Course Objectives¶
The course introduces the mathematical theory and numerical methods for PDE-constrained optimal control problems.
The focus is on:
rigorous formulation of optimal control problems governed by PDEs;
derivation of first-order (and selected second-order) optimality conditions;
finite element discretization of optimal control problems;
implementation of numerical algorithms using deal.II.
At the end of the course, students will be able to develop and analyze research-level prototype codes for PDE-constrained optimization.
Prerequisites¶
Advanced calculus
Partial differential equations (elliptic and parabolic)
Numerical methods for PDEs (finite element methods)
Programming in C++ and Python is helpful for the laboratories, but not required for the lectures
Basic familiarity with Linux and scientific software tools.
Previous experience with deal.II is helpful but not mandatory.
Numerical Laboratories (deal.II, Python)¶
The laboratories are an integral part of the course and include:
distributed control of the Poisson equation;
state–adjoint formulation and reduced gradient methods;
box constraints on the control (projection methods, PDAS);
time-dependent optimal control problems;
analysis of computational performance and scalability.
The codes:
are written in C++ and Python;
are based on deal.II and lightweight Python prototypes;
are designed as starting points for extensions and final projects;
will be provided with detailed documentation and comments for you to understand and modify.
The repository contains two complementary laboratory tracks:
Python notebooks in
jupyterbook/codes, used for rapid prototyping, visualization, and reduced models;a C++/deal.II testbench in
codes/dealii, used for finite element implementations of PDE-constrained optimal control problems.
The deal.II testbench is organized as a teaching-oriented environment for laboratories on:
state equations, adjoints, and optimality systems;
distributed and boundary control for elliptic problems;
reduced formulations and all-at-once formulations;
constrained optimization algorithms and solver experiments;
verification, testing, and reproducible computational experiments.
The corresponding documentation is collected in the Jupyter Book under the section deal.II laboratories.
References¶
F. Tröltzsch, Optimal Control of Partial Differential Equations, AMS, 2010
A. Manzoni, A. Quarteroni, S. Salsa, Optimal Control of Partial Differential Equations, Springer, 2021
J. C. De los Reyes, Numerical PDE-Constrained Optimization, Springer, 2015
Additional references are provided within the individual chapters.
License¶
The content of this repository is licensed under the MIT License. See the LICENSE file for details.
Use the table of contents to navigate the 20 sessions (each ~2h).
Course Topic Summary¶
The course is organized into three main blocks, progressing from foundational concepts to advanced numerical methods and applications. The emphasis is on PDE-constrained optimal control, with systematic integration of theory and numerical implementation.
Block I: Foundations and Linear Elliptic Optimal Control (Lectures 1-6)¶
This block introduces the mathematical framework of optimal control problems governed by PDEs, with a focus on linear-quadratic elliptic problems.
Topics¶
Introduction to optimal control problems: motivation, examples, and applications.
Abstract formulation of PDE-constrained optimization problems.
Control-to-state mapping and reduced cost functional.
Adjoint equations and reduced gradients.
Linear-quadratic optimal control of elliptic PDEs.
Distributed and boundary controls.
Existence and uniqueness of optimal controls.
Control constraints and Karush-Kuhn-Tucker (KKT) conditions.
Conceptual introduction to numerical discretization strategies: optimize-then-discretize vs discretize-then-optimize.
By the end of this block, students understand the continuous optimality system and its analytical properties.
Block II: Numerical Approximation and Parabolic Optimal Control (Lectures 7-12)¶
This block focuses on the numerical solution of optimal control problems and extends the theory to time-dependent (parabolic) PDEs.
Topics¶
Finite element discretization of elliptic optimal control problems.
Reduced vs all-at-once formulations.
Saddle-point systems and block structure.
Iterative solvers and preconditioning strategies.
Numerical treatment of control constraints.
Linear-quadratic optimal control of parabolic PDEs.
Adjoint equations backward in time.
Space-time discretization strategies.
Numerical algorithms for time-dependent optimal control problems.
Computational aspects and scalability considerations.
Laboratories in this block introduce deal.II-based implementations for elliptic and parabolic problems.
Block III: Nonlinear Problems and Advanced Topics (Lectures 13-20)¶
The final block addresses nonlinear and nonsmooth optimal control problems, together with advanced numerical algorithms and applications.
Topics¶
Optimal control problems governed by semilinear PDEs.
Optimization in Banach spaces.
First-order optimality conditions for nonlinear problems.
Second-order optimality conditions and stability.
Newton and Sequential Quadratic Programming (SQP) methods.
Primal-Dual Active Set (PDAS) methods and semismooth Newton methods.
Nonsmooth problems: box constraints, sparsity-promoting costs.
Selected advanced applications (e.g., flow control, parameter identification).
Discussion of current research directions in PDE-constrained optimization.
This block prepares students to read current research literature and develop research-level numerical codes.
Overall Learning Trajectory¶
The course progresses along the following axis:
continuous theory -> discrete formulations -> algorithms -> implementation -> applications
By the end of the course, students have:
A solid theoretical understanding of PDE-constrained optimal control.
Practical experience with finite element implementations in deal.II.
The ability to critically assess numerical methods and modeling choices.