Finite-dimensional approximation of a class of constrained nonlinear optimal control problems. (English) Zbl 0849.49005

Gunzburger, Max D.; Hou, L. Steven

SIAM J. Control Optimization 34, No. 3, 1001-1043 (1996).

Summary: An abstract framework for the analysis and approximation of a class of nonlinear optimal control and optimization problems is constructed. Nonlinearities occur in both the objective functional and the constraints. The framework includes an abstract nonlinear optimization problem posed on infinite-dimensional spaces and an approximate problem posed on finite-dimensional spaces, together with a number of hypotheses concerning the two problems. The framework is used to show that optimal solutions exist, to show that Lagrange multipliers may be used to enforce the constraints, to derive an optimality system from which optimal states 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 applied to three concrete control or optimization problems and their approximation by finite-element methods. The first involves the von Kármán plate equations of nonlinear elasticity, the second the Ginzburg-Landau equations of superconductivity, and the third the Navier-Stokes equations for incompressible, viscous flows.

MSC:

49J20	Existence theories for optimal control problems involving partial differential equations
65J15	Numerical solutions to equations with nonlinear operators
65N30	Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J65	Nonlinear boundary value problems for linear elliptic equations
76D05	Navier-Stokes equations for incompressible viscous fluids
81Q05	Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
35Q30	Navier-Stokes equations
74B20	Nonlinear elasticity

Keywords:

nonlinear partial differential equations; finite-dimensional approximation; nonlinear optimal control; approximation; finite-element methods; von Kármán plate equations; Ginzburg-Landau equations; Navier-Stokes equations