E. Sontag's work spans control and dynamical systems theory, theoretical
computer science, machine learning, cancer and immunology, and molecular,
synthetic, and computational biology.
His work in nonlinear control theory led in the late 1980s to mid 1990s to the
introduction of the concept of inputtostate stability (ISS), a stability
theory notion for nonlinear systems, and various related variants (iISS,
etc.), as well as in 1981 the first formulation of controlLyapunov functions,
and later a ``universal formula'' for smooth stabilization.
At the time, he also developed tools for stabilization of linear systems under
actuator saturation.
In the 1980s, he pioneered techniques for the analysis of nonlinear systems
based on commutative algebra and algebraic geometry, for observability and
minimal realizations.
In the early to mid 1980s, he developed notions of computational complexity
for control systems, and published one of the first papers on hybrid
(piecewise linear) systems.
In the late 1990s, Sontag turned to a focus on discontinuous stabilization, a
topic that had originated in joint work with Sussmann in 1980.
Other work in the 1980s to early 1990s dealt with the foundations of
observation spaces, identifiability, and input/output equations for nonlinear
behaviors, and the finalizing of a major textbook in Mathematical Control
Theory.
Sontag has a longstanding interest in biology and neuroscience, starting with
a book on AI that he authored while an undergraduate.
In the 1990s, he established basic results on representability,
identifiability, and superTuring computability by neural networks, as well as
``PAC learning'' sample estimates for dynamical systems, rates of
approximation in function approximation, and a foundational result on
neuralnetwork feedback control (``two hidden layers are necessary'').
Since the year 2000, he has increasingly been motivated by molecular biology,
and has published in systems biology (and more pure biology) journals on the
topics of bacterial chemotasis, cell cycle, immune recognition, interactions
between cancers and infections, chemotherapyinduced resistance, metastasis,
epigenetics, ribosome flow models, and tumor heterogeneity.
He has also worked on population models (LotkaVolterra species competition,
epidemiology).
Another recent major direction of work has been in synthetic biology, and
specifically the design of genetic and posttranslational circuits for gene
copy number compensation, rejection of disturbances, and Boolean computation.
More generally in systems biology, he has worked on fundamental mathematical
principles of monotone systems, stochastic models, and chemical network
theory.
Some specifics:

Starting with his PhD thesis, Sontag developed the foundations
of observability, identifiability, and realizations of discretetime (and
later continuoustime) algebraic nonlinear systems, based in the idea of
studying the dual adjoint dynamics on "system observables". This work was the
subject of his Invited Address to the 1994 International Congress of
Mathematicians.

Following on the work of Kalman and coauthors, Sontag made some of the most
significant advances to linear systems over rings, in which the entries of the
systems matrices (A,B,C) belong to rings of operators (useful in studying the
stabilization of delay differential systems), integers (noroundoff), or
polynomial functions of parameters (useful in the stabilization of
parametrized families of systems). For example his 1982 paper (with
Khargonekar) [4a] presented the first state space formulas for stable proper
coprime factorizations, required for controller parameterization in the
DesoerYoulaVidyasagar sense, framed naturally in the language of
systems over rings.

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. He thus demonstrated that aiming for results that replicate those in
linear control theory might produce feedback laws without
noiserobustness. But they also showed that timeperiodic feedback
stabilization is possible, at least in dimension one, which Coron subsequently
greatly extended. Sontag then developed a whole line of research on
stabilization by discontinuous feedback, in which a particularly remarkable
achievement is the TAC paper that showed that every (asymptotically)
controllable system can be stabilized by discontinuous feedback.

Another direction largely initiated by Sontag and Sussmann was 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 their work with a brillant approach with nested
saturations, for the chain of integrators. In turn, Sontag and Sussmann
extended that construction to the general case, and Sontag and students went
on to prove additional robustness and ISS properties of these designs.

Arguably, the bestknown contribution of Sontag's control theory work is the
notion of inputtostate stability (ISS). This paradigm has been employed in
tens of thousands of papers by other authors and dozens of books, in both
further theoretical developments and applications.
Prior to the 1989 introduction of ISS, two main streams of nonlinear systems
theory remained disjoint: while classical Lyapunov stability theory dealt with
state space models, with a very limited capability to incorporate
disturbances, SandbergZames inputoutput theory more suitable to
engineering needs was unable to benefit from rich insights gained by
Lyapunov theory. Thus, responses to inputs and responses to initial
conditions existed in separate control theory "universes". The new notion of
ISS seamlessly integrated Lyapunov and I/O theory. The first first paper was
followed with at least a decade of further theoretical advances, and in the
hands of control designoriented researchers ISS became a tool for synthesis
of control algorithms robust to disturbances and unmodeled dynamics.
 Probably the next bestknown contribution of Sontag's control work
wwre control Lyapunov functions (cLf's).
Nonsmooth cLf's were introduced by him in 1981 (Arstein soon after
developed the idea of smooth cLf's), as well as the "universal formula" for
stabilization. These notion, together with ISS and its derivatives, allowed Lyapunov functions to be applied as a
control synthesis tool, and not merely an analysis tool, through innovations
(by other authors) such as backstepping, then forwarding, and inverse optimal
control, "safe control", control barrier functions, and "safeguards", which
have been key to areas ranging from stable biped walking to automated and
driverless cars.

