Volume 23, Number 4, October-December 2017
|Page(s)||1667 - 1714|
|Published online||28 September 2017|
Stability of integral delay equations and stabilization of age-structured models
1 Department of Mathematics, National Technical University of Athens, Zografou Campus, 15780 Athens, Greece.
2 Department of Mechanical and Aerospace Engineering, University of California, San Diego, La Jolla, CA 92093-0411, USA.
Received: 6 October 2015
Revised: 20 April 2016
Accepted: 5 October 2016
We present bounded dynamic (but observer-free) output feedback laws that achieve global stabilization of equilibrium profiles of the partial differential equation (PDE) model of a simplified, age-structured chemostat model. The chemostat PDE state is positive-valued, which means that our global stabilization is established in the positive orthant of a particular function space–a rather non-standard situation, for which we develop non-standard tools. Our feedback laws do not employ any of the (distributed) parametric knowledge of the model. Moreover, we provide a family of highly unconventional Control Lyapunov Functionals (CLFs) for the age-structured chemostat PDE model. Two kinds of feedback stabilizers are provided: stabilizers with continuously adjusted input and sampled-data stabilizers. The results are based on the transformation of the first-order hyperbolic partial differential equation to an ordinary differential equation (one-dimensional) and an integral delay equation (infinite-dimensional). Novel stability results for integral delay equations are also provided; the results are of independent interest and allow the explicit construction of the CLF for the age-structured chemostat model.
Mathematics Subject Classification: 34K20 / 35L04 / 35L60 / 93D20 / 34K05 / 93C23
Key words: First-order hyperbolic partial differential equation / age-structured models / chemostat / integral delay equations / nonlinear feedback control
© EDP Sciences, SMAI 2017
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