4. A Least-Squares Finite Element Method for Optimization and Control Problems

Abstract

4.1 Introduction Optimization and control problems for systems governed by PDEs arise in many applications. Experimental studies of such problems go back 100 years [20]. Computational approaches have been applied since the advent of the computer age. Most of the efforts in the latter direction have employed elementary optimization strategies, but, more recently, there has been considerable practical and theoretical interest in the application of sophisticated local and global optimization strategies, e.g., Lagrange multiplier methods, sensitivity or adjoint-based gradient methods, quasi-Newton methods, evolutionary algorithms, etc. The optimal control or optimization problems we consider consist of • state variables, i.e., variables that describe the system being modeled; • control variables or design parameters, i.e., variables at our disposal that can be used to affect the state variables; • a state system, i.e., PDEs relating the state and control variables; and • a functional of the state and control variables whose minimization is the goal.