Eduardo Sontag, elected to the National Academy of Sciences, applies math to real-world problems, from engineering to immunology and biology.
Eduardo Sontag has earned a place among the world’s top scientists by solving applied mathematical problems in a pure theorist’s manner.
A Northeastern University distinguished professor of electrical and computer engineering and bioengineering, and affiliate professor of mathematics and chemical engineering, Sontag was recently elected to the National Academy of Sciences.
“I like to think of big ideas in basic concepts,” says Sontag, who was recognized for his distinguished and continuing contributions to original research in applied mathematics. “I like to think very rigorously about things in an abstract way, but very carefully, logically.”
But he also has a profound curiosity about many different fields, especially biology and neuroscience.
“You can transfer ideas from one field to another,” he says. “I attend lectures in all kinds of fields, and I often see analogies between engineering and biology.”
Sontag has written more than 500 research papers, book chapters and monographs. His concept of input-to-state stability for nonlinear systems — those where outputs aren’t directly proportional to inputs — has become foundational in control theory. His work has been cited nearly 65,000 times and referenced in hundreds of patents.
“Eduardo Sontag’s work has profoundly shaped modern control theory, bridging rigorous mathematics with systems biology, neural networks and nonlinear dynamics — illuminating paths where engineering meets the complexity of life,” says Mario Sznaier, professor of electrical and computer engineering at Northeastern.
Sontag grew up in Buenos Aires, Argentina. He was drawn to science and mathematics early on, he says, both for the intellectual pursuit and their power to understand and solve problems in health, engineering and society.
“Mathematics provides a common language to talk about many things, even in very different fields,” he says.
He studied at the University of Buenos Aires, where mathematical training was based on the strict and pure French school of mathematics.
“That made me think in a certain way that is different from the way that a lot of other people think,” Sontag says.
He believes breaking problems down to their simplest form is key to understanding fundamental ideas and underlying principles.
“Take away everything that is very special about the problem,” Sontag says. “Think of it very abstractly, very generally, and that often allows you to understand things better. You don’t get distracted by the details.”
Still, once the theoretical work is done, Sontag says, it is important to collaborate with practitioners on its application, to get feedback and refine it further.
Sontag’s interests have always spanned disciplines. As an undergraduate in 1972, he published a book in Spanish on artificial intelligence, exploring neural networks and whether machines could surpass human intelligence.
That same year, he was invited to the University of Florida by Rudolf Kalman — pioneer of the Kalman filter, a key algorithm in aerospace, robotics and defense. Kalman inspired Sontag’s shift toward practical applications of elegant mathematics.
After earning his Ph.D., Sontag joined Rutgers University as an assistant professor of mathematics. His research would go on to span control theory, theoretical computer science, machine learning, cancer and immunology, and molecular, synthetic and computational biology. Nonlinear systems have been the unifying thread across all his work.
Control theory is about keeping systems on track, Sontag says, despite unexpected changes. He calls it the science of feedback — monitoring outputs, comparing them to goals and adjusting behavior to stay on course.
Linear systems behave predictably: inputs yield proportional outputs. Nonlinear systems, such as a wildfire in a forest or a crowd evacuating a stadium, are more unpredictable. These systems can have multiple outcomes, Sontag says, sudden shifts or endlessly looping behaviors that are never quite the same.
In 1892, Russian mathematician Aleksandr Lyapunov developed the mathematical theory of stability of motion. It helps understand whether an isolated nonlinear dynamical system, once slightly disturbed, will return to its original state.
Building on this, in 1979, Sontag introduced input-to-state stability, a concept that incorporates the influence of external inputs over time. These inputs could be, for example, traffic approaching a self-driving car or wind gusts impacting an airplane under autopilot.
Sontag’s concept helps bridge the gap between theoretical models and messy real-world systems. It essentially states that if inputs remain small, the system’s behavior won’t spiral out of control. It formalizes robustness: systems may be nudged, but they recover, making them reliable even under imperfect conditions.
In the 1980s and early 1990s, Sontag simultaneously worked on theoretical computer science, machine learning and neuroscience. But a lecture in the late 1990s on gene expression and molecular biology shifted his focus toward understanding disease through the lens of mathematics and systems engineering.
Around the same time he began exploring immunology: how the immune system reacts to different pathogens or tumor cells and how its behavior could be modeled mathematically.
“That got me started talking to more biologists,” Sontag says.
Today, a major focus of his work is synthetic biology, specifically, designing tiny biological systems inside cells that can adjust to gene expression changes, block out unwanted interference and carry out simple logical decisions.
In another collaboration with Northeastern colleagues funded by the Office of Naval Research, Sontag is working on mathematical modeling to ensure safety in data-driven machine-learning systems like self-driving cars.
“You design things based on data and machine learning, but you have no guarantees that they’re going to be safe,” he explains. “So how do you analyze those for safety?”
He also continues to explore more pure mathematical control theory questions.
“As our technologies and scientific understanding become more complex and more intertwined with the real world, the importance of these ideas only grows,” Sontag says. “Whether we’re trying to cure diseases, build intelligent machines or understand climate dynamics, feedback and stability are at the core of making systems reliable, adaptable and robust.”