Boundary control of a linear distributed parameter bioprocess. (English) Zbl 1036.93033
The characteristic conditions for exponential stability of a linear distributed parameter bio-process are discussed. The substrate and bio-mass concentrations are controlled using a feedback signal at the input boundary and measurements at the output boundary. It is shown that the system is exponentially stabilizable by a proper selection of the feedback gain. It is proved that the used closed-loop infinitesimal Hilbert space operator generates a uniformly bounded semigroup which relates the initial and actual distributed states. It is also shown that the decay rate of the given open-loop distributed parameter system can be calculated by letting the feedback gain tend to zero.
Reviewer: Ingmar Randvee (Tallinn)
MSC:
| 93C20 | Control/observation systems governed by partial differential equations |
| 93D15 | Stabilization of systems by feedback |
| 93D20 | Asymptotic stability in control theory |
| 93C95 | Application models in control theory |
Keywords:
boundary control; linear bioprocesses; exponential stability; distributed parameter system; bounded semigroup; decay rateReferences:
| [1] | Bourrel, S.; Dochain, D., Stability analysis of two linear distributed parameter bioprocess models, Math. Comput. Modelling Dyn. Systems, 6, 3, 267-281 (2000) · Zbl 0968.93037 |
| [2] | Huang, F. L., Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Differential Equations, 1, 1, 43-56 (1985) · Zbl 0593.34048 |
| [3] | C.Z. Xu, J.P. Gauthier, I. Kupka, Exponential stability of the heat exchanger equation, Proceedings of the Second European Control Conference, Vol. 1, Groningen, 1993, pp. 303-307.; C.Z. Xu, J.P. Gauthier, I. Kupka, Exponential stability of the heat exchanger equation, Proceedings of the Second European Control Conference, Vol. 1, Groningen, 1993, pp. 303-307. |
| [4] | Kunimatsu, N.; Sano, H., Stability analysis of heat-exchanger equations with boundary feedbacks, IMA J. Math. Control Inform., 15, 317-330 (1998) · Zbl 0917.93034 |
| [5] | R.F. Curtain, H.J. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory, Texts in Applied Mathematics, Vol. 21, Springer, New York, 1995.; R.F. Curtain, H.J. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory, Texts in Applied Mathematics, Vol. 21, Springer, New York, 1995. · Zbl 0839.93001 |
| [6] | A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, Vol. 44, Springer, New York, 1983.; A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, Vol. 44, Springer, New York, 1983. · Zbl 0516.47023 |
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