Finite element approximation of the nonlinear Kirchhoff string with boundary control

This paper addresses the finite element approximation problem of a nonlinear Kirchhoff string with the nonlinear boundary control. The nonlinear boundary control is the negative feedback of the transversal velocity of the string at one end, which satisfies a sector constrain. Based on the equivalent variational formulation of the nonlinear string equations, the finite element approximation of the Kirchhoff string with boundary control input is derived in the Lagrange polynomial space of degree 1. Simulation examples are presented to show the effectiveness of our main result.