FINITE-DIMENSIONAL APPROXIMATION OF A CLASS OF CONSTRAINED NONLINEAR OPTIMAL-CONTROL PROBLEMS

An abstract framework for the analysis and approximation of a class of nonlinear optimal control and optimization problems is constructed. N onlinearities occur in both the objective functional and the constrain ts. The framework includes an abstract nonlinear optimization problem posed on infinite-dimensional spaces, an approximate problem posed on finite-dimensional spaces, together with a number of hypotheses concer ning the two problems. The framework is used to show that optimal solu tions exist, to show that La,orange multipliers may be used to enforce the constraints, to derive an optimality system from which optimal st ates and controls may be deduced, and to derive existence results and error estimates for solutions of the approximate problem. The abstract framework and the results derived from that framework are then applie d to three concrete control or optimization problems and their approxi mation by finite-element methods. The first involves the von Karman pl ate equations of nonlinear elasticity, the second the Ginzburg-Landau equations of superconductivity, and the third the Navier-Stokes equati ons for incompressible, viscous flows.