# Necessary Optimality Conditions for a Class of Control Problems with State Constraint

^{*}

## Abstract

**:**

## 1. Introduction

- (i)
- it has a finite number of inclusion-maximal boundary intervals,
- (ii)
- an implication holds that if $g(x(t))=0$ for a certain t, then t belongs to the closure of some boundary interval of u, 2
- (iii)
- the conditions of nontangentiality

- (iv)
- there is an open set ${X}_{u}\subset {R}^{n}$ containing all points $x(t)$ such that $g(x(t))=0$, and there is a ${C}^{1}$ function $w:{X}_{u}\to R$ such that

## 2. The One-Spike Control Variation and Trajectory Variation

- (i)
- ${u}^{\epsilon}(t)=w({x}^{\epsilon}(t))$ if $g({x}^{\epsilon}({t}_{1}))=0$ for some ${t}_{1}\le t$, and u has no exit points in $[{t}_{1},t]$,
- (ii)
- ${u}^{\epsilon}(t)=u({t}_{\mathrm{en}}-)$ if $g(x(t))=0$ and $g({x}^{\epsilon}(t))<0$, where ${t}_{\mathrm{en}}$ is the greatest entry point of u less than or equal to t,
- (iii)
- ${u}^{\epsilon}(t)=u(t)$ otherwise.

**Lemma**

**1.**

**Proof.**

## 3. The Adjoint Function and The One-Spike Necessary Optimality Condition

**Lemma**

**2.**

**Theorem**

**1.**

- (i)
- $\Delta H(t,v)\ge 0$for every$t\in [0,T]$and every$v\in {U}_{t}$,
- (ii)
- the function$[0,T]\ni t\mapsto \chi (t)=H(\psi (t),x(t),u(t))$is constant.

**Proof.**

**Corollary**

**1.**

## 4. The Two-Spike Necessary Optimality Condition

**Lemma**

**3.**

**Proof.**

**Lemma**

**4.**

**Proof.**

**Theorem**

**2**

**(main result).**

- (i)
- if$\dot{g}(x({t}_{1}),{v}_{1})<0$and$\Delta H({t}_{1},{v}_{1})=0$, then$\Delta H({t}_{2},{v}_{2})\ge 0$,
- (ii)
- if$\dot{g}(x({t}_{2}),{v}_{2})=0$and$\Delta H({t}_{2},{v}_{2})<0$, then$\dot{g}(x({t}_{1}),{v}_{1})\ge 0$,
- (iii)
- if$\dot{g}(x({t}_{1}),{v}_{1})<0$and$\Delta H({t}_{2},{v}_{2})<0$, then$\dot{g}(x({t}_{2}),{v}_{2})>0$and

**Proof.**

## 5. A Geometrical Interpretation and a Minimum Condition

**Theorem**

**3.**

- (i)
- $-{\scriptscriptstyle \frac{1}{2}}\pi \le {\varphi}_{\mathrm{min}}(t)\le {\varphi}_{\mathrm{max}}(t)\le \pi $for every$t\in \theta $,
- (ii)
- ${\varphi}_{\mathrm{max}}({t}_{1})-{\varphi}_{\mathrm{min}}({t}_{2})\le \pi $for every pair${t}_{1},{t}_{2}\in \theta $such that${t}_{1}\le {t}_{2}$.

## 6. Example 1

**Example**

**1a.**

**Example**

**1b.**

## 7. The Control Affine Case

**Lemma**

**5.**

**Theorem**

**4.**

**Corollary**

**2.**

- (i)
- if$\left(g(x(t))<0and{\alpha}_{2}(t)0\right)$or$\left(t\in {\Theta}_{u},\hspace{0.17em}{\alpha}_{1}(t)>0and{\alpha}_{2}(t)0\right)$, then$\mathrm{min}U$exists and$u(t)=\mathrm{min}U$,
- (ii)
- if$\left(g(x(t))<0and{\alpha}_{2}(t)0\right)$or$\left(t\in {\Theta}_{u},\hspace{0.17em}{\alpha}_{1}(t)<0and{\alpha}_{2}(t)0\right)$, then$\mathrm{max}U$exists and$u(t)=\mathrm{max}U$.

**Lemma**

**6.**

**Corollary**

**3.**

**Theorem**

**5.**

- (i)
- if${\alpha}_{1}({t}_{1})({v}_{1}-u({t}_{1}))<0$and${\alpha}_{2}({t}_{1})=0$, then${\alpha}_{2}({t}_{2})({v}_{2}-u({t}_{2}))\ge 0$,
- (ii)
- if${\alpha}_{1}({t}_{2})=0$and${\alpha}_{2}({t}_{2})({v}_{2}-u({t}_{2}))<0$, then${\alpha}_{1}({t}_{1})({v}_{1}-u({t}_{1}))\ge 0$,
- (iii)
- if${\alpha}_{1}({t}_{1})({v}_{1}-u({t}_{1}))<0$and${\alpha}_{2}({t}_{2})({v}_{2}-u({t}_{2}))<0$, then${\alpha}_{1}({t}_{2})\ne 0$and$p({t}_{1})\le p({t}_{2})<0$.

