Conditions for Optimality of Singular Controls in Dynamic Systems with Retarded Control

In this chapter, we consider an optimal control problem with retarded control and study a larger class of singular (in the classical sense) controls. For the optimality of singular controls, the various necessary conditions in the recurrent forms are obtained. These conditions contain also the analogs of Kelly, Koppa-Mayer, Gabasov, and equality-type conditions. While proving the main results, the Legendre polynomials are used as variations of control

ON THE ISSUE OF PLANNING SOWING AGRICULTURAL CROPS WITH THE MINIMUM RISK UNDER THE PRESENCE OF VARIOUS AGROCLIMATIC CONDITIONS

The present paper deals with one problem of quantitative controlling the seeding of the sown area by agricultural crops in different agroclimatic conditions. The considered problem is studied from the standpoint of three strategies: from the seeding planning perspective aiming at minimal risk associated with possible unfavourable agroclimatic conditions (a probabilistic approach is used); from the perspective of obtaining the maximum crops sales profit (a deterministic approach is used); from the perspective of obtaining the maximum crops harvest. For the considered problem, mathematical models are constructed (one probabilistic model and two deterministic models, respectively), their analytical solutions are found, and then, using a specific example, the application of the constructed and solved mathematical models is illustrated as well as the obtained numerical results are analysed.

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

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

Integral necessary condition of optimality of the second order for control problems described by system of integro-differential equations with delay

We consider the optimal control problem that is described by the system of integro-differential equations of the Volterra type with delay and multipoint performance criterion. The first and the second variations of the performance criterion are calculated under the hypothesis that the control domain is open. The necessary condition of the first order optimality in the form analogous to the Euler equations is deduced from the equality of the first variation of performance criterion and zero along the optimal process. Next, the implicit necessary condition of the second order optimality is obtained, which helps to establish rather general but constructively verified necessary condition for the second order optimality. The obtained results are applicable for constructing easy-verifying necessary conditions of optimality for the singular (in the usual sense) controls

Existence and uniqueness of solutions for first-order nonlinear differential equations with two-point and integral boundary conditions

In this article, we study the existence of solutions to boundary-value problems for ordinary differential equations with two-point and integral boundary conditions. Existence and uniqueness results are obtained by using well known fixed point theorems. Some illustrative examples are also discussed

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

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