| Publications by Eduardo D. Sontag in year 2025 |
| Books and proceedings |
| Articles in journal or book chapters |
| Numerical ``direct'' approaches to time-optimal control often fail to find solutions that are singular in the sense of the Pontryagin Maximum Principle. These approaches behave better when searching for saturated (bang-bang) solutions. In previous work by one of the authors, singular solutions were theoretically shown to exist for the time-optimal problem for two-link manipulators under hard torque constraints. The theoretical results gave explicit formulas, based on Lie theory, for singular segments of trajectories, but the global structure of solutions remains unknown. In this work, we show how to effectively combine these theoretically found formulas with the use of general-purpose optimal control softwares. By using the explicit formula given by theory in the intervals where the numerical solution enters a singular arcs, we not only obtain an algebraic expression for the control in that interval, but we are also able to remove artifacts present in the numerical solution. In this way, the best features of numerical algorithms and theory complement each other and provide a better picture of the global optimal structure. We showcase the technique on a 2 degrees of freedom robotic arm example, and also propose a way of extending the analyzed method to robotic arms with higher degrees of freedom through partial feedback linearization, assuming the desired task can be mostly performed by a few of the degrees of freedom of the robot and imposing some prespecified trajectory on the remaining joints. |
| In previous work, we have developed an approach to understanding the long-term dynamics of classes of chemical reaction networks, based on rate-dependent Lyapunov functions. In this paper, we show that stronger notions of convergence can be established by proving contraction with respect to non-standard norms. This enables us to show that such networks entrain to periodic inputs. We illustrate our theory with examples from signaling pathways and genetic circuits. |
| This work explores the integration of machine learning (ML) and mechanistic models (MM). While ML has demonstrated remarkable success in data-driven modeling across engineering, biology, and other scientific fields, MM remain essential for their interpretability and capacity to extrapolate beyond observed conditions based on established principles such as chemical kinetics and physiological processes. However, MM can be labor-intensive to construct and often rely on simplifying assumptions that may not fully capture real-world complexity. It is thus desirable to combine MM and ML approaches so as to enable more robust predictions, enhanced system insights, and improved handling of sparse or noisy data. A key challenge when doing so is ensuring that ML components do not disregard mechanistic information, potentially leading to overfitting or reduced interpretability. To address that challenge, this paper introduces the idea of Partially Uncertain Model Structures (PUMS) and investigates conditions that discourage the ML components from ignoring mechanistic constraints. This work also introduces the concept of embedded Physics-Informed Neural Networks (ePINNs), which consist of two loss-sharing neural networks that seamlessly blend ML and MM components. This work arose in the study of the context problem in synthetic biology. Engineered genetic circuits may exhibit unexpected behavior in living cells due to resource sharing. To illustrate the advantages of the ePINNs approach, this paper applies the framework to a gene network model subject to resource competition, demonstrating the effectiveness of this hybrid modeling approach in capturing complex system interactions while maintaining physical consistency. |
| This work studies relationships between monotonicity and contractivity, and applies the results to establish that many reaction networks are weakly contractive, and thus, under appropriate compactness conditions, globally convergent to equilibria. Verification of these properties is achieved through a novel algorithm that can be used to generate cones for an accompanying monotone system. The results given here allow a unified proof of global convergence for several classes of networks that had been previously studied in the literature. |
| There is growing recognition that phenotypic plasticity enables cancer cells to adapt to various environmental conditions. An example of this adaptability is the persistence of an initially sensitive population of cancer cells in the presence of therapeutic agents. Understanding the implications of this drug-induced resistance is essential for predicting transient and long-term tumor tumor dynamics subject to treatment. This paper introduces a mathematical model of this phenomenon of drug-induced resistance which provides excellent fits to time-resolved in vitro experimental data. From observational data of total numbers of cells, the model unravels the relative proportions of sensitive and resistance subpopulations, and quantifies their dynamics as a function of drug dose. The predictions are then validated using data on drug doses which were not used when fitting parameters. The model is then used, in conjunction with optimal control techniques, in order to discover dosing strategies that might lead to better outcomes as quantified by lower total cell volume. |
