# The Problem of Invariance in Nonlinear Discrete-Time Dynamic Systems

## Abstract

**:**

## 1. Introduction and Problem Statement

## 2. Algebra of Functions

**m**and

**M**.

**0**and

**1**exist; they correspond to the identity and constant functions, respectively, in the sense that for every vector function $\mathsf{\alpha}\in S$, $0\le \mathsf{\alpha}\le 1$.

**m**and

**M**.

**Lemma**

**1**

**m**and

**M**. It is known from [19,23] that there exists the function $\mathsf{\gamma}$ satisfying the condition $(\mathsf{\alpha}\times u(t))\oplus f\cong \mathsf{\gamma}(f)$; define $m(\mathsf{\alpha})\cong \mathsf{\gamma}$. When $\mathsf{\beta}(f(x(t),u(t)))$ can be transformed into the form

## 3. Problem Solution

**Theorem**

**1.**

**Proof.**

**Theorem**

**2**

**[19,23].**Given${\mathsf{\phi}}_{0}$, compute recursively for$i\ge 0$, based on the formula

**Theorem**

**3.**

**Proof.**

Algorithm |

Step 1. Set ${\mathsf{\beta}}_{0}:=\mathsf{\rho}(h)$ and $i:=0$.Step 2. Compute the function ${\mathsf{\gamma}}_{i}=M({\mathsf{\beta}}_{i})$.Step 3. If the components of the vector function ${\mathsf{\gamma}}_{i}$ can be expressed in terms of the function $h\times {\mathsf{\beta}}_{0}\times \dots \times {\mathsf{\beta}}_{i}$, then go to Step 5. Otherwise, go to Step 4.Step 4. Find the vector function ${\mathsf{\beta}}_{i+1}$ with minimal number of components such that $h\times {\mathsf{\beta}}_{0}\times \dots \times {\mathsf{\beta}}_{i+1}\le {\mathsf{\gamma}}_{i}$, set $i:=i+1$ and go to Step 2.Step 5. Define $\mathsf{\phi}:={\mathsf{\beta}}_{0}\times \dots \times {\mathsf{\beta}}_{i}$. |

## 4. Logic–Dynamic Approach

**Step 1.**Remove the term $\mathsf{\Psi}(x(t),u(t))$ from the original system (6).

**Step 2.**Solve the problem of IWS to the unknown input for the linear part under some linear restriction. This restriction is necessary to find out whether or not the nonlinear term can be designed based on the linear solution.

**Step 3.**Supplement the solution, obtained at Step 2, by the transformed nonlinear term.

## 5. Solvability Conditions of Invariance

## 6. Problem Solution

## 7. Applications

## 8. Conclusions

## Funding

## Conflicts of Interest

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Zhirabok, A.
The Problem of Invariance in Nonlinear Discrete-Time Dynamic Systems. *Symmetry* **2020**, *12*, 1241.
https://doi.org/10.3390/sym12081241

**AMA Style**

Zhirabok A.
The Problem of Invariance in Nonlinear Discrete-Time Dynamic Systems. *Symmetry*. 2020; 12(8):1241.
https://doi.org/10.3390/sym12081241

**Chicago/Turabian Style**

Zhirabok, Alexey.
2020. "The Problem of Invariance in Nonlinear Discrete-Time Dynamic Systems" *Symmetry* 12, no. 8: 1241.
https://doi.org/10.3390/sym12081241