# Toward Applications of Linear Control Systems on the Real World and Theoretical Challenges

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## Abstract

**:**

## 1. Introduction

**Theorem**

**1.**

## 2. Matrix Groups Dynamics and Systems

- 1.
- $\left[P,Q\right]=-\left[Q,P\right]$ skew-symmetric, and the Jacobi identity
- 2.
- $[X,[Y,Z\left]\right]+[Z,[X,Y\left]\right]+[Y,[Z,X\left]\right]=0,$ for any $X,Y,Z\in \mathfrak{g}.$

- Abelian, if for any $X,Y\in \mathfrak{g}$, $\left[X,Y\right]=0$.
- Nilpotent, if $\exists k\ge 1:\mathbf{a}{\mathbf{d}}^{1}=\left[\mathfrak{g},\mathfrak{g}\right]\supset ...\supset \mathbf{a}{\mathbf{d}}^{k+1}=\left[\mathbf{a}{\mathbf{d}}^{k},\mathfrak{g}\right]=0$.
- Solvable, if $\exists k\ge 1:\mathbf{a}{\mathbf{d}}^{1}\supset ...\supset \mathbf{a}{\mathbf{d}}^{\left(k\right)}=\left[\mathbf{a}{\mathbf{d}}^{(k-1)},\mathbf{a}{\mathbf{d}}^{(k-1)}\right]=0.$
- Semisimple, if the largest solvable subalgebra $\mathfrak{r}\left(\mathfrak{g}\right)$ of $\mathfrak{g}$ is trivial.
- Finite semi-simple center if any semi-simple subalgebra has a trivial center.

#### 2.1. The Notion of Linear Control System on G

- for any $g\in G,$$\exists u$ such that $\phi (g,u,t)\in \mathcal{C},$ $t\ge 0$
- $\mathcal{C}\subset cl\left(\mathcal{A}\right(g),$ for any $g\in \mathcal{C}$
- $\mathcal{C}$ is maximal with respect 1 and 2.

#### 2.2. The Classical Linear Control System on ${\mathbb{R}}^{n}$

#### 2.3. The $\mathcal{D}$-Decomposition of $\mathfrak{g}$

## 3. Linear Control Systems and Controllability

#### 3.1. The Dimension 2

#### 3.1.1. The Abelian Structure

**Theorem**

**2.**

- $1.$
- ${\mathsf{\Sigma}}_{{\mathbb{R}}^{2}}$ is controllable ⇔ Kalman rank condition and $Spe{c}_{Ly}\left(\mathcal{D}\right)=\left\{0\right\}.$
- $2.$
- Around the identity element 0, there exists an unique control set with non-empty interior, given by$$\mathcal{C}=\phantom{\rule{4.pt}{0ex}}cl\left(\mathcal{A}\right)\cap {\mathcal{A}}^{*}.$$

#### 3.1.2. The Solvable Structure

**Theorem**

**3**

- Let us consider$${\mathsf{\Sigma}}_{G}:\left\{\begin{array}{c}\dot{x}=ux\hfill \\ \dot{y}=by\hfill \end{array}\right.,\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}where\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}u\in \mathsf{\Omega}=\left[\phantom{\rule{4.pt}{0ex}}{u}_{*},{u}^{*}\right],$$$$\phi (t,(x,y),u)=({\mathrm{e}}^{tu}x,{\mathrm{e}}^{tb}y),\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}t\in \mathbb{R}.$$As a matter of fact, the only control set is given by $\mathcal{C}={\mathbb{R}}_{+}\times \left\{0\right\}$.
- Let us consider the linear system$${\mathsf{\Sigma}}_{G}:\left\{\begin{array}{c}\dot{x}=0\hfill \\ \dot{y}=a(x-1)+ux\hfill \end{array}\right.,\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}where\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}u\in \mathsf{\Omega}=\left[\phantom{\rule{4.pt}{0ex}}{u}_{*},{u}^{*}\right],$$$$\phi \left(\right(x,y),u,t)=(x,(a(x-1)+ux)t+y),\phantom{\rule{4.pt}{0ex}}t\in \mathbb{R}.$$Here, there are an infinity number of control sets. In fact, for any $x\in \mathbb{R}$$${\mathcal{C}}_{x}=\left\{x\right\}\times \mathbb{R},\phantom{\rule{4.pt}{0ex}}\mathrm{when}\phantom{\rule{4.pt}{0ex}}x\in (1-\epsilon ,1+\epsilon ),\phantom{\rule{4.pt}{0ex}}\mathrm{and}\phantom{\rule{4.pt}{0ex}}\mathrm{any}\phantom{\rule{4.pt}{0ex}}y\in \mathbb{R}$$
- Consider the linear system$${\mathsf{\Sigma}}_{G}:\left\{\begin{array}{c}\dot{x}=ux\hfill \\ \dot{y}=by+ux\hfill \end{array}\right.,\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\mathrm{where},b<0,\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}u\in \mathsf{\Omega}=\left[\phantom{\rule{4.pt}{0ex}}{u}_{*},{u}^{*}\right],$$

