TY - GEN
T1 - On local exponential stability of equilibrium profiles of nonlinear distributed parameter systems
AU - HASTIR, A.
AU - Winkin, J. J.
AU - Dochain, D.
N1 - Publisher Copyright:
Copyright © 2021 The Authors. This is an open access article under the CC BY-NC-ND license.
PY - 2021/6/1
Y1 - 2021/6/1
N2 - Local exponential (exp.) stability of nonlinear distributed parameter, i.e. infinite- dimensional state space, systems is considered. A weakened concept of Frechet differentiability ((y,X)-Frechet differentiability) for nonlinear operators defined on Banach spaces is proposed, including the introduction of an alternative space (Y) in the analysis. This allows more freedom in the manipulation of norm-inequalities leading to adapted Frechet differentiability conditions that are easier to check. Then, provided that the nonlinear semigroup generated by the nonlinear dynamics is Frechet-different iable in the new sense, appropriate local exp. stability of the equilibria for the nonlinear system is established. In particular, the nonlinear semigroup has to be Frechet differentiate on Y and (Y, X)-Frechet differentiate in order to go back to the original state space X. This approach may be called "perturbation-based"since exp. stability is also deduced from exp. stability of a linearized version of the nonlinear semigroup. Under adapted Frechet differentiability assumptions, the main result establishes that local exp. stability of an equilibrium for the nonlinear system is guaranteed as long as the exp. stability holds for the linearized semigroup. The same conclusion holds regarding instability. The theoretical results are illustrated on a convection-diffusion-reaction system.
AB - Local exponential (exp.) stability of nonlinear distributed parameter, i.e. infinite- dimensional state space, systems is considered. A weakened concept of Frechet differentiability ((y,X)-Frechet differentiability) for nonlinear operators defined on Banach spaces is proposed, including the introduction of an alternative space (Y) in the analysis. This allows more freedom in the manipulation of norm-inequalities leading to adapted Frechet differentiability conditions that are easier to check. Then, provided that the nonlinear semigroup generated by the nonlinear dynamics is Frechet-different iable in the new sense, appropriate local exp. stability of the equilibria for the nonlinear system is established. In particular, the nonlinear semigroup has to be Frechet differentiate on Y and (Y, X)-Frechet differentiate in order to go back to the original state space X. This approach may be called "perturbation-based"since exp. stability is also deduced from exp. stability of a linearized version of the nonlinear semigroup. Under adapted Frechet differentiability assumptions, the main result establishes that local exp. stability of an equilibrium for the nonlinear system is guaranteed as long as the exp. stability holds for the linearized semigroup. The same conclusion holds regarding instability. The theoretical results are illustrated on a convection-diffusion-reaction system.
KW - Distributed parameter systems - nonlinear systems - equilibrium - exponential stability
UR - http://www.scopus.com/inward/record.url?scp=85117886449&partnerID=8YFLogxK
U2 - 10.1016/j.ifacol.2021.06.097
DO - 10.1016/j.ifacol.2021.06.097
M3 - Conference contribution
AN - SCOPUS:85117886449
VL - 54
T3 - IFAC-PapersOnLine
SP - 390
EP - 396
BT - Proceedings of MTNS 2020
T2 - 24th International Symposium on Mathematical Theory of Networks and Systems, MTNS 2020
Y2 - 23 August 2021 through 27 August 2021
ER -