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

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## Abstract

**:**

## 1. Introduction

## 2. Problem Statement

## 3. The Cost-Functional Increment Formulas

## 4. Variational Maximum Principle

**Theorem**

**1.**

- We choose an arbitrary admissible control $u=u\left(t\right)$. Then, we calculate $y(t,u)$ and $\psi (s,t)$.

## 5. Illustrative Example

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

ODE | ordinary differential equation |

## References

- Demidenko, N. Optimal control of thermal-engineering processes in tube furnaces. Chem. Petrol. Eng.
**2006**, 42, 128–130. [Google Scholar] [CrossRef] - Petukhov, A. Modeling of threshold effects in social systems based on nonlinear dynamics. Cybern. Phys.
**2019**, 8, 277–287. [Google Scholar] [CrossRef] - Vasiliev, O. Optimization Methods; World Federation Publishers Company Inc.: Atlanta, GA, USA, 1996. [Google Scholar]
- Arguchintsev, A.V.; Poplevko, V.P. Optimal control of initial conditions in canonical hyperbolic system of the first-order based on non-standard increment formulas. Rus. Math.
**2008**, 52, 1–7. [Google Scholar] [CrossRef] - Rozonoer, L.I. LS Pontryagin’s maximum principle in the theory of optimum systems. Part I. Autom. Remote Contr.
**1959**, 20, 1288–1302. [Google Scholar] - Rozhdestvenskiyi, B.L.; Yanenko, N.N. Systems of Quasilinear Equations and Their Applications to Gas Dynamics; Nauka: Moscow, Russia, 1968. [Google Scholar]
- Godunov, S.K. Equations of Mathematical Physics; Nauka: Moscow, Russia, 1979. [Google Scholar]
- LeVeque, R. Finite Volume Methods for Hyperbolic Problems (Cambridge Texts in Applied Mathematics); Cambridge University Press: Cambridge, UK, 2002. [Google Scholar] [CrossRef]
- Dafermos, C.M. Hyperbolic Conservation Laws in Continuum Physics (Grundlehren der Mathematischen Wissenschaften), 4th ed.; Springer: Berlin/Heidelberg, Germany, 2016; Volume 325. [Google Scholar]
- Rao, A.V. A survey of numerical methods for optimal control. Adv. Astron. Sci.
**2009**, 135, 1–32. [Google Scholar] - Golfetto, W.A.; Silva Fernandes, S. A review of gradient algorithms for numerical computation of optimal trajectories. J. Aerosp. Technol. Manag.
**2012**, 4, 131–143. [Google Scholar] [CrossRef][Green Version] - Biral, F.; Bertolazzi, E.; Bosetti, P. Notes on numerical methods for solving optimal control problems. IEEJ J. Ind. Appl.
**2016**, 5, 154–166. [Google Scholar] [CrossRef][Green Version] - Srochko, V.A.; Antonik, V.G. Optimality conditions for extremal controls in bilinear and quadratic problems. Russ. Math.
**2016**, 60, 75–80. [Google Scholar] [CrossRef] - Srochko, V.A.; Aksenyushkina, E.V. Parameterization of some linear systems control problems. Bull. Irkutsk. State Univ. Ser. Math.
**2019**, 30, 83–98. [Google Scholar] [CrossRef]

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**MDPI and ACS Style**

Arguchintsev, A.; Poplevko, V. An Optimal Control Problem by a Hybrid System of Hyperbolic and Ordinary Differential Equations. *Games* **2021**, *12*, 23.
https://doi.org/10.3390/g12010023

**AMA Style**

Arguchintsev A, Poplevko V. An Optimal Control Problem by a Hybrid System of Hyperbolic and Ordinary Differential Equations. *Games*. 2021; 12(1):23.
https://doi.org/10.3390/g12010023

**Chicago/Turabian Style**

Arguchintsev, Alexander, and Vasilisa Poplevko. 2021. "An Optimal Control Problem by a Hybrid System of Hyperbolic and Ordinary Differential Equations" *Games* 12, no. 1: 23.
https://doi.org/10.3390/g12010023