#### 3.2. The Dimension 3

#### 3.2.1. The Abelian Structure

**Theorem**

**4.**

- ${\mathsf{\Sigma}}_{{\mathbb{R}}^{3}}$ is controllable ⇔ Kalman rank condition and $Spe{c}_{Ly}\left(\mathcal{D}\right)=\left\{0\right\}.$
- Around the identity element 0, there exists an unique control set with non-empty interior, given by$$\mathcal{C}=\phantom{\rule{4.pt}{0ex}}cl\left(\mathcal{A}\right)\cap {\mathcal{A}}^{*}.$$

#### 3.2.2. The Nilpotent Structure

**Theorem**

**5**

**Theorem**

**6**

#### 3.2.3. The Solvable Structure

- $1.$
- The one control vector,$$\dot{g}=\mathcal{X}\left(g\right)+u{Y}^{1}\left(g\right),\phantom{\rule{4.pt}{0ex}}\mathrm{where}\phantom{\rule{4.pt}{0ex}}{Y}_{1}=(1,0).$$
- $2.$
- The two control vectors,$$\dot{g}=\mathcal{X}\left(g\right)+{u}_{1}{Y}^{1}\left(g\right)+{u}_{2}{Y}^{2}\left(g\right),\mathrm{with}\phantom{\rule{4.pt}{0ex}}{Y}_{1}=(1,0),{Y}_{2}=(0,w),\mathrm{some}\phantom{\rule{4.pt}{0ex}}w\in {\mathbb{R}}^{2}$$

**Theorem**

**7**

- If $G={R}_{2}$: ${\mathsf{\Sigma}}_{G}$ is controllable $\iff {\mathfrak{g}}^{0}\simeq \mathfrak{aff}\left(\mathbb{R}\right)$
- If $G={E}_{2}$ or ${R}_{3}$: ${\mathsf{\Sigma}}_{G}$ is controllable ⇔$\mathfrak{g}={\mathfrak{g}}^{0}$ and ${\mathcal{D}}^{*}\not\equiv 0$
- If $G={R}_{3,\lambda}$: ${\mathsf{\Sigma}}_{G}$ is controllable $\iff \lambda =1$ and ${\mathcal{D}}^{*}$ has a pair of complex eigenvalues
- If $G={R}_{3,\lambda}^{\prime}$: ${\mathsf{\Sigma}}_{G}$ is controllable.

**Theorem**

**8**

**Theorem**

**9.**

- If $G={R}_{2}$ : ${\mathsf{\Sigma}}_{G}$ is controllable ⇔$dim{\mathfrak{g}}^{0}>1$ or $dim{\mathfrak{g}}^{0}=1$ and $\mathsf{\Delta}\simeq \mathfrak{aff}\left(\mathbb{R}\right)$
- If $G={E}_{2},$ or ${R}_{3,\lambda}^{\prime}$: ${\mathsf{\Sigma}}_{G}$ is controllable
- If $G={R}_{3}$: ${\mathsf{\Sigma}}_{G}$ is controllable ⇔$\mathfrak{g}={\mathfrak{g}}^{0}$
- If $G={R}_{3,\lambda}$: ${\mathsf{\Sigma}}_{G}$ is controllable ⇔$ker{\mathcal{D}}^{*}\not\subset \mathsf{\Delta}$ or $\mathcal{D}$ has a pair of complex eigenvalues.

