MATHEMATICAL CONTROL AND doi:10.3934/mcrf.2020015 RELATED FIELDS UNIFORM INDIRECT BOUNDARY CONTROLLABILITY OF SEMI-DISCRETE 1-d COUPLED WAVE EQUATIONS Abdeladim El Akri * and Lahcen Maniar Universit´ e Cadi Ayyad, Facult´ e des Sciences Semlalia, LMDP, UMMISCO (IRD- UPMC) Marrakech 40000, B.P. 2390, Maroc (Communicated by Sylvain Ervedoza) Abstract. In this paper, we treat the problem of uniform exact boundary controllability for the finite-difference space semi-discretization of the 1-d cou- pled wave equations with a control acting only in one equation. First, we show how, after filtering the high frequencies of the discrete initial data in an ap- propriate way, we can construct a sequence of uniformly (with respect to the mesh size) bounded controls. Thus, we prove that the weak limit of the afore- mentioned sequence is a control for the continuous system. The proof of our results is based on the moment method and on the construction of an explicit biorthogonal sequence. 1. Introduction. This work concerns the indirect boundary controllability prop- erties for the finite difference approximation of the 1-d coupled wave equations. To clarify our aim we start by introducing the problem of the indirect controllability in the continuous setting. Let T> 0 and consider the 1-d linear coupled wave equations                    y tt - y xx + αq =0 for (x, t) ∈ (0, 1) × (0,T ) q tt - q xx + αy =0 for (x, t) ∈ (0, 1) × (0,T ) y(0,t)=0,y(1,t)= v(t) for t ∈ (0,T ) q(0,t)=0,q(1,t)=0 for t ∈ (0,T ) y(0) = y 0 ,y 0 (0) = y 1 for x ∈ (0, 1) q(0) = q 0 ,q 0 (0) = q 1 for x ∈ (0, 1), (1) where α ∈ R is the coupling constant and (y 0 ,y 1 ,q 0 ,q 1 ) ∈ L 2 (0, 1) × H -1 (0, 1) × H 1 0 (0, 1) × L 2 (0, 1) are the initial conditions and v ∈ L 2 (0,T ) is the control function. Here the subscript t stands for the partial derivative with respect to time variable while subscript x stands for the space variable. The indirect null boundary controllability problem (which is equivalent to the indirect exact boundary controllability) for system (1) can be formulated as follows: 2010 Mathematics Subject Classification. Primary: 93B05, 35L05, 30E05; Secondary: 65M06. Key words and phrases. Coupled wave equations, uniform indirect exact boundary controllabil- ity, space semi-discretization, finite differences, moment problem, biorthogonal sequence, filtered spaces. The first author would like to thank S. Micu for fruitful discussions on several parts of this paper during his visit to Craiova University. * Corresponding author: Abdeladim El Akri. 1