CT23 Panel Explores Future Directions of Applied Mathematics and Control Theory

By Lina Sorg

Though applications of control theory were initially limited to process regulation for engineering and the physical sciences, concepts from this field now contribute directly to computer science, biology, and the social sciences. In recent years, the utilization of control theory has continued to expand to automotive systems, aerospace engineering, energy systems, nanotechnology, cellular regulation, and even the smart grid. During an interactive discussion at the 2023 SIAM Conference on Control and Its Applications, which took place in Philadelphia, Pa., this July, six panelists shared their thoughts on the future of applied mathematics and control. Eduardo Casas (Universidad de Cantabria), Eduardo Cerpa (Pontificia Universidad Católica de Chile), Hélène Frankowska (Sorbonne Université and CNRS), Sonia Martinez (University of California, San Diego), Robert McCann (University of Toronto), and Benedetto Piccoli (Rutgers University) all contributed to an insightful dialogue about the potential directions of these disciplines.

Casas opened the conversation by affirming that control theory still boasts plenty of interesting open problems. He noted that researchers in the 1960s originally focused on control problems with linear partial differential equations (PDEs), whereas present-day practitioners primarily investigate the control of nonlinear PDE systems — which are more complicated. “As science evolves, the models become more complex,” Casas said. “They are going to introduce many new open questions, so we need to develop methods and ideas in both control and applied mathematics.”

Frankowska echoed Casas’ comments, adding that numerous questions still remain from the 1960s. In light of the intricacy of current problems, she spoke about the value of straightforward models that allow researchers to quickly propose efficient solutions. Frankowska also touched on the importance of teamwork, since many control problems require input from engineers, doctors, and experts in other domains.

Piccoli agreed with Frankowska’s remarks on the necessity of interdisciplinary partnerships. As scientists explore novel ideas and tackle issues with higher levels of complexity and dimensionality, the nature of research and collaboration must change accordingly. In general, groups that embody a wide variety of skillsets can more successfully address problems at larger scales. “What you see happening is that you’re moving from the single person to team science in many different terms,” Piccoli said. “We are a group of people with different expertise, and we are exchanging ideas to progress our own opinions.”

Martinez concurred with the need for multidisciplinary approaches that yield novel techniques for complex systems, and specifically identified data science as an influential field with potential opportunities for control and other applied math subsets. “We now live in a world of data science, and we can contribute a lot to this literature as applied mathematicians,” she said. “The tools that we work with—such as stochastic systems and differential equations—are going to lead to algorithms that are tractable and can learn and provide solutions to these complex systems, which will be very useful.”

McCann, who considers himself to be both a pure and applied mathematician, brought an alternative perspective to the discussion. “It’s exciting to see the progress that technology brings, but also the basic insights that pure math was able to offer before the technology was accessible,” he said. He touted the existing space for creative insight in the control domain but expressed some apprehension about the influence of artificial intelligence (AI). “Human understanding is really important,” McCann continued. “Humans are very interested in the question of ‘why.’ I don’t care if the machine understands, I’m not satisfied until humans understand too. I love using mathematics to tell stories and I think that there is a boundless future as long as we don’t allow machines to take over from us.”

Cerpa emphasized education as a necessary pathway to preserve existing knowledge and advance the future of control. He encouraged teachers to challenge their students with forward-thinking coursework that will prepare them for demanding, real-world scenarios. He also reiterated McCann’s thoughts on AI. “We need to transmit the message that AI isn’t everything,” Cerpa said. “We can do other things that are very valuable — AI isn’t taking over everything that we do.”

Next, a panel attendee inquired about the future of theoretical versus applied aspects of control theory, which complement each other but require different kinds of expertise. Frankowska stated that researchers should ideally be familiar with both types in order to effectively move forward. “That’s why you need to work with a group where some people are more applied and some are more theoretical,” she said, but admitted that it can be hard to coalesce the personal approaches of different researchers. “It’s probably the most difficult thing in the scientific lab — to be open to everything. Most of us can’t always do it because mathematics is so time consuming.”

Frankowska proceeded to comment on the possible challenges that are associated with interdisciplinary partnerships at universities, where faculty promotions are predominantly based on published papers rather than long-term interdisciplinary projects. “Teams are difficult to have when you’re in a university because people are working in different directions,” she said, adding that she would like to see a favorable attitude shift towards leadership and teamwork in science and mathematics departments. 

McCann mentioned that pure and applied mathematics are often housed in different academic departments, which is not necessarily a structural advantage. “Everyone at the university is enthusiastic about interdisciplinary research until you try to publish it,” he said. “When you talk to physicists and economists, they’re not always particularly open to input from mathematicians. And I don’t know what we can do as a discipline to try to change that.”

Nevertheless, the panel concluded positively with a brief overview of the many current and future opportunities for modeling in control — including potential contributions to climate change studies. “The future is related to the evolution of science and the new models that are going to appear,” Casas said. “There’s work here for everybody.”

Lina Sorg is the managing editor of SIAM News.