Around certain critical cases in stability studies in hydraulic engineering

Vladimir Răsvan

Archivum Mathematicum (2023)

  • Issue: 1, page 109-116
  • ISSN: 0044-8753

Abstract

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It is considered the mathematical model of a benchmark hydroelectric power plant containing a water reservoir (lake), two water conduits (the tunnel and the turbine penstock), the surge tank and the hydraulic turbine; all distributed (Darcy-Weisbach) and local hydraulic losses are neglected,the only energy dissipator remains the throttling of the surge tank. Exponential stability would require asymptotic stability of the difference operator associated to the model. However in this case this stability is “fragile” i.e. it holds only for a rational ratio of the two delays, with odd numerator and denominator also. Otherwise this stability is critical (non-asymptotic and displaying an oscillatory mode).

How to cite

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Răsvan, Vladimir. "Around certain critical cases in stability studies in hydraulic engineering." Archivum Mathematicum (2023): 109-116. <http://eudml.org/doc/298968>.

@article{Răsvan2023,
abstract = {It is considered the mathematical model of a benchmark hydroelectric power plant containing a water reservoir (lake), two water conduits (the tunnel and the turbine penstock), the surge tank and the hydraulic turbine; all distributed (Darcy-Weisbach) and local hydraulic losses are neglected,the only energy dissipator remains the throttling of the surge tank. Exponential stability would require asymptotic stability of the difference operator associated to the model. However in this case this stability is “fragile” i.e. it holds only for a rational ratio of the two delays, with odd numerator and denominator also. Otherwise this stability is critical (non-asymptotic and displaying an oscillatory mode).},
author = {Răsvan, Vladimir},
journal = {Archivum Mathematicum},
keywords = {neutral functional differential equations; energy Lyapunov functional; asymptotic stability; water hammer},
language = {eng},
number = {1},
pages = {109-116},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Around certain critical cases in stability studies in hydraulic engineering},
url = {http://eudml.org/doc/298968},
year = {2023},
}

TY - JOUR
AU - Răsvan, Vladimir
TI - Around certain critical cases in stability studies in hydraulic engineering
JO - Archivum Mathematicum
PY - 2023
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
IS - 1
SP - 109
EP - 116
AB - It is considered the mathematical model of a benchmark hydroelectric power plant containing a water reservoir (lake), two water conduits (the tunnel and the turbine penstock), the surge tank and the hydraulic turbine; all distributed (Darcy-Weisbach) and local hydraulic losses are neglected,the only energy dissipator remains the throttling of the surge tank. Exponential stability would require asymptotic stability of the difference operator associated to the model. However in this case this stability is “fragile” i.e. it holds only for a rational ratio of the two delays, with odd numerator and denominator also. Otherwise this stability is critical (non-asymptotic and displaying an oscillatory mode).
LA - eng
KW - neutral functional differential equations; energy Lyapunov functional; asymptotic stability; water hammer
UR - http://eudml.org/doc/298968
ER -

References

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  1. Abolinia, V.E., Myshkis, A.D., Mixed problem for an almost linear hyperbolic system in the plane, Mat. Sb. (N.S.) 50(92) (4) (1960), 423–442, (Russian). (1960) 
  2. Aronovich, G.V., Kartvelishvili, N.A., Lyubimtsev, Ya.K., Water hammer and surge tanks, Nauka, Moscow USSR, 1968, (Russian). (1968) 
  3. Cooke, K.L., A linear mixed problem with derivative boundary conditions, Seminar on Differential Equations and Dynamical Systems (III), University of Maryland, College Park, 1970. (1970) 
  4. Escande, L., Dat, J., Piquemal, J., Stabilité d’une chambre d’équilibre placée à la jonction de deux galeries alimentées par des lacs situés à la même cote, C.R. Acad. Sci. Paris 261 (1965), 2579–2581. (1965) 
  5. Halanay, A., Popescu, M., Une propriété arithmétique dans l’analyse du comportement d’un systéme hydraulique comprenant une chambre d’équilibre avec étranglement, C.R. Acad. Sci. Paris Série II 305 (15) (1987), 1227–1230. (1987) 
  6. Hale, J.K., 10.1016/0022-247X(69)90175-9, J. Math. Anal. Appl. 26 (1969), 39–69. (1969) DOI10.1016/0022-247X(69)90175-9
  7. Hale, J.K., Verduyn Lunel, S.M., Introduction to Functional Differential Equations, vol. 99, Applied Mathematical Sciences, Springer-Verlag, New York, 1993. (1993) Zbl0787.34002
  8. Haraux, A., Systémes dynamiques dissipatifs et applications, Recherches en Mathématiques appliquées, vol. 17, Masson, Paris, 1991. (1991) 
  9. Răsvan, V., Augmented Validation and a Stabilization Approach for Systems with Propagation, Systems Theory: Perspectives, Applications and Developments, Nova Science Publishers, New York, 2014. (2014) 
  10. Răsvan, V., 10.7494/OpMath.2022.42.4.605, Opuscula Math. 42 (4) (2022), 605–633. (2022) MR4449109DOI10.7494/OpMath.2022.42.4.605
  11. Răsvan, V., 10.14232/ejqtde.2022.1.19, Electron. J. Qual. Theory Differ. Equ. 19 (2022), 1–32. (2022) MR4417616DOI10.14232/ejqtde.2022.1.19
  12. Saperstone, S.H., Semidynamical Systems in Infinite Dimensional Spaces, vol. 37, Applied Mathematical Sciences, Springer, New York-Heidelberg-Berlin, 1981. (1981) 

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