Discrete-Continuous Systems with Parameters: Method for Improving Control and Parameters

Abstract

The paper presents one of the classes of controlled systems capable of changing their structure over time. The general name for such systems is hybrid. In the article it is discusses the so-called discrete-continuous systems containing parameters. It is a two-level hierarchical model. The upper level of this model is represented by a discrete system. At the lower level continuous controlled systems operate in turn. All these systems contain parameters and are linked by the functional. In recent decades hybrid systems have been the subject of active research both of the systems themselves and of a wide range of problems for them, using various methods that reflect the views of scientific schools and directions. At the same time the most diverse mathematical apparatus is presented in the research. In this case a generalization of Krotov’s sufficient optimality conditions is used. Their advantage is the possibility of preserving the classical assumptions about the properties of objects that appear in the formulation of the optimal control problem. For the optimal control problem considered in this paper for discrete-continuous systems with parameters an analogue of Krotov’s sufficient optimality conditions is proposed. Two theorems are formulated. On their base an easy-to-implement algorithm for improving control and parameters is built. A theorem on its functional convergence is given. This algorithm contains a vector system of linear equations for conjugate variables, which always has a solution. It is guarantees a solution to the original problem. The algorithm is tested on an illustrative example. Calculations and graphs are presented.