Exact controllability of the suspension bridge model proposed by Lazer and McKenna

In this paper we give a sufficient condition for the exact controllability of the following model of the suspension bridge equation proposed by Lazer and McKenna in [A.C. Lazer, P.J. McKenna, Large-amplitude periodic oscillations in suspension bridges: Some new connections with nonlinear analysis, SIAM Rev. 32 (1990) 537-578]: {wtt+cwt+dwxxxx+kw+=p(t,x)+u(t,x)+f(t,w,u(t,x)),00, c>0, k>0, the distributed control u∈L2(0,t1;L2(0,1)), p:R×[0,1]→R is continuous and bounded, and the non-linear term f:[0,t1]×R×R→R is a continuous function on t and globally Lipschitz in the other variables, i.e., there exists a constant l>0 such that for all x1,x2,u1,u2∈R we have ‖f(t,x2,u2)−f(t,x1,u1)‖⩽l{‖x2−x1‖+‖u2−u1‖},t∈[0,t1]. To this end, we prove that the linear part of the system is exactly controllable on [0,t1]. Then, we prove that the non-linear system is exactly controllable on [0,t1] for t1 small enough. That is to say, the controllability of the linear system is preserved under the non-linear perturbation −kw++p(t,x)+f(t,w,u(t,x)).