# 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

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**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).

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**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