Many phenomena arising in engineering, life sciences, and other disciplines of science are usually formulated mathematically by differential equations, in particular partial differential equations (PDEs). Often one wants to achieve a desired behavior of the solutions of these PDEs which can be achieved by designing appropriate control functions in order to manipulate the solutions. This leads to optimization problems with PDE constraints.

This course is designed for graduate students majoring in mathematics as well as mathematically inclined graduate students. At the end of this course, the student will be able to

• recognize and formulate problems involving optimization with PDE constraint
• understand and apply theory for optimization problems in Banach spaces
• explain the basic properties of the relevant functional spaces, in particular Sobolev spaces
• derive necessary and sufficient optimality conditions for PDE constrained problems
• derive efficient discretization techniques as well as the numerical solution of the arising large scale linear systems of equations
• understand the optimization methods for the numerical solution of PDE constrained problems
• use the computational tools available to solve optimization problems with PDE constraints on computers.