# Optimal Control and Positional Controllability in a One-Sector Economy

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Statement of the Dynamic Model and Maximization Problem

## 3. Properties of Solution of System (2)

**Lemma**

**1.**

**Proof.**

**Lemma**

**2.**

**Proof.**

**Lemma**

**3.**

**Proof.**

## 4. Pontryagin Maximum Principle

#### Singular Regime

- (i)
- performing a standard change of variables $\dot{\chi}=0$, $\chi \left(0\right)=0$, where $\chi $ is a new auxiliary variable, leads, on the one hand, to an increase by one in the order of the original system (2), and, on the other hand, makes such a system autonomous (see [8]);
- (ii)
- introducing a new objective function $\tilde{J}\left(u(\xb7)\right)=-J\left(u(\xb7)\right)$ leads the considered problem (5) to the problem on minimum:$$\tilde{J}\left(u(\xb7)\right)=-y\left(T\right)\to \underset{u(\xb7)\in \mathsf{\Omega}\left(T\right)}{min}.$$

**Lemma**

**4.**

## 5. Numerical Results

## 6. Suboptimal Control

## 7. Position Controller for the Terminal Control Problem

**Lemma**

**5.**

## 8. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Kolemaev, V.A. Mathematical Economics; Yuniti-Dana: Moscow, Russia, 2015. [Google Scholar]
- Grass, D.; Caulkins, J.P.; Feichtinger, G.; Tragler, G.; Behrens, D.A. Optimal Control of Nonlinear Processes. With Applications in Drugs, Corruptions, and Terror; Springer: Berlin/Heidelberg, Germany, 2008. [Google Scholar]
- Ramsey, F.P. A mathematical theory of saving. Econ. J.
**1928**, 38, 543–559. [Google Scholar] [CrossRef] - Seierstad, A.; Sydsæter, K. Optimal Control Theory with Economic Applications; North-Holland: Amsterdam, The Netherlands, 1987. [Google Scholar]
- Shell, K. Applications of Pontryagin’s maximum principle to economics. In Mathematical Systems Theory and Economics 1; Springer: Berlin, Germany, 1969; pp. 241–292. [Google Scholar]
- Spear, S.E.; Young, W. Optimum savings and optimal growth: Ramsey–Mavlinvaud–Koopmans nexus. Macroecon. Dyn.
**2014**, 18, 215–243. [Google Scholar] [CrossRef] - Pontryagin, L.S. Ordinary Differential Equations; Addison-Wesley Publishing Company: London, UK, 1962. [Google Scholar]
- Pontryagin, L.S.; Boltyanskii, V.G.; Gamkrelidze, R.V.; Mishchenko, E.F. Mathematical Theory of Optimal Processes; John Wiley & Sons: New York, NY, USA, 1962. [Google Scholar]
- Vasiliev, F.P. Optimization Methods; Factorial Press: Moscow, Russia, 2002. [Google Scholar]
- Afanasiev, V.N.; Kolmanovskii, V.; Nosov, V.R. Mathematical Theory of Control Systems Design; Springer: Dordrecht, The Netherlands, 1996. [Google Scholar]
- Schättler, H.; Ledzewicz, U. Optimal Control for Mathematical Models of Cancer Therapies: An Application of Geometric Methods; Springer: New York, NY, USA; Heidelberg, Germany; Dordrecht, The Netherlands; London, UK, 2015. [Google Scholar]
- Zelikin, M.I.; Borisov, V.F. Theory of Chattering Control: With Applications to Astronautics, Robotics, Economics and Engineering; Birkhäuser: Boston, MA, USA, 1994. [Google Scholar]
- Bonnans, F.; Martinon, P.; Giorgi, D.; Grélard, V.; Maindrault, S.; Tissot, O.; Liu, J. BOCOP 2.2.0—User Guide. Available online: http://bocop.org (accessed on 27 June 2019).
- Kolesnikov, A.; Veselov, G.; Kolesnikov, A.; Monti, A.; Ponci, F.; Santi, E.; Dougal, R. Synergetic synthesis of Dc-Dc boost converter controllers: Theory and experimental analysis. In Proceedings of the 17th Annual IEEE Applied Power Electronics Conference and Exposition, Dallas, TX, USA, 10–14 March 2002; pp. 409–415. [Google Scholar]

**Figure 1.**Upper row: graphs of the optimal solutions ${k}^{*}\left(t\right)$, ${v}_{1}^{*}\left(t\right)$ and ${v}_{2}^{*}\left(t\right)$; lower row: the graph of the optimal control ${u}^{*}\left(t\right)$, the phase portrait ${v}_{2}^{*}\left({k}^{*}\right)$, and the phase portrait ${v}_{2}^{*}\left({k}^{*}\right)$ in the neighborhood of the singular arc.

**Figure 2.**Graph of the approximating control $\tilde{u}\left(t\right)$ (

**left**) and graph of the corresponding solution $\tilde{k}\left(t\right)$ (

**right**).

**Figure 3.**Graphs of solutions $k\left(t\right)$ and ${v}_{2}\left(t\right)$ to system (14) and trajectory ${v}_{2}\left(k\right)$ under control (23).

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Grigorenko, N.; Luk’yanova, L.
Optimal Control and Positional Controllability in a One-Sector Economy. *Games* **2021**, *12*, 11.
https://doi.org/10.3390/g12010011

**AMA Style**

Grigorenko N, Luk’yanova L.
Optimal Control and Positional Controllability in a One-Sector Economy. *Games*. 2021; 12(1):11.
https://doi.org/10.3390/g12010011

**Chicago/Turabian Style**

Grigorenko, Nikolai, and Lilia Luk’yanova.
2021. "Optimal Control and Positional Controllability in a One-Sector Economy" *Games* 12, no. 1: 11.
https://doi.org/10.3390/g12010011