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Anthony Hastir (UNamur)

October 3, 2019 @ 13:00 - 14:00

Title: On stability and control of nonlinear infinite-dimensional systems

Abstract: We consider nonlinear dynamical systems whose state evolves in an infinite-dimensional space, typically a Hilbert space. Stability properties of an equilibrium of such systems can be deduced by looking at a Gâteaux/Fréchet linearization of the nonlinear dynamics around that equilibrium. From « Al Jamal, R., Morris, K., 2018, Linearized stability of partial differential equations with application to stabilization of the Kuramoto-Sivashinsky equation, SIAM Journal on Control and Optimization 56 (1), 120–147 », it is known that, if the Gâteaux/Fréchet linearization generates an exponentially stable C 0 -semigroup that is Fréchet differentiable at the equilibrium, the nonlinear dynamics will generate a locally exponentially stable C 0 – semigroup of nonlinear operators. However, checking the Fréchet differentiability of nonlinear operators is really challenging when they are unbounded. This observation leads us to propose a weakenend definition of Fréchet differentiability by considering different norms. This allows more flexibility in the manipulation of norm inequalities, providing Fréchet differentiability conditions that can be verified more easily. Under some appropriate assumptions, we show how to get Fréchet differentiability of the nonlinear semigroup in a weaker sense in order to establish appropriate local exponential stability of the nonlinear system.
We show also that every linear and bounded perturbation added to the nonlinear dynamics preserves the Fréchet differentiability property of the nonlinear C 0 -semigroup generated by these perturbed dynamics. For instance, this property is of primary importance when the system is controlled by means of a feedback law. Indeed, it implies that every stabilizing feedback for the linearized dynamics will locally exponentially stabilize the nonlinear dynamics around an equilibrium. This allows to design a stabilizing controller on a linearized model instead of a nonlinear one.
Our results are applied to a nonisothermal axial dispersion tubular reactor model for which we consider distributed LQ-optimal regulation of the temperature.


October 3, 2019
13:00 - 14:00
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