This course explores the theory and methods specific to optimization problems constrained by partial differential equations (PDEs). We address how to formulate and solve optimization problems where the state variables are governed by PDEs. The course covers theoretical aspects, numerical techniques, and applications in various fields such as fluid dynamics, structural optimization, and control theory.

Prerequisites: Advanced Calculus, Hilbert spaces, Linear Algebra, Basic knowledge in PDEs and optimization.

1. Introduction to PDE-Constrained Optimization
2. Mathematical Preliminaries
   1. Differentiability in Banach and Hilbert spaces
   2. State differentiability for elliptic equations
3. Optimality Conditions
4. Numerical Methods for PDEs (overview)
5. Numerical Optimization Techniques (overview)
6. Solving PDE-Constrained Optimization Problems
   1. Discretize-then-optimize vs. optimize-then-discretize approaches
   2. Implementation of adjoint methods
7. Applications and Case Studies
8. Advanced Topics and Research Frontiers

Course slides and reading material will be available from the instructor.

Reference Books:

• M. Hinze, R. Pinnau, M. Ulbrich, and S. Ulbrich, "Optimization with PDE Constraints," Springer.
• A. B. Lawson, "Solving PDE-Constrained Optimization Problems," SIAM.
• A. Borzì and V. Schulz, "Computational Optimization of Systems Governed by Partial Differential Equations," SIAM.
• J. Nocedal and S. Wright, "Numerical Optimization," Springer.

1. Midterm Exam: 25%
2. Project: 25%
3. Final Exam: 50%