| This paper introduces the notion of cumulative dose response (cDR). The cDR is the area under the plot of a response variable, an integral taken over a fixed time interval and seen as a function of an input parameter. This work was motivated by the accumulation of cytokines resulting from T cell stimulation, where a non-monotonic cDR has been observed experimentally. However, the notion is of general applicability. A surprising conclusion is that incoherent feedforward loops studied in the systems biology literature, though capable of non-monotonic dose responses, can be mathematically shown to always result in monotonic cDR. |
| We give a proof of an extension of the Hartman-Grobman theorem to nonhyperbolic but asymptotically stable equilibria of vector fields. Moreover, the linearizing topological conjugacy is (i) defined on the entire basin of attraction if the vector field is complete, and (ii) a $C^{k\geq 1}$ diffeomorphism on the complement of the equilibrium if the vector field is $C^k$ and the underlying space is not $5$-dimensional. We also show that the $C^k$ statement in the $5$-dimensional case is equivalent to the $4$-dimensional smooth Poincar\'{e} conjecture. |
| Linear immersions (or Koopman eigenmappings) of a nonlinear system have wide applications in prediction and control. In this work, we study the non-existence of one-to-one linear immersions for nonlinear systems with multiple omega-limit sets. While previous research has indicated the possibility of discontinuous one-to-one linear immersions for such systems, it remained uncertain whether continuous one-to-one linear immersions are attainable. Under mild conditions, we prove that any continuous one-to-one immersion to a class of systems including linear systems cannot distinguish different omega-limit sets, and thus cannot be one-to-one. Furthermore, we show that this property is also shared by approximate linear immersions learned from data as sample size increases and sampling interval decreases. Multiple examples are studied to illustrate our results. |
| Since its introduction by Briat, Gupta and Khammash, the antithetic feedback controller design has attracted considerable attention in both theoretical and experimental systems biology. The case in which the plant is a two-dimensional linear system (making the closed-loop system a four-dimensional nonlinear system) has been analyzed in much detail. This system has a unique equilibrium~$e$ but, depending on parameters, it may exhibit periodic orbits. An interesting question is for what parameter values periodic orbits exist. Another open question is whether other dynamical behaviors, such as chaotic attractors, might be possible for some parameter choices. We show that, for any parameter choices, every compact omega-limit set that does not include~$e$ is a periodic solution. We also show that if the Jacobian of the vector field at the equilibrium is unstable then a (non-trivial) periodic orbit exists. The analysis is based on the theory of strongly~$2$-cooperative systems. |
| Powerful distributed computing can be achieved by communicating cells that individually perform simple operations. We have developed design software to divide a large genetic circuit across cells as well as the genetic parts to implement the subcircuits in their genomes. These tools were demonstrated using a 2-bit version of the MD5 hashing algorithm, an early predecessor to the cryptographic functions underlying cryptocurrency. One iteration requires 110 logic gates, which were partitioned across 66 strains of Escherichia coli, requiring the introduction of a total of 1.1 Mb of recombinant DNA into their genomes. The strains are individually experimentally verified to integrate their assigned input signals, process this information correctly, and propagate the result to the cell in the next layer. This work demonstrates the potential to obtain programmable control of multicellular biological processes. |
| Bispecific T Cell Engagers (BTC) constitute an exciting antibody design in immuno-oncology that acts to bypass antigen presentation and forms a direct link between cancer and immune cells in the tumor microenvironment (TME). By design, BTCs are efficacious only when the drug is bound to both immune and cancer cell targets, and therefore approaches to maximize drug-target trimer in the TME should maximize the drug's efficacy. In this study, we quantitatively investigate how the concentration of ternary complex and its distribution depend on both the targets' specific properties and the design characteristics of the BTC, and specifically on the binding kinetics of the drug to its targets. A simplified mathematical model of drug-target interactions is considered here, with insights from the "three-body" problem applied to the model. Parameter identifiability analysis performed on the model demonstrates that steady-state data, which is often available at the early pre-clinical stages, is sufficient to estimate the binding affinity of the BTC molecule to both targets. The model is used to analyze several existing antibodies that are either clinically approved or are under development, and to explore the common kinetic features. We conclude with a discussion of the limitations of the BTCs, such as the increased likelihood of cytokine release syndrome, and an assessment for a full quantitative pharmacology model that accounts for drug distribution into the peripheral compartment. |
