TY - GEN

T1 - Frequency domain conditions for the existence of Bohr almost periodic solutions in evolution equations

AU - Reitmann, Volker

N1 - Copyright: Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2007

Y1 - 2007

N2 - We consider a control problem for the heating process of an elastic plate. The heat flux within the plate is modeled by the heat equation with nonlinear Neumann boundary conditions according to Newton's law. As input at a part of the boundary we take the nonlinearly transformed and modulated heat production of a separate heater which is given by a nonlinear Duffing-type ODE. This ODE depends on measurements of the temperature within the plate and on Bohr resp. Stepanov almost periodic in time forcing terms. The physical problem is generalized to a bifurcation problem for non-autonomous evolution systems in rigged Hilbert spaces. Using Lyapunov functionals, invariant cones and monotonicity properties of the nonlinearities in certain Sobolev spaces, we derive frequency domain conditions for the existence and uniqueness of an asymptotically stable and almost periodic in time temperature field.

AB - We consider a control problem for the heating process of an elastic plate. The heat flux within the plate is modeled by the heat equation with nonlinear Neumann boundary conditions according to Newton's law. As input at a part of the boundary we take the nonlinearly transformed and modulated heat production of a separate heater which is given by a nonlinear Duffing-type ODE. This ODE depends on measurements of the temperature within the plate and on Bohr resp. Stepanov almost periodic in time forcing terms. The physical problem is generalized to a bifurcation problem for non-autonomous evolution systems in rigged Hilbert spaces. Using Lyapunov functionals, invariant cones and monotonicity properties of the nonlinearities in certain Sobolev spaces, we derive frequency domain conditions for the existence and uniqueness of an asymptotically stable and almost periodic in time temperature field.

KW - Control closed-loop

KW - Frequency domains

KW - Partial differential equations

KW - Periodic motion

KW - Stability analysis

UR - http://www.scopus.com/inward/record.url?scp=79960986408&partnerID=8YFLogxK

U2 - 10.3182/20070829-3-ru-4912.00041

DO - 10.3182/20070829-3-ru-4912.00041

M3 - Conference contribution

AN - SCOPUS:79960986408

SN - 9783902661302

T3 - IFAC Proceedings Volumes (IFAC-PapersOnline)

SP - 240

EP - 244

BT - 3rd IFAC Workshop "Periodic Control Systems", PSYCO'2007 - Final Program and Abstracts

PB - International Federation of Automatic Control

ER -