Mathematical Control and Related Fields doi:10.3934/mcrf.2024011 EXPONENTIAL STABILIZATION OF A DYNAMIC FRICTIONAL CONTACT PROBLEM FOR VISCOELASTIC MATERIALS WITH NORMAL DAMPED RESPONSE AND INFINITE MEMORY Imane Ouakil 1 , Benyattou Benabderrahmane 2 , Yamna Boukhatem 2 and Baowei Feng *3 1 Laboratory of Pure and Applied Mathematics, Mohamed Boudiaf University 28000 M’sila, Algeria 2 National Higher School of Mathematics, Mahelma, 16093 Sidi Abdellah, Algiers, Algeria 3 Department of Mathematics, Southwestern University of Finance and Economics 611130 Chengdu, China (Communicated by Sorin Micu) Abstract. We consider a dynamic frictional contact for viscoelastic materi- als. The contact is modeled by normal damped response. A nonlinear consti- tutive law with infinite memory is given. The global existence of solution is proved based on nonlinear semigroup theory. Moreover, we apply the multiplier method to establish the exponential stability of the system. 1. Introduction. Frictional contact phenomena arise in industry and daily life. Consequently, an extensive literature has been devoted to investigate contact prob- lems involving friction. Their modeling, mathematical analysis and numerical anal- ysis have been studied by a large number of researchers. We refer the reader to e.g. [32, 40, 21, 43, 3]. In the modeling of contact problems, various contact conditions have been con- sidered for example the normal damped response condition which is the normal compliance condition but this last is expressed in the term of normal velocity. This contact condition describes the contact with a lubricated foundation. Moreover, it assigns a reactive normal pressure. The latter depends on normal velocity on the contact surface, see [13, 34]. In [20], Han et al. dealt with a frictionless viscoelastic unilateral contact problem with normal damped response condition. They proved the existence and uniqueness of a weak solution to the corresponding problem using surjectivity results for particular operators. In [11], a frictionless contact problem in thermoelasticity with a normal damped response is considered. The existence of a unique weak solution is proved using Faedo-Galerkin method. For the frictional case with a normal damped response, the literature is extensive. In [33], a viscoelas- tic contact problem with nonmonotone normal damped response is considered. the unique weak solvability result for the corresponding model is established by using 2020 Mathematics Subject Classification. Primary: 93D20, 93D15; Secondary: 35A01, 35B40, 74M15. Key words and phrases. Contact problem, exponential stability, frictional contact, nonlinear semigroup theory, existence. * Corresponding author: Baowei Feng. 1