symmetry S S Article Sliding Mode Control and Geometrization Conjecture in Seismic Response Ligia Munteanu 1 , Dan Dumitriu 1 , Cornel Brisan 2 , Mircea Bara 2 , Veturia Chiroiu 1, *, Nicoleta Nedelcu 1 and Cristian Rugina 1   Citation: Munteanu, L.; Dumitriu, D.; Brisan, C.; Bara, M.; Chiroiu, V.; Nedelcu, N.; Rugina, C. Sliding Mode Control and Geometrization Conjecture in Seismic Response. Symmetry 2021, 13, 353. https://doi.org/10.3390/sym13020353 Academic Editor: Christos Volos Received: 31 January 2021 Accepted: 15 February 2021 Published: 22 February 2021 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). 1 Institute of Solid Mechanics, Romanian Academy, Bucures , ti 010141, Romania; ligia.munteanu@imsar.ro (L.M.); dan.dumitriu@imsar.ro (D.D.); nicoleta.nedelcu@imsar.ro (N.N.); rugina.cristian@imsar.ro (C.R.) 2 Department of Mechatronics and Machine Dynamics, Technical University from Cluj-Napoca, Cluj-Napoca 400114, Romania; cornel.brisan@mdm.utcluj.ro (C.B.); mircea.bara@mdm.utcluj.ro (M.B.) * Correspondence: veturia.chiroiu@imsar.ro or veturiachiroiu@yahoo.com; Tel.: +40-745989267 Abstract: The purpose of this paper is to study the sliding mode control as a Ricci flow process in the context of a three-story building structure subjected to seismic waves. The stability conditions result from two Lyapunov functions, the first associated with slipping in a finite period of time and the second with convergence of trajectories to the desired state. Simulation results show that the Ricci flow control leads to minimization of the displacements of the floors. Keywords: Ricci flow; Ricci solitons; geodesics; seismic waves 1. Introduction In recent years, the algorithms applied to control building systems subjected to seismic loads have been studied extensively [1,2]. The sliding mode control arises as a variable control, which constrains the structure to lie within a neighborhood of the switching function [3]. A series of phenomena in mechanics and other fields exhibit discontinuities, with respect to the current state in the differential equations that describe their behavior. An example is the Coulomb dry friction process, whose resistance force depends on the motion direction. Additionally, for the automatic control, minimizing the power consumed for the control purposes or restricting the range of variation of the control parameters leads to interruptions [4]. The aim of the control is to tailor the structure behavior with respect to a choice of the switching function, in terms of insensitivity or any uncertainties [5]. Optimization and control of the sliding modes are discussed in [6]. The sliding mode is associated with a discontinuity surface s whenever the distances to this surface and the velocity of its change are of opposite signs, i.e., when [6] lim s→−0 . s > 0 and lim s→+0 . s < 0. A discontinuous dynamic system may be described by the equation . x = f ( x, t), (1) where x ∈ R n is the state vector, t is time, and f ( x, t) has discontinuities in the form of points, lines, or surfaces within the (n + 1)-dimensional space ( x, t). For example, for the equation m .. x + P( . x)+ kx = 0, (2) where x is the displacement, m the mass, and k is the spring rigidity and P( . x)=  P 0 , for . x > 0 −P 0 , for . x < 0 , (3) Symmetry 2021, 13, 353. https://doi.org/10.3390/sym13020353 https://www.mdpi.com/journal/symmetry