**Corollary**

**4.**

## 8. Example 2: The Pendulum on a Cart

- (i)
- $u({t}_{2})={u}_{\mathrm{max}}$ if ${\alpha}_{2}({t}_{1})=0$ and ${\alpha}_{2}({t}_{2})<0$,
- (ii)
- $u({t}_{2})={u}_{\mathrm{min}}$ if ${\alpha}_{2}({t}_{1})=0$ and ${\alpha}_{2}({t}_{2})>0$,
- (iii)
- $p({t}_{1})\le p({t}_{2})<0$ if ${\alpha}_{2}({t}_{2})\ne 0$ and ${u}_{\mathrm{min}}<u({t}_{2})<{u}_{\mathrm{max}}$.

**Example**

**2a.**

**Example**

**2b.**

## 9. Connections with Some Classical Results

**Theorem**

**6**

**([3], Theorem 5.1).**

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Pontryagin, L.S.; Boltyanskii, V.G.; Gamkrelidze, R.V.; Mishchenko, E.F. The Mathematical Theory of Optimal Processes; Nauka: Moscow, Russia, 1961; (in Russian, first English language edition: Wiley & Sons, Inc., New York, 1962). [Google Scholar]
- Karamzin, D.; Pereira, F.L. On a Few Questions Regarding the Study of State-Constrained Problems in Optimal Control. J. Optimiz. Theory App.
**2019**, 180, 235–255. [Google Scholar] [CrossRef] - Maurer, H. On the Minimum Principle for Optimal Control Problems with State Constraints; Universität Münster: Münster, Germany, 1979. [Google Scholar]
- Hartl, R.F.; Sethi, S.P.; Vickson, R.G. A survey of the maximum principles for optimal control problems with state constraints. SIAM Rev.
**1995**, 37, 181–218. [Google Scholar] [CrossRef] - Arutyunov, A.V.; Karamzin, D.Y.; Pereira, F.L. The maximum principle for optimal control problems with state constraints by R.V. Gamkrelidze: Revisited. J. Optimiz. Theory Appl.
**2011**, 149, 474–493. [Google Scholar] [CrossRef] - Bonnans, J.F. Course on Optimal Control. Part I: The Pontryagin approach (Version of 21 August 2019). Available online: http://www.cmap.polytechnique.fr/~bonnans/notes/oc/ocbook.pdf (accessed on 30 December 2020).
- Bourdin, L. Note on Pontryagin Maximum Principle with Running State Constraints and Smooth Dynamics—Proof based on the Ekeland Variational Principle; University of Limoges: Limoges, France, 2016; Available online: https://arxiv.org/pdf/1604.04051v1.pdf (accessed on 30 December 2020).
- Dmitruk, A.; Samylovskiy, I. On the relation between two approaches to necessary optimality conditions in problems with state constraints. J. Optimiz. Theory Appl.
**2017**, 173, 391–420. [Google Scholar] [CrossRef] - Vinter, R. Optimal Control; Birkhäuser: Boston, MA, USA, 2000. [Google Scholar]
- Korytowski, A.; Szymkat, M. On convergence of the Monotone Structural Evolution. Control Cybern.
**2016**, 45, 483–512. [Google Scholar] - Bonnans, J.F. The shooting approach to optimal control problems. IFAC Proc. Vol.
**2013**, 46, 281–292. [Google Scholar] [CrossRef][Green Version] - Bonnans, J.F.; Hermant, A. Well-posedness of the shooting algorithm for state constrained optimal control problems with a single constraint and control. SIAM J. Control Optim.
**2007**, 46, 1398–1430. [Google Scholar] [CrossRef][Green Version] - Chertovskih, R.; Karamzin, D.; Khalil, N.T.; Pereira, F.L. Regular path-constrained time-optimal control problems in three-dimensional flow fields. Eur. J. Control
**2020**, 56, 98–106. [Google Scholar] [CrossRef][Green Version] - Cortez, K.; de Pinho, M.R.; Matos, A. Necessary conditions for a class of optimal multiprocess with state constraints. Int. J. Robust Nonlinear Control
**2020**, 30, 6021–6041. [Google Scholar] [CrossRef]

1 | PC(0,T; U) is the space of all functions [0,T] → U which have a finite number of discontinuities, are right-continuous in [0,T[, left-continuous at T, and have a finite left-hand limit at every point. |

2 | Controls leading to state trajectories with boundary touch points are not verifiable. |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Korytowski, A.; Szymkat, M. Necessary Optimality Conditions for a Class of Control Problems with State Constraint. *Games* **2021**, *12*, 9.
https://doi.org/10.3390/g12010009

**AMA Style**

Korytowski A, Szymkat M. Necessary Optimality Conditions for a Class of Control Problems with State Constraint. *Games*. 2021; 12(1):9.
https://doi.org/10.3390/g12010009

**Chicago/Turabian Style**

Korytowski, Adam, and Maciej Szymkat. 2021. "Necessary Optimality Conditions for a Class of Control Problems with State Constraint" *Games* 12, no. 1: 9.
https://doi.org/10.3390/g12010009