formula by analogous arguments. Under the assumption that the set is convex, a special increment of the quality criterion for admissible control is defined.The methods of calculus of variations and optimal control, the theory of difference equations are used. The result is obtained for the optimality of a special, first-order control, in the case of convexity of the set. The case when the minimized functional is linear is considered. In this case, a necessary and sufficient condition is obtained for the optimality of the admissible control.Thanks to the research results, it is possible to obtain the necessary first-order optimality conditions in terms of directional derivatives in the stepwise problem of optimal control of discrete two-parameter systems. As well as the necessary conditions of optimality of the second order in the case of convexity of the control domain and the necessary optimality conditions of special controls.The theoretical results obtained in the work are of interest in the theory of optimal control of step systems and can be used in the further development of the theory of necessary optimality conditions for step control problems.
Highlights
Discrete dynamic models of controlled systems represent an important class of mathematical models both theoreti cally and in practice, which makes it possible to cover a very wide range of real objects
An important place is occupied by optimal con trol problems described by discrete multidimensional and, in particular, two-parameter systems
Many real processes are described by multiparameter, and, in particular, twoparameter systems [1,2,3]
Summary
Discrete dynamic models of controlled systems represent an important class of mathematical models both theoreti cally and in practice, which makes it possible to cover a very wide range of real objects. A number of image processing models in the state space are described by difference equations, which are a difference analogue of a system of second-order hyperbolic equations with Goursat – Darboux boundary conditions [10]. Such models are called Fornasini-Marquezini systems [11, 12]. A number of optimal control problems for discrete two-parameter systems were studied, for example, in [14]. The object of research is the linear optimal control problem described by discrete two-parameter systems under the assumption that the controlled process is stepwise. The aim of research is finding sufficient and necessary conditions under various conditions of optimality
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