Systems with hysteresis in the feedback loop: existence,regularity and asymptotic behaviour of solutions

E-ISSN： 1262-3377|9|issue|169-196

ISSN： 1292-8119

Source： ESAIM: Control, Optimisation and Calculus of Variations, Vol.9, Iss.issue, 2010-03, pp. : 169-196

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Abstract

An existence and regularity theorem is proved for integral equationsof convolution type which contain hysteresis nonlinearities. Onthe basis of this result, frequency-domain stability criteria arederived for feedback systems with a linear infinite-dimensionalsystem in the forward path and a hysteresis nonlinearity in thefeedback path. These stability criteria are reminiscent of theclassical circle criterion which applies to static sector-boundednonlinearities. The class of hysteresis operators underconsideration contains many standard hysteresis nonlinearitieswhich are important in control engineering such as backlash (orplay), plastic-elastic (or stop) and Prandtl operators. Whilst themain results are developed in the context of integral equations ofconvolution type, applications to well-posed state space systemsare also considered.