Inverse Problems and Optimal Control in Non Linear Mechanics

The purpose of this article is to present applications of optimal control theory in order to solve inverse problems in non linear mechanics. The constitutive equations of the material is known. In the first class of inverse problem, knowing the initial state and the final state of the body, we propose a method to estimate the history of loading and the internal state such as plastic strains knowing the mechanical quantities on the boundary at the final state. Finally using of results on cyclic respons of an anelastic system to cyclic loading, an optimal control approach is proposed to determine the asymptotic behaviour. The classical theorems on elastic or plastic shakedown give conditions for the existence of a periodic answer but are not able to determine explicitly the cyclic answer. This new approach gives the limit cyclic behaviour dependeing on the initial conditions. 1. ON HISTORY IN VISCOPLASTICITY. In this case, the displacement u,the strain ε, the stress σ are functions of position and time in a domain Ω. Let us denoted tf the final state. Assuming we know both the initial state and the displacement um at time tf over the part ΓT of the boundary ∂Ω, and u = 0 over Γu = ∂Ω/ΓT we search an history of loading applied on the boundary ΓT and an internal state α such that the external displacement at time tf is closed to um. The local behaviour is defined by a free energy w(ε, α) and we assume that the internal state evolution is determined by a normality rule α̇ = ∂Φ ∂A associated with the potential convex Φ(A), with A = − ∂w ∂α . The optimal loading history is defined as the minimum of the cost function: