# Control, Optimization and Modeling

In many areas of human endeavor, including medicine, biology and engineering, as well as finance and the social sciences, mathematical models help us understand what is happening, predict what will happen and determine how to improve the outcome. The need for higher-quality models integrating more diverse phenomena leads to the consideration of ever more complex and challenging mathematical systems.

Faculty at NC State are looking at models for a wide range of phenomena, including mechanical systems, aerospace systems, medical diagnosis and equipment design, groundwater flow, biological systems, and spread of disease. They are working on this modeling together with scientists, engineers and practitioners from the different areas.

Given a model, one can ask, how do we fit the model to data (parameter estimation)? What does the model predict (simulation)? Often, there is an ongoing process of comparing model performance to data and experiment.

Once one has a good model, optimization questions arise. What is the best way to measure data? What is the best performance that the system is capable of? What is the worst that can happen? Can the system be redesigned to improve performance? How can the effect of disturbances be reduced (to improve robustness)? To get the system to perform the way we want, often we have to control it (optimal control). For example, how do we control a vehicle to get where it needs to go efficiently? How do we administer a drug to reduce cost and improve health?

The mathematical models can be deterministic or stochastic; involve discrete or continuous time; use ordinary, partial, integral or delay differential equations; or contain a combination of several types of systems. There may be constraints.

Faculty who work in the Control, Optimization and Modeling group also frequently have expertise in numerical analysis, analysis, scientific computing, probability and various applications.

Lorena Bociu

Associate Professor

Partial Differential Equations: qualitative and quantitative properties of solutions to systems of PDEs, as well as associated stabilization and controllability issues. Applications of these problems range from nonlinear acoustics to fluid dynamics.

Patrick Combettes

Distinguished Professor

Convex analysis, convex optimization, monotone operators, computational data science, applied nonlinear analysis.

Mohammad Farazmand

Assistant Professor

Applied dynamical systems, control, optimization and modeling, data-driven methods

Mansoor Haider

Professor

Modeling biological soft tissues; Biomechanics, mechanobiology, biomedical imaging, wound healing and tissue engineering; Continuum mechanics, applied and numerical PDEs, unsupervised machine learning.

Kazufumi Ito

Professor

Optimal control and inverse problems for partial differential equations, control of Navier-Stokes equations, numerical partial differential equations, nonlinear semigroup theory, dynamical systems in Banach spaces, stochastic differential equations and applications, applied functional analysis.

Paata Ivanisvili

Assistant Professor

Areas of Expertise: Analysis, Probability, Convex geometry, Discrete approximation theory, Functional inequalities.

Tim Kelley

Drexel Professor

Nonlinear equations, multilevel methods, large-scale and multi-model optimization, flow in porous media, nano-scale electronics and sensing, radiative and neutron transfer, optimal control, integral equations, partial differential equations.

Sharon Lubkin

Professor

Modeling biological systems; continuum mechanics of tissues; morphogenesis; mixture models; transport.

Ryan Murray

Assistant Professor

Calculus of variations, partial differential equations, theory of statistics and machine learning, control theory, optimization, game theory, continuum mechanics, applied analysis.

Tien Khai Nguyen

Assistant Professor

Nonlinear PDEs, optimal control problems and differential Games, nonsmooth analysis and geometric measure theory.

Mette Olufsen

Professor

Mathematical biology, cardiovascular physiology, inverse problems, parameter estiamtion, differential equations, cardiovascular fluid mechanics.

Tao Pang

Professor, Director of Financial Mathematics Graduate Program

Stochastic control, probability, mathematical finance.

Andrew Papanicolaou

Assistant Professor

My research interests are computational finance and stochastic systems for control and optimization. Currently I am working on problems involving non-Markovian and high-dimensional optimizations. These problems were previously unsolvable due to the immensity of their computational demands. The applications of this work include financial data analysis and the challenges associate with these highly complex data sets. My background is in probability theory and nonlinear filtering. Among the newer problems that I am considering, are issues related to financial data and how machine learning methods can be applied.

Dávid Papp

Associate Professor

Optimization theory and algorithms. Applications in medicine, healthcare and engineering.