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This paper presents a time-invariant extremum seeking controller (ESC) for nonlinear autonomous systems with limit cycles. For this time-invariant ESC, we propose a method to prove the closed loop system has an asymptotically stable limit cycle. The method is based on a perturbation theorem for maps, and, unlike existing techniques that use averaging and singular perturbation tools, it is not limited to weakly nonlinear systems. We use a typical example system to show that our method does indeed establish asymptotic stability of the limit cycle with minimal amplitude. Utilizing the example, we provide a general guide for analytic computations that are required to apply our method. The corresponding Mathematica code is available as supplementary material. 

This paper develops an analytic framework to design both stress-controlled and displacement-controlled T -periodic loadings which make the quasistatic evolution of a one-dimensional network of elastoplastic springs converging to a unique periodic regime. The solution of such an evolution problem is a function t ↦( e ( t ), p ( t )), where e i ( t ) is the elastic elongation and p i ( t ) is the relaxed length of spring i , defined on [ t 0 , ∞ ) by the initial condition ( e ( t 0 ), p ( t 0 )). After we rigorously convert the problem into a Moreau sweeping process with a moving polyhedron C ( t ) in a vector space E of dimension d , it becomes natural to expect (based on a result by Krejci) that the elastic component t ↦ e ( t ) always converges to a T -periodic function as t → ∞ . The achievement of this paper is in spotting a class of loadings where the Krejci’s limit doesn’t depend on the initial condition ( e ( t 0 ), p ( t 0 )) and so all the trajectories approach the same T -periodic regime. The proposed class of sweeping processes is the one for which the normals of any d different facets of the moving polyhedron C ( t ) are linearly independent. We further link this geometric condition to mechanical properties of the given network of springs. We discover that the normals of any d different facets of the moving polyhedron C ( t ) are linearly independent, if the number of displacement-controlled loadings is two less the number of nodes of the given network of springs and when the magnitude of the stress-controlled loading is sufficiently large (but admissible). The result can be viewed as an analogue of the high-gain control method for elastoplastic systems. In continuum theory of plasticity, the respective result is known as Frederick-Armstrong theorem. 

We offer a finite-time stability result for Moreau sweeping processes on the plane with periodically moving polyhedron. The result is used to establish the convergence of stress evolution of a simple network of elastoplastic springs to a unique cyclic response in just one cycle of the external displacement-controlled cyclic loading. The paper concludes with an example showing that smoothing the vertices of the polyhedron makes finite-time stability impossible.