PDE-constrained optimization refers to the optimization of systems governed by PDEs. The simulation problem is to solve the PDEs for the state variables (e.g., displacement, velocity, temperature, electric field, magnetic field, species concentration), given appropriate data (e.g., geometry, coefficients, boundary conditions, initial conditions, source functions). The optimization problem seeks to determine some of these data—the decision variables—given performance goals in the form of an objective function and possibly inequality or equality constraints on the behavior of the system. Since the behavior of the system is modeled by the PDEs, they appear as (usually equality) constraints in the optimization problem. We will refer to these PDE constraints as the state equations.

Let u represent the state variables, d the decision variables, the objective function, c the residual of the state equations, and h the residual of the inequality constraints. We can then state the general form of a PDE-constrained optimization problem as min u,d J (u,d) subject to c (u,d)=0, h (u,d)≥0 . The PDE-constrained optimization problem (16.1) can represent an optimal design, optimal control, or inverse problem, depending on the nature of the objective function and decision variables. The decision variables correspondingly represent design, control, or inversion variables.