Well-posedness and stability for a nonlinear Euler-Bernoulli beam equation

We study the well-posedness and stability for a nonlinear Euler-Bernoulli beam equation modeling railway track deflections in the framework of input-to-state stability (ISS) theory. More specifically, in the presence of both distributed in-domain and boundary disturbances, we prove first the existence and uniqueness of a classical solution by using the technique of lifting and the semigroup method, and then establish the Lʳ -integral input-to-state stability estimate for the solution whenever r ∈ [2,∞] by constructing a suitable Lyapunov functional with the aid of Sobolev-like inequalities, which are used to deal with the boundary terms. We provide an extensive extension of relevant work presented in the existing literature.

Uncontrolled Keywords

• Euler-Bernoulli beam equation
• input-to-state stability
• integral input-to-state stability
• boundary disturbance
• Lyapunov method