Optimal spiral-like solutions near a singular extremal in a two-input control problem

Larisa Manita; Mariya Ronzhina

doi:10.3934/dcdsb.2021187

Article Contents

2022, Volume 27, Issue 6: 3325-3343. Doi: 10.3934/dcdsb.2021187

This issue Previous Article Asymptotic behavior for the solutions to a bistable-bistable reaction diffusion equation Next Article Matrix measures, stability and contraction theory for dynamical systems on time scales

Optimal spiral-like solutions near a singular extremal in a two-input control problem

Larisa Manita^1, , and
Mariya Ronzhina^2,

1.
HSE University, Moscow State Institute of Electronics and Mathematics, Moscow, Russia
2.
National University of Oil and Gas ``Gubkin University", Moscow, Russia

^* Corresponding author: Larisa Manita
^* Corresponding author: Larisa Manita

Received: January 2020

Revised: May 2021

Early access: July 19, 2021

Published online: July 19, 2021

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Abstract

We study an optimal control problem affine in two-dimensional bounded control, in which there is a singular point of the second order. In the neighborhood of the singular point we find optimal spiral-like solutions that attain the singular point in finite time, wherein the corresponding optimal controls perform an infinite number of rotations along the circle $ S^{1} $. The problem is related to the control of an inverted spherical pendulum in the neighborhood of the upper unstable equilibrium.

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Mathematics Subject Classification: Primary: 34H05, 49N60; Secondary: 93C05.

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Figure 1. Optimal chattering solutions in P1

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Figure 2. Solutions of the blown-up Hamiltonian system that lie on $ Q $ and tend to $ \xi^{0} $

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Figure 3. The inverted spherical pendulum

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