Stability of integral delay equations and stabilization of age-structured models

Karafyllis, Iasson; Krstic, Miroslav

ESAIM: Control, Optimisation and Calculus of Variations, Tome 23 (2017) no. 4, pp. 1667-1714.

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.

Classification : 34K20, 35L04, 35L60, 93D20, 34K05, 93C23
Mots clés : First-order hyperbolic partial differential equation, age-structured models, chemostat, integral delay equations, nonlinear feedback control

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