Research

In more detail

• Realizations & observability

  Starting with his Ph.D. thesis, Sontag developed the foundations of observability, identifiability, and realizations of discrete-time (and later continuous-time) algebraic nonlinear systems, based on studying the dual adjoint dynamics on "system observables". This work was the subject of his Invited Address to the 1994 International Congress of Mathematicians.

• Systems over rings

  Following the work of Kalman and coauthors, Sontag made significant advances to linear systems over rings, in which the entries of the system matrices (A, B, C) belong to rings of operators (useful for delay differential systems), integers (no round-off), or polynomial functions of parameters (useful for parametrized families of systems). His 1982 paper with Khargonekar gave the first state-space formulas for stable proper coprime factorizations, required for controller parameterization in the Desoer–Youla–Vidyasagar sense, framed naturally in the language of systems over rings.

• Feedback stabilization

  With his Rutgers colleague Sussmann, in a 1980 CDC paper Sontag showed that it is not possible to stabilize every controllable system by continuous feedback, demonstrating that aiming to replicate linear-theory results might produce feedback laws without noise-robustness. They also showed that time-periodic feedback stabilization is possible, at least in dimension one, which Coron subsequently greatly extended. Sontag then developed a line of research on stabilization by discontinuous feedback, culminating in the TAC paper showing that every asymptotically controllable system can be stabilized by discontinuous feedback.

• Bounded feedback

  Another direction largely initiated by Sontag and Sussmann was the stabilization of linear systems by bounded feedback. A first paper announced the problem's full solution, but with a construction that was not practically implementable. Teel improved the work with a brilliant approach using nested saturations for the chain of integrators; Sontag and Sussmann in turn extended that construction to the general case, and Sontag and his students went on to prove additional robustness and ISS properties of these designs.

• Input-to-state stability

  Arguably the best-known contribution of Sontag's control-theory work is the notion of input-to-state stability (ISS), employed in tens of thousands of papers and dozens of books. Before its 1989 introduction, two streams of nonlinear systems theory remained disjoint: classical Lyapunov stability theory dealt with state-space models with limited capability to incorporate disturbances, while Sandberg–Zames input-output theory —more suitable to engineering needs— could not benefit from the insights of Lyapunov theory. Responses to inputs and to initial conditions existed in separate "universes". ISS seamlessly integrated Lyapunov and I/O theory, and in the hands of design-oriented researchers became a tool for synthesizing control algorithms robust to disturbances and unmodeled dynamics.

• Control-Lyapunov functions

  Probably the next best-known contribution is the notion of control-Lyapunov functions (cLfs). Non-smooth cLfs were introduced by Sontag in 1981 (Artstein soon after developed the idea of smooth cLfs), together with the "universal formula" for stabilization. These notions, along with ISS and its derivatives, allowed Lyapunov functions to serve as a control-synthesis tool rather than merely an analysis tool, through later innovations such as backstepping, forwarding, inverse optimal control, control barrier functions, and safety filters, key to areas ranging from stable biped walking to driverless cars.

• Hybrid systems

  Sontag was a pioneer in the now-popular field of hybrid systems. His 1980 paper on piecewise linear systems showed their power as controllers, combining continuous variables with discrete mathematics and computer-science ideas toward a comprehensive approach to the regulation of nonlinear systems.

• Computational complexity

  In the late 1980s Sontag began the study of computational complexity in nonlinear control. He proved that deciding nonlinear controllability for bilinear systems is NP-hard, a fundamental limitation establishing that the search for efficiently testable necessary and sufficient conditions could not succeed.

• Neural networks & learning

  Sontag's interest in neural networks, already evident in a book he published as an undergraduate, culminated in the 1990s in foundational work on feedforward and recurrent networks and computational learning theory; he gave the first results on the sample complexity and VC dimension of analog neural networks. A remarkable paper showed that what are now called "deep neural networks" are necessary to stabilize non-holonomic systems, surprising given the era's focus on single-hidden-layer architectures. With his student Siegelmann he developed a theory of analog computing, showing that even analog computers could not solve certain problems unless a condition akin to "P=NP" held. Many of these papers appeared at the most selective computer-science conferences (FOCS, STOC, COLT, NIPS).

• Toward biology

  Since around the year 2000, Sontag has turned much of his research to questions inspired by systems biology. Feedback is as central to living systems as to engineering, from homeostatic regulation of temperature, pressure, or chemical levels to the delicate interactions between infections, tumors, and the immune system. In contrast to engineering systems governed by long-understood physical laws, biological feedback models contain substantial uncertainty and noise, the very things living organisms overcome to survive. His effort has focused on understanding what is special about biological control systems, and the analysis of biological problems often leads to fundamental new concepts in control theory that are later applied elsewhere.

