{"ID":2921550,"CreatedAt":"2026-06-02T02:42:49.606572591Z","UpdatedAt":"2026-06-03T00:47:32.987482086Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2606.00969","arxiv_id":"2606.00969","title":"Nonsmooth High-Order Averaging Theory with Application to Extremum Seeking Optimization and Control","abstract":"In this paper, we introduce a higher-order averaging theory and method for a wide range of nonsmooth systems that are generally characterized by the classical averaging canonical form. Utilizing tools from generalized derivatives theory, we provide a nonsmooth near-identity transformation analogous to the one in smooth averaging theory. Additionally, we exploit sharp calculus rules from lexicographic differentiation theory to provide a closed formula for nonsmooth first-order averaging, and for the first time in the literature, nonsmooth second-order averaging. In fact, our approach recovers the smooth averaging results, without needing to check, if the system under consideration is smooth. Equipped with a nonsmooth second-order averaging theory, we generalize literature results and introduce a class of control-affine extremum seeking systems that tolerate nonsmoothness in the vector fields and/or the objective function by analyzing its stability based on a closed formula analogous to first-order Lie bracket approximations available in the smooth literature. We provide numerical simulation results involving complicated nonsmooth functions to demonstrate the effectiveness of our approach.","short_abstract":"In this paper, we introduce a higher-order averaging theory and method for a wide range of nonsmooth systems that are generally characterized by the classical averaging canonical form. Utilizing tools from generalized derivatives theory, we provide a nonsmooth near-identity transformation analogous to the one in smooth...","url_abs":"https://arxiv.org/abs/2606.00969","url_pdf":"https://arxiv.org/pdf/2606.00969v1","authors":"[\"Hesham Abdelfattah\",\"Sameh A. Eisa\",\"Peter Stechlinski\"]","published":"2026-05-31T03:00:45Z","proceeding":"math.OC","tasks":"[\"math.OC\",\"math.DG\",\"math.DS\"]","methods":"[]","has_code":false}