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

In the late 1980s, Sontag started the study of computational complexity in
nonlinear control. He proved that deciding nonlinear controllability for
bilinear systems is NPhard, a fundamental limitation not expected at that
time, thus establishing that the search for efficiently testable necessary and
sufficient conditions (an area of great activity at the time) could not
possibly succeed.

Sontag's interest in neural networks (already evident in an Artificial
Intelligence book that he published as an undergraduate) culminated in the
1990s in his examination of the foundations of feedforward and recurrent
neural networks and more generally computational learning theory; he
provided 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 nonholonomic systems, which was
surprising given the focus on single hidden layer architectures for control at
the time. Going further, he developed a theory of analog computing
(with his student Siegelmann), showing that even analog computers would not be
able to solve certain computational problems unless a condition similar to
"P=NP" would hold. Many of these papers were accepted for presentation at the
most selective computer science conferences (FOCS, STOC, COLT, NIPS).

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 it is in engineering, from
homeostatic mechanisms for temperature, pressure, or chemical levels, to the
delicate interactions between infections, tumors, and the immune system. In
contrast to engineering systems governed by physical laws understood at least
a century ago, the feedback models in biology contain substantial uncertainty
and noise  the very things that living organisms overcome to survive.
Sontag's effort has focused on understanding what is special about biological
control systems.
The analysis of biological problems often leads to the discovery of
fundamental new concepts in control theory, which may be subsequently applied
in other fields.

In a 2001 paper, Sontag showed how stability of a popular model of immune cell
activation could be established by using chemical reaction network (CRN)
theory, in the process extending certain stability proofs of CNR theory from
local to global and quantifying robustness properties. In a line of very
successful followups with Angeli and others, he vastly expanded the knowledge
of dynamics of biochemical networks, which has been followed up by the efforts
of a large community.

Also with Angeli, and again motivated by biological signaling networks,
Sontag introduced in a 2004 paper the idea of monotone systems with inputs, in
the process advancing classical results by Hirsch, Smith, Smale, and others in
mathematics.
The introduction of inputs (and outputs) made possible the analysis of large
systems through decomposition into smaller monotone subsystems, for example,
through new smallgain theorems (different than those for ISS), an approach
that would have been impossible if considering only classical isolated
dynamical systems. This work led to an explosion of interest by the control
community in monotone systems and systems decomposable into monotone systems,
with applications much beyond the original biological motivations.

Much of Sontag's recent research has focused on the interactions between
therapies, immune system components, and tumors and/or infections. Resistance
is often viewed as the result of Darwinian selection of either preexisting
(standing variation) or duringtreatment (de novo) random genetic or
epigenetic modifications. However, experimental evidence suggests that the
progression to resistance need not occur randomly, but instead may be induced
by the therapeutic agent itself. This "Lamarckian" process of resistance
induction can be a result of genetic changes, or can occur through epigenetic
alterations that cause otherwise drugsensitive cancer cells to undergo
"phenotype switching". This relatively novel notion of resistance further
complicates the already challenging task of designing treatment protocols that
minimize the risk of evolving resistance. In an effort to better understand
treatment resistance, Sontag developed in (with Greene and Gevertz) a
mathematical modeling framework that incorporates both random and druginduced
resistance. the model demonstrates that the ability (or lack thereof) of a
drug to induce resistance can result in qualitatively different responses to
the same drug dose and delivery schedule. This model has already had an
impact, forming the basis of experimental work such as the DNA barcoding work
at Brock’s lab at UT.

Sontag also considered a related question, that of systemic
resistance in the context of metronomic chemotherapy. In work with Waxman's
lab at BU, he developed a mathematical model to elucidate the underlying
complex interactions between tumor growth, immune system activation, and
therapymediated immunogenic cell death. Our model is conceptually simple, yet
it provides a surprisingly excellent fit to empirical data obtained from a
GL261 mouse glioma model treated with cyclophosphamide on a metronomic
schedule. Strikingly, a fixed set of parameters, not adjusted for individuals
nor for drug schedule, excellently recapitulates experimental data across
various drug regimens. Additionally, the model predicts peak immune activation
times, rediscovering experimental data that had not been used in parameter
fitting or in model construction. The validated model was then used to predict
expected tumorimmune dynamics for novel drug administration schedules, and it
suggested that immunostimulatory and immunosuppressive intermediates are
responsible for the observed phenomena of resistance and immune cell
recruitment, and thus for variation of responses with respect to different
schedules of drug administration.

