On the optimal control of some nonsmooth distributed parameter systems arising in mechanics

Abstract

Variational inequalities are an important mathematical tool for modelling free boundary problems that arise in different application areas. Due to the intricate nonsmooth structure of the resulting models, their analysis and optimization is a difficult task that has drawn the attention of researchers for several decades. In this paper we focus on a class of variational inequalities, called of the second kind, with a twofold purpose. First, we aim at giving a glance at some of the most prominent applications of these types of variational inequalities in mechanics, and the related analytical and numerical difficulties. Second, we consider optimal control problems constrained by these variational inequalities and provide a thorough discussion on the existence of Lagrange multipliers and the different types of optimality systems that can be derived for the characterization of local minima. The article ends with a discussion of the main challenges and future perspectives of this important problem class.

Topology Optimization of Elasto-Plastic Structures in Contact

“…The generalized plastic stress may take values only in a closed convex set K of admissible generalized stresses. For a given yield function ϕ this set is defined as [5,7]…”

“…For the sake of sensitivity analysis let us introduce the variational formulation of this contact problem. We shall use the dual rather than primal variational formulation of the contact problem (2)- (12) with von Mises yield function ϕ in (5). This formulation is based on the generalized stress tensor Σ = (σ, χ) rather than on the generalized strain tensor (ε p , ξ).…”

“…Therefore the original structural optimization problem has to be regularized either using the filter technique or different penalization techniques. Optimal control problems for elasto-plastic structures have been studied in a series of papers [4,5] where primal and dual formulations as well as first order necessary optimality conditions are provided. Optimal control or structural optimization of contact problems for elasto-plastic materials have been considered in [2,9,20].…”

On the Shape Differentiability of Objectives: A Lagrangian Approach and the Brinkman Problem

“…Constraints arise in a variety of applications. The constraint operator (see (3)) may become: the trace operator under contact conditions [1][2][3], the jump operator for cracks and anticracks [4][5][6], the gradient operator in plasticity [7], the divergence operator under incompressibility conditions [8][9][10], a state-constraint in mathematical programs with equilibrium constraints [11,12], and the like. The constraint problems are related to parameter identification problems (see the theory in References [13][14][15] and application to biological systems in Reference [16]), to inverse problems by the mean of observation data used in mathematical physics [17,18] and in acoustics [19][20][21], to overdetermined and free-boundary problems [22,23].…”

Numerical solution of bilateral obstacle optimal control problem, where the controls and the obstacles coincide

“…Introduction. Variational inequalities and related optimal control problems have been recognized as suitable mathematical models for dealing with many problems arising in different fields, such as shape optimization theory, image processing and mechanics, (see for example [7], [6], [10], [13], [15], [16], [25], [31] and [32]).…”