Geodesic
Systems with hysteresis in the feedback loop : existence, regularity and asymptotic behaviour of solutions
ESAIM: Control, Optimisation and Calculus of Variations, Tome 9 (2003), pp. 169-196

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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.

DOI : 10.1051/cocv:2003007
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 
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     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},
     pages = {169--196},
     publisher = {EDP-Sciences},
     volume = {9},
     year = {2003},
     doi = {10.1051/cocv:2003007},
     mrnumber = {1957097},
     zbl = {1076.45004},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/cocv:2003007/}
}
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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, Tome 9 (2003), pp. 169-196. doi: 10.1051/cocv:2003007

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