An existence and regularity theorem is proved for integral equations of convolution type which contain hysteresis nonlinearities. On the basis of this result, frequency-domain stability criteria are derived for feedback systems with a linear infinite-dimensional system in the forward path and a hysteresis nonlinearity in the feedback path. These stability criteria are reminiscent of the classical circle criterion which applies to static sector-bounded nonlinearities. The class of hysteresis operators under consideration contains many standard hysteresis nonlinearities which are important in control engineering such as backlash (or play), plastic-elastic (or stop) and Prandtl operators. Whilst the main results are developed in the context of integral equations of convolution type, applications to well-posed state space systems are also considered.

Classification: 45M05, 45M10, 47J40, 93C10, 93C25, 93D05, 93D10, 93D25

Keywords: absolute stability, asymptotic behaviour, frequency-domain stability criteria, hysteresis, infinite-dimensional systems, integral equations, regularity of solutions

@article{COCV_2003__9__169_0, author = {Logemann, Hartmut and Ryan, Eugene P.}, title = {Systems with hysteresis in the feedback loop : existence, regularity and asymptotic behaviour of solutions}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, publisher = {EDP-Sciences}, volume = {9}, year = {2003}, pages = {169-196}, doi = {10.1051/cocv:2003007}, zbl = {1076.45004}, mrnumber = {1957097}, language = {en}, url = {http://www.numdam.org/item/COCV_2003__9__169_0} }

Logemann, Hartmut; Ryan, Eugene P. Systems with hysteresis in the feedback loop : existence, regularity and asymptotic behaviour of solutions. ESAIM: Control, Optimisation and Calculus of Variations, Volume 9 (2003) , pp. 169-196. doi : 10.1051/cocv:2003007. http://www.numdam.org/item/COCV_2003__9__169_0/

[1] Hysteresis operators, in Phase Transitions and Hysteresis, edited by A. Visintin. Springer-Verlag, Berlin (1994) 1-38. | MR 1321830 | Zbl 0836.35065

,[2] Hysteresis and Phase Transitions. Springer-Verlag, New York (1996). | MR 1411908 | Zbl 0951.74002

and ,[3] Almost Periodic Functions, 2nd Edition. Chelsea Publishing Company, New York (1989). | Zbl 0672.42008

,[4] Stability results of Popov-type for infinite-dimensional systems with applications to integral control, Mathematics Preprint 01/09. University of Bath (2001). Proc. London Math. Soc. (to appear). Available at http://www.maths.bath.ac.uk/MATHEMATICS/preprints.html | MR 1974399 | Zbl 1032.93061

, and ,[5] Well-posedness of triples of operators in the sense of linear systems theory, in Control and Estimation of Distributed Parameter System, edited by F. Kappel, K. Kunisch and W. Schappacher. Birkhäuser Verlag, Basel (1989) 41-59. | MR 1033051 | Zbl 0686.93049

and ,[6] Volterra Integral and Functional Equations. Cambridge University Press, Cambridge (1990). | MR 1050319 | Zbl 0695.45002

, and ,[7] Stability of Motion. Springer-Verlag, Berlin (1967). | MR 223668 | Zbl 0189.38503

,[8] Systems with Hysteresis. Springer-Verlag, Berlin (1989). | Zbl 0665.47038

and .[9] Low-gain integral control of infinite-dimensional regular linear systems subject to input hysteresis, in Advances in Mathematical Systems Theory, edited by F. Colonius et al. Birkhäuser, Boston (2001) 255-293. | MR 1787350

and ,[10] Time-varying and adaptive integral control of infinite-dimensional regular linear systems with input nonlinearities. SIAM J. Control Optim. 38 (2000) 1120-1144. | MR 1760063 | Zbl 0968.93074

and ,[11] Mathematical models for hysteresis. SIAM Rev. 35 (1993) 94-123. | MR 1207799 | Zbl 0771.34018

, and ,[12] Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer-Verlag, New York (1983). | MR 710486 | Zbl 0516.47023

,[13] Realization theory in Hilbert space. Math. Systems Theory 21 (1989) 147-164. | MR 977021 | Zbl 0668.93018

,[14] Infinite-dimensional linear systems with unbounded control and observation: A functional analytic approach. Trans. Amer. Math. Soc. 300 (1987) 383-431. | MR 876460 | Zbl 0623.93040

,[15] Well-Posed Linear Systems. Book manuscript (in preparation). Available at http://www.abo.fi/~staffans/

,[16] $J$-energy preserving well-posed linear systems. Int. J. Appl. Math. Comput. Sci. 11 (2001) 1361-1378. | MR 1885509 | Zbl 1008.93024

,[17] Quadratic optimal control of stable well-posed linear systems. Trans. Amer. Math. Soc. 349 (1997) 3679-3715. | MR 1407712 | Zbl 0889.49023

,[18] Transfer functions of regular linear systems, Part II: The system operator and the Lax-Phillips semigroup. Trans. Amer. Math. Soc. 354 (2002) 3229-3262. | Zbl 0996.93012

and ,[19] Nonlinear Systems Analysis, 2nd Edition. Prentice Hall, Englewood Cliffs, NJ (1993). | Zbl 0759.93001

,[20] Transfer functions of regular linear systems, Part I: Characterization of regularity. Trans. Amer. Math. Soc. 342 (1994) 827-854. | MR 1179402 | Zbl 0798.93036

,[21] The representation of regular linear systems on Hilbert spaces, in Control and Estimation of Distributed Parameter System, edited by F. Kappel, K. Kunisch and W. Schappacher. Birkhäuser Verlag, Basel (1989) 401-416. | MR 1033074 | Zbl 0685.93040

,[22] The conditions for absolute stability of a control system with a hysteresis-type nonlinearity. Soviet Phys. Dokl. 8 (1963) 235-237 (translated from Dokl. Akad. Nauk SSSR 149 (1963) 288-291). | MR 166025 | Zbl 0131.31703

,