| Motivated by the growing use of Artificial Intelligence (AI) tools in control design, this paper takes the first steps towards bridging the gap between results from Direct Gradient methods for the Linear Quadratic Regulator (LQR), and neural networks. More specifically, it looks into the case where one wants to find a Linear Feed-Forward Neural Network (LFFNN) feedback that minimizes a LQR cost. This paper starts by computing the gradient formulas for the parameters of each layer, which are used to derive a key conservation law of the system. This conservation law is then leveraged to prove boundedness and global convergence of solutions to critical points, and invariance of the set of stabilizing networks under the training dynamics. This is followed by an analysis of the case where the LFFNN has a single hidden layer. For this case, the paper proves that the training converges not only to critical points but to the optimal feedback control law for all but a set of measure-zero of the initializations. These theoretical results are followed by an extensive analysis of a simple version of the problem (the ``vector case''), proving the theoretical properties of accelerated convergence and robustness for this simpler example. Finally, the paper presents numerical evidence of faster convergence of the training of general LFFNNs when compared to traditional direct gradient methods, showing that the acceleration of the solution is observable even when the gradient is not explicitly computed but estimated from evaluations of the cost function. |
| Conference articles |
| We consider a general class of translation-invariant systems with a specific category of output nonlinearities motivated by biological sensing. We show that no dynamic output feedback can stabilize this class of systems to an isolated equilibrium point. To overcome this fundamental limitation, we propose a simple control scheme that includes a low-amplitude periodic forcing function akin to so-called "active sensing" in biology, together with nonlinear output feedback. Our analysis shows that this approach leads to the emergence of an exponentially stable limit cycle. These findings offer a provably stable active sensing strategy and may thus help to rationalize the active sensing movements made by animals as they perform certain motor behaviors. |
| The development of genetic memory devices in synthetic biology is a challenging process that requires extensive analysis and characterization. In mammalian systems, this complexity is compounded by the need for a small DNA payload for efficient delivery into the cell. Previous genetic memory devices have relied exclusively on protein-based regulation, which are limited by their large size; in this paper, we propose a microRNA-based multistable network, which effectively halves the payload size for more efficient delivery. We demonstrate that the system can be multistable, and use formal methods to characterize constraints on design parameters that guarantee multistability. Our results provide a new genetic network topology that can achieve multistability and demonstrate the use of formal methods in the design of sophisticated genetic network architectures against non-convex top-level specifications. |
| This work explores generalizations of the Polyak-Lojasiewicz inequality (PLI) and their implications for the convergence behavior of gradient flows in optimization problems. Motivated by the continuous-time linear quadratic regulator (CT-LQR) policy optimization problem -- where only a weaker version of the PLI is characterized in the literature -- this work shows that while weaker conditions are sufficient for global convergence to, and optimality of the set of critical points of the cost function, the "profile" of the gradient flow solution can change significantly depending on which "flavor" of inequality the cost satisfies. After a general theoretical analysis, we focus on fitting the CT-LQR policy optimization problem to the proposed framework, showing that, in fact, it can never satisfy a PLI in its strongest form. We follow up our analysis with a brief discussion on the difference between continuous- and discrete-time LQR policy optimization, and end the paper with some intuition on the extension of this framework to optimization problems with L1 regularization and solved through proximal gradient flows. |
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