Controllability of a string submitted to unilateral constraint

Abstract This article studies the controllability property of a homogeneous linear string of length one, submitted to a time dependent obstacle (described by the function { ψ ( t ) } 0 ⩽ t ⩽ T ) located below the extremity x = 1 . The Dirichlet control acts on the other extremity x = 0 . The string is modelled by the wave equation y ″ − y x x = 0 in ( t , x ) ∈ ( 0 , T ) × ( 0 , 1 ) , while the obstacle is represented by the Signorini's conditions y ( t , 1 ) ⩾ ψ ( t ) , y x ( t , 1 ) ⩾ 0 , y x ( t , 1 ) ( y ( t , 1 ) − ψ ( t ) ) = 0 in ( 0 , T ) . The characteristic method and a fixed point argument allow to reduce the problem to the analysis of the solutions at x = 1 . We prove that, for any T > 2 and initial data ( y 0 , y 1 ) ∈ H 1 ( 0 , 1 ) × L 2 ( 0 , 1 ) with ψ ( 0 ) ⩽ y 0 ( 1 ) , the system is null controllable with controls in H 1 ( 0 , T ) . Two distinct approaches are used. We first introduce a penalized system in y ϵ , transforming the Signorini's condition into the simpler one y ϵ , x ( t , 1 ) = ϵ − 1 [ y ϵ ( t , 1 ) − ψ ( t ) ] − , ϵ being a small positive parameter. We construct explicitly a family of controls of the penalized problem, uniformly bounded with respect to ϵ in H 1 ( 0 , T ) . This enables us to pass to the limit and to obtain a control for the initial equation. A more direct approach, based on differential inequalities theory, leads to a similar positive conclusion. Numerical experiments complete the study.