Determination of an unknown shear force in cantilever Kirchhoff–Love plate from measured final data with application to atomic force microscope

Abstract

Abstract We present a new and feasible formulation for the inverse problem of identifying the unknown shear force acting on the inaccessible tip of the nonhomogeneous micro-cantilever plate which is a key component in most atomic force microscopes. The mathematical modelling of this phenomena leads to the inverse problem of determining the shear force g(x 2, t) acting on the inaccessible boundary x 1 = ℓ 1 in a system governed by the variable coefficient damped Kirchhoff–Love plate equation ρ ( x ) h ( x ) u t t + μ ( x ) u t + D ( x ) ( u x 1 x 1 + ν u x 2 x 2 ) x 1 x 1 + D ( x ) ( ν u x 1 x 1 + u x 2 x 2 ) x 2 x 2 + 2 ( 1 − ν ) D ( x ) u x 1 x 2 x 1 x 2 = F ( x , t ) , where (x, t) ∈ Ω T ≔ Ω × (0, T), x ≔ (x 1, x 2) and Ω ≔ ( 0 , ℓ 1 ) × ( 0 , ℓ 2 ) ⊂ R 2 , subject to clamped-free boundary conditions, from final time measured output (displacement) u T (x) ≔ u(x, T). We prove that under appropriate conditions, this inverse problem has a unique solution. We introduce the input–output operator and prove that it is a linear compact and Lipschitz continuous operator. Based on these results, the existence of a quasi-solution of the inverse problem, as a solution of the minimization problem for the Tikhonov functional, is proved. Furthermore, an explicit gradient formula for the Fréchet derivative of the Tikhonov functional is derived. Moreover, it is proved that the Fréchet derivative is Lipschitz continuous. These results provide a mathematical basis for gradient based computational algorithm.