Electronic Journal of Qualitative Theory of Differential Equations 2018, No. 40, 1–16; https://doi.org/10.14232/ejqtde.2018.1.40 www.math.u-szeged.hu/ejqtde/ Global stability in a system using echo for position control Dedicated to Professor László Hatvani on the occasion of his 75th birthday Ferenc A. Bartha and Tibor Krisztin  Bolyai Institute, University of Szeged, Aradi vértanúk tere 1, Szeged, H–6720, Hungary Received 27 January 2018, appeared 26 June 2018 Communicated by Gergely Röst Abstract. We consider a system of equations describing automatic position control by echo. The system can be reduced to a single differential equation with state-dependent delay. The delayed terms come from the control mechanism and the reaction time. H.-O. Walther [Differ. Integral Equ. 15(2002), No. 8, 923–944] proved that stable periodic motion is possible for large enough reaction time. We show that, for sufficiently small reaction lag, the control is perfect, i.e., the preferred position of the system is globally asymptotically stable. Keywords: state-dependent delay, reaction lag, functional differential equation, asymp- totic stability. 2010 Mathematics Subject Classification: 34K20, 34K25, 34K35. 1 Introduction Figure 1.1: A control system H.-O. Walther [10] considered the following idealized model of a control system depicted on Figure 1.1. An object moves along a line and attempts to control its position relative to an obstacle by approximating its position through sending and receiving reflected signals. The obstacle is positioned at x = −w < 0, and the goal of the mechanism is to achieve (asymptotically as time goes to infinity) the ideal position x = 0 while avoiding collision with  Corresponding author. Email: krisztin@math.u-szeged.hu