# Lie Geometric Methods in the Study of Driftless Control Affine Systems with Holonomic Distribution and Economic Applications

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Lie Geometric Methods in Optimal Control

#### 2.1. Control Affine Systems

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Theorem**

**1.**

**Definition**

**4.**

**Theorem**

**2.**

#### 2.2. Lie Algebroids

**Definition**

**5.**

- a)
- ${C}^{\infty}\left(M\right)$-module of sections$\Gamma \left(E\right)$is equipped with a Lie algebra structure${[\xb7,\xb7]}_{E}$;
- b)
- $\sigma :E\to TM$is a bundle map, called the anchor, which induces a Lie algebra homomorphism from the Lie algebra of sections$(\Gamma \left(E\right),{[\xb7,\xb7]}_{E})$to the Lie algebra of vector fields$\left(\mathcal{X}\right(M),[\xb7,\xb7\left]\right)$, satisfying the Leibniz rule:

- ${1}^{\circ}$${[\xb7,\xb7]}_{E}$ is a ℝ-bilinear operation,
- ${2}^{\circ}$${[\xb7,\xb7]}_{E}$ is skew-symmetric, i.e., ${[{s}_{1},{s}_{2}]}_{E}=-{[{s}_{2},{s}_{1}]}_{E},\phantom{\rule{1.em}{0ex}}\forall {s}_{1},{s}_{2}\in \Gamma \left(E\right)$,

**Example**

**1.**

**Example**

**2.**

#### 2.3. The Prolongation of a Lie Algebroid

- (i)
- The associated vector bundle is $(\mathcal{T}{E}^{*},{\tau}_{1},{E}^{*})$, where $\mathcal{T}{E}^{*}=\cup {\mathcal{T}}_{{u}^{*}}{E}^{*}$, ${u}^{*}\in {E}^{*}$,$${\mathcal{T}}_{{u}^{*}}{E}^{*}=\{({u}_{x},{v}_{{u}^{*}})\in {E}_{x}\times {T}_{{u}^{*}}{E}^{*}|\sigma ({u}_{x})={T}_{{u}^{*}}\tau ({v}_{{u}^{*}}),\tau \left({u}^{*}\right)=x\in M\},$$
- (ii)
- The Lie algebra structure ${[\xb7,\xb7]}_{\mathcal{T}{E}^{*}}$ on $\phantom{\rule{4pt}{0ex}}\Gamma \left(\mathcal{T}{E}^{*}\right)$ is defined as follows: If ${\rho}_{1},{\rho}_{2}\in \Gamma \left(\mathcal{T}{E}^{*}\right)$ are such that ${\rho}_{i}\left({u}^{*}\right)=({X}_{i}\left(\tau \left({u}^{*}\right)\right),{U}_{i}\left({u}^{*}\right))$, where ${X}_{i}\in \Gamma \left(E\right),{U}_{i}\in \chi \left({E}^{*}\right)$ and $\sigma ({X}_{i}\left(\tau \left({u}^{*}\right)\right)={T}_{{u}^{*}}\tau \left({U}_{i}\left({u}^{*}\right)\right)$, $\phantom{\rule{4pt}{0ex}}i=1,2$, then$${[{\rho}_{1},{\rho}_{2}]}_{\mathcal{T}{E}^{*}}({u}^{*})=({[{X}_{1},{X}_{2}]}_{\mathcal{T}{E}^{*}}(\tau \left({u}^{*}\right)),{[{U}_{1},{U}_{2}]}_{\mathcal{T}{E}^{*}}({u}^{*})).$$
- (iii)
- The anchor map is the projection ${\sigma}^{1}:\mathcal{T}{E}^{*}\to T{E}^{*}$, ${\sigma}^{1}(u,v)=v$.

**Theorem**

**3.**

**Proof.**

## 3. Application to Optimal Control

## 4. Economic Application

#### 4.1. The Case ${\beta}_{1}={\beta}_{2}={\beta}_{3}=0$

**Theorem**

**4.**

**Proof.**

#### 4.2. The Case ${\beta}_{1},{\beta}_{2},{\beta}_{3}>0$

**Theorem**

**5.**

**Proof.**

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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Popescu, L.; Militaru, D.; Tică, G.
Lie Geometric Methods in the Study of Driftless Control Affine Systems with Holonomic Distribution and Economic Applications. *Mathematics* **2022**, *10*, 545.
https://doi.org/10.3390/math10040545

**AMA Style**

Popescu L, Militaru D, Tică G.
Lie Geometric Methods in the Study of Driftless Control Affine Systems with Holonomic Distribution and Economic Applications. *Mathematics*. 2022; 10(4):545.
https://doi.org/10.3390/math10040545

**Chicago/Turabian Style**

Popescu, Liviu, Daniel Militaru, and Gabriel Tică.
2022. "Lie Geometric Methods in the Study of Driftless Control Affine Systems with Holonomic Distribution and Economic Applications" *Mathematics* 10, no. 4: 545.
https://doi.org/10.3390/math10040545