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Stabilization of a One-Dimensional Dam-River System: Nondissipative and Noncollocated Case

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Abstract

In this paper, we consider a one-dimensional dam-river system, described by a diffusive-wave equation and often used in hydraulic engineering to model the dynamic behavior of the unsteady flow in a river for shallow water when the flow variations are not important. We propose an integral boundary control which leads to a nondissipative closed-loop system with noncollocated actuators and sensors; hence, two main difficulties arise: first, how to show the C 0-semigroup generation and second, how to achieve the stability of the system. To overcome this situation, the Riesz basis methodology is adopted to show that the closed-loop system generates an analytic semigroup. Concerning the stability, the shooting method is applied to assign the spectrum of the system in the open left-half plane and ensure its exponential stability as well as the output regulation. Numerical simulations are presented for a family of system parameters.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Sultan Qaboos University, Muscat, Sultanate of Oman, United Arab Emirates

    B. Chentouf

  2. Department of Mathematics, Beijing Institute of Technology, Beijing, 100081, China

    J. M. Wang

Authors
  1. B. Chentouf
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  2. J. M. Wang
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Corresponding author

Correspondence to B. Chentouf.

Additional information

Communicated by F.E. Udwadia.

The authors express their sincere thanks to Boumenir Amin for valuable comments and suggestions. The first author acknowledges the support of Sultan Qaboos University. The second author was supported by the National Natural Science Foundation of China.

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Chentouf, B., Wang, J.M. Stabilization of a One-Dimensional Dam-River System: Nondissipative and Noncollocated Case. J Optim Theory Appl 134, 223–239 (2007). https://doi.org/10.1007/s10957-007-9223-z

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  • DOI: https://doi.org/10.1007/s10957-007-9223-z

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