Adaptive prescribed performance control for wave equations with dynamic boundary and multiple parametric uncertainties

Zaihua Xu; Jian Li; Zaihua Xu; Jian Li

doi:10.3934/math.2024148

AIMS Mathematics

2024, Volume 9, Issue 2: 3019-3034. doi: 10.3934/math.2024148

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Research article

Adaptive prescribed performance control for wave equations with dynamic boundary and multiple parametric uncertainties

Zaihua Xu ^{1
,
,},
Jian Li ²

1.
Institute of Dynamics and Control Science, Shandong Normal University, Jinan, 250014, China
2.
School of Mathematics and Information Science, Yantai University, Yantai, 264005, China

Received: 01 November 2023 Revised: 07 December 2023 Accepted: 19 December 2023 Published: 02 January 2024
MSC : 93C20, 93D21

In modern engineering, the dynamics of many practical problems can be described by hyperbolic distributed parameter systems. This paper is devoted to the adaptive prescribed performance control for a class of typical uncertain hyperbolic distributed parameter systems, since uncertainties are inevitable in practice. The systems in question simultaneously have unknown in-domain spatially varying damping coefficient and unknown boundary constant damping coefficient. Moreover, dynamic boundary condition is considered in the present paper. These characteristics make the control problem in the paper essentially different from those in the related works. To solve the problem, using adaptive technique based projection operator, backstepping method developed for ODEs and Lyapunov stability theories, a powerful adaptive prescribed performance control scheme is proposed to successfully guarantee that all states of the resulting closed-loop system are bounded, furthermore, the original system state converges to an arbitrary prescribed small neighborhood of the origin. Compared with the existing results, the developed control schemes can not only effectively handle the serious uncertainties, but also overcome the technical difficulties in the infinite-dimensional backstepping control design method caused by the dynamic boundary condition and guarantee prescribed performance.
- wave equations,
- prescribed performance control,
- parametric uncertainties,
- dynamic boundary condition,
- adaptive technique
Citation: Zaihua Xu, Jian Li. Adaptive prescribed performance control for wave equations with dynamic boundary and multiple parametric uncertainties[J]. AIMS Mathematics, 2024, 9(2): 3019-3034. doi: 10.3934/math.2024148

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Abstract

In modern engineering, the dynamics of many practical problems can be described by hyperbolic distributed parameter systems. This paper is devoted to the adaptive prescribed performance control for a class of typical uncertain hyperbolic distributed parameter systems, since uncertainties are inevitable in practice. The systems in question simultaneously have unknown in-domain spatially varying damping coefficient and unknown boundary constant damping coefficient. Moreover, dynamic boundary condition is considered in the present paper. These characteristics make the control problem in the paper essentially different from those in the related works. To solve the problem, using adaptive technique based projection operator, backstepping method developed for ODEs and Lyapunov stability theories, a powerful adaptive prescribed performance control scheme is proposed to successfully guarantee that all states of the resulting closed-loop system are bounded, furthermore, the original system state converges to an arbitrary prescribed small neighborhood of the origin. Compared with the existing results, the developed control schemes can not only effectively handle the serious uncertainties, but also overcome the technical difficulties in the infinite-dimensional backstepping control design method caused by the dynamic boundary condition and guarantee prescribed performance.

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