In work with Zloza's lab at Rutgers, a different type of resistance was
studied; this work provided a novel mechanistic model of interactions between
a nononcogenic viral lung infection (A/H1N1/PR8), distal B16F10 skin
melanoma, T cells, and checkpoint inhibition therapy, which might explain the
increased tumor growth observed in the Zloza's lab.

Since the early 1990s, many authors have independently suggested that
self/nonself recognition by the immune system might be modulated by the rates
of change of antigen challenges, in addition to the antigen identity. In
recent work, Sontag introduced a very simple mathematical model which predicts
that exponentially increasing antigen stimulation (tumor growth, acute
infections, doubling vaccine dose in successive boosters) will enhance immune
response, and which recovers, in particular, behaviors observed 3040 years
ago (“twozone tumor tolerance” phenomenon).

The advent of the field of (molecular, systems) "synthetic biology" at the
turn of the 21st Century was motivated by two distinct, but mutually
reinforcing, goals. A first goal is to design novel –or reengineer existing–
molecular biological systems, extending or modifying the behavior of
organisms, and to control them to perform new tasks. Through the de novo
construction of simple elements and circuits, the field aims to develop a
systematic approach to obtaining new cell behaviors in a predictable and
reliable fashion. Applications include targeted drug delivery, immunotherapy
(e.g., redesign of cytotoxic T cells to fight tumors), renewable energy (e.g.,
biofuels through ethanolproducing bacteria), reengineering bacterial
metabolism for waste recycling, biosensing (e.g., detecting environmental
pathogens or toxins), tissue homeostasis (e.g., designing selfdifferentiating
β cells in type 1 diabetes treatments), and computing applications (e.g.,
molecular computing). A second, and no less important, goal is to improve the
quantitative and qualitative understanding of basic natural
phenomena. Experimental validation in theoretical biology is hampered by the
practical difficulties that one faces when attempting to test the validity of
models, due to interlocking regulatory loops in tightly controlled naturally
occurring cellular systems. A very useful approach to the testing of
(mathematical) models of a biological system is to design and construct
synthetic versions of the hypothesized system . Discrepancies between expected
behavior and observed behavior serve to highlight either research issues that
need more studying, or knowledge gaps and inaccurate assumptions in
models. Synthetic, as opposed to naturally occurring, networks have the
advantage that they can be designed specifically to test hypotheses, thus
affording a "clean playground" in which to test models. Systems can be built
with exactly the components that are believed to be relevant to a given
biological signal processing and/or control function. In this fashion, the
combinatorial effects of multiple feedforward and feedback loops, or the steps
involved in complex cellular responses to stimuli, can be deconstructed and
analyzed. The basic knowledge gained from synthetic constructs thus helps
advance our understanding of natural complex systems. Sontag's lab has
contributed to
various areas of synthetic biology, ranging from the theoretical foundations
of modular interconnections ("retroactivity" and "resource competition"
papers, in collaboration with Del Vecchio's lab at MIT) to the design of distributed
multicell computations (collaboration with Voigt's lab at MIT) to the analysis and design of transcriptional gene regulatory
networks, proteasebased enzymatic biosensors (with Khare's lab at Rutgers)
and
cellfree biological systems (with Noireaux's lab at Minnesota).

The different areas of Sontag's systems biology work share an emphasis on
fundamental principles of signal processing, feedback, and control, and
benefit from the rich set of conceptual and practical tools developed from his
expertise in feedback control theory and other areas of applied mathematics,
computer science, and engineering. An example is the work on network inference
of biological pathways. The web of interactions between genes, proteins,
metabolites, and small molecules gives rise to complex circuits that enable
cells to process information, store memories, and respond to external signals,
and the reverse engineering problem in aims to unravel these interactions,
employing experimental data such as changes in concentrations of active
proteins, mRNA levels, and so forth. Steadystate "Modular Response Analysis"
(delevloped with Kholodenko in the early 2000s)
is by now widely applied, and it allows one to discover
fundamental limitations of perturbationbased approaches to network
reconstruction (work with Gunewardena's lab at Harvard Medical School).
Complementing this, the transient
behavior of systems subject to external stimuli can provide deep insights into
network structure (work with Rahi's lab at EPFL), especially when coupled to logsensing
and other input invariances ("fold change detection" work with Alon at
the Weizmann)