An Optimal Control Problem by a Hybrid System of Hyperbolic and Ordinary Differential Equations

Vasilisa Poplevko,Alexander Arguchintsev

doi:10.3390/g12010023

Abstract

This paper deals with an optimal control problem for a linear system of first-order hyperbolic equations with a function on the right-hand side determined from controlled bilinear ordinary differential equations. These ordinary differential equations are linear with respect to state functions with controlled coefficients. Such problems arise in the simulation of some processes of chemical technology and population dynamics. Normally, general optimal control methods are used for these problems because of bilinear ordinary differential equations. In this paper, the problem is reduced to an optimal control problem for a system of ordinary differential equations. The reduction is based on non-classic exact increment formulas for the cost-functional. This treatment allows to use a number of efficient optimal control methods for the problem. An example illustrates the approach.

Highlights

Office of PhD Programs, Irkutsk State University, K
This paper deals with an optimal control problem for a linear system of first-order hyperbolic equations with a function on the right-hand side determined from controlled bilinear ordinary differential equations
General optimal control methods are used for these problems because of bilinear ordinary differential equations

Summary

Problem Statement

We consider an optimization problem for a system of first-order hyperbolic equations with a linear right-hand side:. We assume the function y(t) in the right-hand side of the system (1) is determined by a controlled system of ordinary differential equations dy. We assume that the matrices Φ(s, t), B(u, t), and C (t) and the vectors f (s, t), d(u(t), t) are continuous functions of their arguments everywhere in the corresponding domains. We understand a generalized solution of the boundary-value problem (1)–(3) as a function that satisfies a system of integral equations, where the integration is performed along characteristics of the initial hyperbolic system (1): xi (s, t) = xi (ξ (i ). Note that generalized solutions of hyperbolic balance laws [8,9] and the concept of entropy are well-suited to prove existence and uniqueness results in corresponding functional spaces. The approach based on characteristics of hyperbolic systems is suitable for optimal control methods, estimations of convergence, and so forth

The Cost-Functional Increment Formulas

Variational Maximum Principle

Illustrative Example

Conclusions