A direct approach to the finite element solution of elliptic optimal control problems

Abstract

A general framework is developed for the finite element solution of optimal control problems governed by elliptic nonlinear partial differential equations. Typical applications are steady-state problems in nonlinear continuum mechanics, where a certain property of the solution (a function of displacements, temperatures, etc.) is to be minimized by applying control loads. In contrast to existing formulations, which are based on the ''adjoint state,'' the present formulation is a direct one, which does not use adjoint variables. The formulation is presented first in a general nonlinear setting, then specialized to a case leading to a sequence of quadratic programming problems, and then specialized further to the unconstrained case. Linear governing partial differential equations are also considered as a special case in each of these categories.

Solution of static optimal control problems in non‐linear elasticity via quadratic programming

“…In this paper the direct QP method for static (elliptic) optimal control problems, which has been developed in Reference [15] in a theoretical setting, was applied to two-dimensional problems in non-linear elasticity. It was demonstrated that the QP approach to optimal control is indeed e ective.…”

“…We denote the total number of displacement degrees of freedom by N u , and the total number of control degrees of freedom by N s . Using this discretization results in a system of non-linear algebraic equations, of the form (see Reference [15])…”

Optimal control of radiating panels via sequential quadratic programming

“…There are four main error sources involved in the FE-SQP algorithm described above: The first two errors are associated with the approximation of the continuous problem by a discrete one, whereas the last two errors are defined purely on the discrete (finite dimensional) level. Each of these errors is discussed separately Optimal control of radiating panels in Givoli (1999), which also provides basic convergence testing of the scheme for a one-dimensional model problem. A particularly interesting question is that of the interaction between the Newton iteration process and the QP minimization process; it is not obvious that this integration necessarily leads to convergence.…”

“…However, in Givoli and Patlashenko (2000) we show, albeit in a simplified situation, that the Newton scheme combined with the QP solver retains its quadratic convergence property. For more details, see Givoli (1999) and Givoli and Patlashenko (2000).…”

“…Now we consider the case where there are no constraints except those related to the required regularity of the control functions. (Regarding the latter requirement, see Givoli, 1999. ) In this case, the controls may be "constrained" through penalty terms in the objective functional.…”