# Maximum Principle and Second-Order Optimality Conditions in Control Problems with Mixed Constraints

^{1}

^{2}

^{3}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Problem Formulation

**Hypothesis**

**1**

**(H1).**

**Definition**

**1.**

**Proposition**

**1.**

**Definition**

**2.**

## 3. Normality Condition

**Definition**

**3.**

## 4. Main Result

**Theorem**

**1.**

**Lemma**

**1.**

**Proof.**

**Proof**

**to Theorem 1.**

## 5. Abnormal Case

**Hypothesis**

**2**

**(H2).**

**Hypothesis**

**3**

**(H3).**

**Theorem**

**2.**

**Proof.**

- (a)
- ${e}^{\prime}\left({p}_{i}\right)\delta {p}_{i}=0$;
- (b)
- ${r}_{x}^{\prime}({x}_{i}\left(t\right),{u}_{i}\left(t\right),t)\delta {x}_{i}\left(t\right)+{r}_{u}^{\prime}({x}_{i}\left(t\right),{u}_{i}\left(t\right),t)\delta u\left(t\right)=0$$\mathrm{for}\phantom{\rule{0.166667em}{0ex}}\mathrm{a}.\mathrm{a}.\phantom{\rule{0.166667em}{0ex}}t\in {T}_{i}$.

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**MDPI and ACS Style**

Arutyunov, A.V.; Karamzin, D.Y.; Pereira, F.L.
Maximum Principle and Second-Order Optimality Conditions in Control Problems with Mixed Constraints. *Axioms* **2022**, *11*, 40.
https://doi.org/10.3390/axioms11020040

**AMA Style**

Arutyunov AV, Karamzin DY, Pereira FL.
Maximum Principle and Second-Order Optimality Conditions in Control Problems with Mixed Constraints. *Axioms*. 2022; 11(2):40.
https://doi.org/10.3390/axioms11020040

**Chicago/Turabian Style**

Arutyunov, Aram V., Dmitry Yu. Karamzin, and Fernando Lobo Pereira.
2022. "Maximum Principle and Second-Order Optimality Conditions in Control Problems with Mixed Constraints" *Axioms* 11, no. 2: 40.
https://doi.org/10.3390/axioms11020040