• Reaction networks

  In a 2001 paper, Sontag showed how the stability of a popular model of immune cell activation could be established using chemical reaction network (CRN) theory, in the process extending certain stability proofs of CRN theory from local to global and quantifying robustness. In successful follow-ups with Angeli and others, he vastly expanded knowledge of the dynamics of biochemical networks, spurring a large community effort.

• Monotone systems

  Also with Angeli, and again motivated by biological signaling networks, Sontag introduced in a 2004 paper the idea of monotone systems with inputs, advancing classical results of Hirsch, Smith, Smale, and others. Introducing inputs and outputs made possible the analysis of large systems through decomposition into smaller monotone subsystems, for example via new small-gain theorems, an approach impossible for isolated dynamical systems. This led to an explosion of interest in monotone and monotone-decomposable systems, with applications well beyond the original biological motivation.

• Therapy & resistance

  Much of Sontag's recent research concerns the interactions between therapies, immune components, and tumors or infections. Resistance is often viewed as Darwinian selection of pre-existing or de novo genetic or epigenetic modifications; yet evidence suggests progression to resistance need not be random, but may be induced by the therapeutic agent itself, a "Lamarckian" process that can occur through phenotype switching. With Greene and Gevertz, Sontag developed a modeling framework incorporating both random and drug-induced resistance. The model demonstrates that a drug's ability (or inability) to induce resistance can produce qualitatively different responses to the same dose and schedule, and has informed experimental work such as the DNA-barcoding studies in Brock's lab at UT.

• Metronomic chemotherapy

  Sontag also studied systemic resistance in the context of metronomic chemotherapy. In work with Waxman's lab at BU, he developed a model of the interactions between tumor growth, immune activation, and therapy-mediated immunogenic cell death. Conceptually simple, it fits empirical data from a GL261 mouse glioma model treated with cyclophosphamide on a metronomic schedule; a single fixed parameter set recapitulates data across regimens and predicts peak immune-activation times not used in fitting. The validated model was then used to predict tumor-immune dynamics for novel schedules.

• Infection & melanoma

  In work with Zloza's lab at Rutgers, a different type of resistance was studied, providing a mechanistic model of interactions among a non-oncogenic viral lung infection (A/H1N1/PR8), distal B16-F10 skin melanoma, T cells, and checkpoint-inhibition therapy, which may explain the increased tumor growth observed experimentally.

• Antigen dynamics

  Since the early 1990s, many authors have suggested that self/nonself recognition may be modulated by the rates of change of antigen challenges, not only by antigen identity. Sontag introduced a simple model predicting that exponentially increasing antigen stimulation (tumor growth, acute infections, doubling vaccine doses in successive boosters) enhances immune response, recovering in particular the "two-zone tumor tolerance" phenomenon observed decades earlier.

• Synthetic biology

  Synthetic biology arose around the turn of the century from two mutually reinforcing goals. The first is to design novel —or re-engineer existing— molecular biological systems to perform new tasks, with applications including targeted drug delivery, immunotherapy (e.g. redesigning cytotoxic T cells against tumors), renewable energy, waste recycling, biosensing, tissue homeostasis, and molecular computing. The second is to improve understanding of natural phenomena: because interlocking regulatory loops make natural systems hard to test, building synthetic versions offers a "clean playground" in which discrepancies between expected and observed behavior expose gaps in models. Sontag's lab has contributed across synthetic biology, from the theoretical foundations of modular interconnection ("retroactivity" and "resource competition", with Del Vecchio's lab at MIT) to distributed multi-cell computation (with Voigt's lab at MIT), transcriptional gene-regulatory networks, protease-based biosensors (with Khare's lab at Rutgers), and cell-free systems (with Noireaux's lab at Minnesota).

• Network inference

  The different areas of Sontag's systems-biology work share an emphasis on the fundamental principles of signal processing, feedback, and control, drawing on tools from control theory and applied mathematics. One example is network inference of biological pathways: the reverse-engineering problem aims to unravel the web of interactions among genes, proteins, metabolites, and small molecules from experimental data. Steady-state "Modular Response Analysis" (developed with Kholodenko in the early 2000s) is now widely applied and reveals fundamental limitations of perturbation-based reconstruction (with Gunawardena's lab at Harvard Medical School). Complementing this, transient responses to external stimuli provide deep insight into network structure (with Rahi's lab at EPFL), especially when coupled to log-sensing and other input invariances ("fold-change detection", with Alon at the Weizmann Institute).