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

Abstract:

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 ∈ ℝ, where t ≥ 0, d > 0, 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)). © 2005 Published by Elsevier Inc.

Año de publicación:

2005

Keywords:

• Suspension bridge equation
• Strongly continuous groups
• exact controllability

Fuente:

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