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)), 0 < x < 1, {w(t, 0) = w(t, 1) = wxx(t, 0) = wxx(t, 1) = 0, t e R, where t ≥ 0, d > 0, c > 0, k > 0, the distributed control u e L2(0, t1; L2(0, 1)), p:R × [0, 1] → R is continuous and bounded, and the non-linear term f: [0, t1] x R x 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( ∥t{∥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)).