#### 3.2.4. The Finite Semi-Simple Center Structure

#### 3.2.5. The Compact Semi-Simple Structure

#### 3.2.6. The Non-Compact Semi-Simple Structure

**Theorem**

**10**

- $\exists \phantom{\rule{4pt}{0ex}}$ a control set $\mathcal{C}$ with nonempty interior around e given by,$$\mathcal{C}=\phantom{\rule{4.pt}{0ex}}cl\left(\mathcal{A}\right)\cap {\mathcal{A}}^{*}$$
- The only possible invariant control set is the whole group, i.e.,$$\mathrm{if}\phantom{\rule{4.pt}{0ex}}\mathcal{C}\phantom{\rule{4.pt}{0ex}}\mathrm{is}\phantom{\rule{4.pt}{0ex}}\mathrm{positive}\phantom{\rule{4.pt}{0ex}}\mathrm{invariant},\phantom{\rule{4.pt}{0ex}}\mathrm{then}\phantom{\rule{4.pt}{0ex}}\mathcal{C}=G.$$

## 4. The Pontryagin Maximum Principle

**Theorem**

**11**

- $\lambda \left(t\right)\ne 0$ for all $t\in [0,T].$ In addition, for almost all $t\in [0,T]$
- ${H}_{u*}(\lambda \left(t\right),g\left(t\right))={max}_{u\in \mathcal{U}}{H}_{u}(\lambda \left(t\right),g\left(t\right))$
- ${H}_{u*}(\lambda \left(t\right),g\left(t\right))\ge 0.$

## 5. Examples

**Example**

**1.**

**Example**

**2.**

**Example**

**3.**

- For ${\mathfrak{r}}_{2}$ consider the basis$$X=M(1,1),Y=M(2,3),Z=M(1,3).$$With the change of variable Y by $Z,$ we obtain ${\mathfrak{r}}_{2}$ with the same bracket rules: $[X,Y]=Y$, $[X,Z]=0$. The left invariant vector fields are given by$${X}_{g}=-y\frac{\partial}{\partial x},{Y}_{g}=-y\frac{\partial}{\partial y},{Z}_{g}=\frac{\partial}{\partial z}.$$$$G=\{g=\left(\begin{array}{ccc}-y& 0& x\\ 0& 1& z\\ 0& 0& 1\end{array}\right):x,z\in \mathbb{R},y<0.\}\u27f7(x,y,z)\in \mathbb{R}\times {\mathbb{R}}_{-}\times \mathbb{R}.$$$$({x}_{1},{y}_{1},{z}_{1})\ast ({x}_{2},{y}_{2},{z}_{2})=({x}_{1}-{y}_{1}{x}_{2},-{y}_{1}{y}_{2},{z}_{1}+{z}_{2}).$$The derivation Lie algebra has dimension 4. Precisely, each derivation$$\mathcal{D}=\left(\begin{array}{ccc}a& b& 0\\ 0& 0& 0\\ 0& c& d\end{array}\right):a,b,c,d\in \mathbb{R},$$$${\mathcal{X}}_{g}=(ax+by+b)\frac{\partial}{\partial x}+(dz-cln(-y))\frac{\partial}{\partial z}.$$
- Consider the Lie algebra with a basis$$X=M(1,3),Y=M(2,3),Z=M(1,2)-M(2,1),$$

## 6. Conclusions

- To compute the control sets when G has dimension three
- To characterize the controllability property when $G=SL(2,\mathbb{R})$
- To study the optimal control problem with quadratic cost.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**MDPI and ACS Style**

Ayala, V.; Torreblanca, M.; Valdivia, W.
Toward Applications of Linear Control Systems on the Real World and Theoretical Challenges. *Symmetry* **2021**, *13*, 167.
https://doi.org/10.3390/sym13020167

**AMA Style**

Ayala V, Torreblanca M, Valdivia W.
Toward Applications of Linear Control Systems on the Real World and Theoretical Challenges. *Symmetry*. 2021; 13(2):167.
https://doi.org/10.3390/sym13020167

**Chicago/Turabian Style**

Ayala, Víctor, María Torreblanca, and William Valdivia.
2021. "Toward Applications of Linear Control Systems on the Real World and Theoretical Challenges" *Symmetry* 13, no. 2: 167.
https://doi.org/10.3390/sym13020167