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.

Accepted:

DOI: 10.1051/cocv/2016069

Keywords: First-order hyperbolic partial differential equation, age-structured models, chemostat, integral delay equations, nonlinear feedback control

^{1}; Krstic, Miroslav

^{2}

@article{COCV_2017__23_4_1667_0, author = {Karafyllis, Iasson and Krstic, Miroslav}, title = {Stability of integral delay equations and stabilization of age-structured models}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {1667--1714}, publisher = {EDP-Sciences}, volume = {23}, number = {4}, year = {2017}, doi = {10.1051/cocv/2016069}, zbl = {1379.35340}, mrnumber = {3716937}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv/2016069/} }

TY - JOUR AU - Karafyllis, Iasson AU - Krstic, Miroslav TI - Stability of integral delay equations and stabilization of age-structured models JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2017 SP - 1667 EP - 1714 VL - 23 IS - 4 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2016069/ DO - 10.1051/cocv/2016069 LA - en ID - COCV_2017__23_4_1667_0 ER -

%0 Journal Article %A Karafyllis, Iasson %A Krstic, Miroslav %T Stability of integral delay equations and stabilization of age-structured models %J ESAIM: Control, Optimisation and Calculus of Variations %D 2017 %P 1667-1714 %V 23 %N 4 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2016069/ %R 10.1051/cocv/2016069 %G en %F COCV_2017__23_4_1667_0

Karafyllis, Iasson; Krstic, Miroslav. Stability of integral delay equations and stabilization of age-structured models. ESAIM: Control, Optimisation and Calculus of Variations, Volume 23 (2017) no. 4, pp. 1667-1714. doi : 10.1051/cocv/2016069. http://archive.numdam.org/articles/10.1051/cocv/2016069/

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