Publications about 'Lyapunov functions'
Articles in journal or book chapters
  1. M.A. Al-Radhawi, D. Angeli, and E.D. Sontag. On structural contraction of biological interaction networks. 2023. Note: To be submitted. Preprint in: arXiv contractions, contractive systems, matrix measures, logarithmic norms.
    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.

  2. M.A. Al-Radhawi, D. Angeli, and E.D. Sontag. A computational framework for a Lyapunov-enabled analysis of biochemical reaction networks. PLoS Computational Biology, pp 16(2): e1007681, 2020. [PDF] Keyword(s): MAPK cascades, Lyapunov functions, stability, chemical networks, chemical rection networks, systems biology, RFM, ribosome flow model.
    This paper deals with the analysis of the dynamics of chemical reaction networks, developing a theoretical framework based only on graphical knowledge and applying regardless of the particular form of kinetics. This paper introduces a class of networks that are "structurally (mono) attractive", by which we mean that they are incapable of exhibiting multiple steady states, oscillation, or chaos by the virtue of their reaction graphs. These networks are characterized by the existence of a universal energy-like function which we call a Robust Lyapunov function (RLF). To find such functions, a finite set of rank-one linear systems is introduced, which form the extremals of a linear convex cone. The problem is then reduced to that of finding a common Lyapunov function for this set of extremals. Based on this characterization, a computational package, Lyapunov-Enabled Analysis of Reaction Networks (LEARN), is provided that constructs such functions or rules out their existence. An extensive study of biochemical networks demonstrates that LEARN offers a new unified framework. We study basic motifs, three-body binding, and transcriptional networks. We focus on cellular signalling networks including various post-translational modification cascades, phosphotransfer and phosphorelay networks, T-cell kinetic proofreading, ERK signaling, and the Ribosome Flow Model.

  3. J. Miller, M.A. Al-Radhawi, and E.D. Sontag. Mediating ribosomal competition by splitting pools. IEEE Control Systems Letters, 5:1555-1560, 2020. [PDF] Keyword(s): systems biology, synthetic biology, ribosomes, RFM, ribosome flow model.
    Synthetic biology constructs often rely upon the introduction of "circuit" genes into host cells, in order to express novel proteins and thus endow the host with a desired behavior. The expression of these new genes "consumes" existing resources in the cell, such as ATP, RNA polymerase, amino acids, and ribosomes. Ribosomal competition among strands of mRNA may be described by a system of nonlinear ODEs called the Ribosomal Flow Model (RFM). The competition for resources between host and circuit genes can be ameliorated by splitting the ribosome pool by use of orthogonal ribosomes, where the circuit genes are exclusively translated by mutated ribosomes. In this work, the RFM system is extended to include orthogonal ribosome competition. This Orthogonal Ribosomal Flow Model (ORFM) is proven to be stable through the use of Robust Lyapunov Functions. The optimization problem of maximizing the weighted protein translation rate by adjusting allocation of ribosomal species is formulated and implemented. Note: publsihed Nov 2020, even though journal reprint says "Nov 2021".

  4. J. L. Mancilla-Aguilar, R. Garcìa, E.D. Sontag, and Y. Wang. Uniform stability properties of switched systems with switchings governed by digraphs. Nonlinear Anal., 63(3):472-490, 2005. [PDF]
    This paper develops characterizations of various uniform stability properties of switched systems described by differential inclusions, and whose switchings are governed by a digraph. These characterizations are given in terms of stability properties of the system with restricted switchings and also in terms of Lyapunov functions.

  5. M. Malisoff and E.D. Sontag. Asymptotic controllability and input-to-state stabilization: the effect of actuator errors. In Optimal control, stabilization and nonsmooth analysis, volume 301 of Lecture Notes in Control and Inform. Sci., pages 155-171. Springer, Berlin, 2004. [PDF] Keyword(s): input to state stability, control-Lyapunov functions, nonlinear control, feedback stabilization, ISS.
    We discuss several issues related to the stabilizability of nonlinear systems. First, for continuously stabilizable systems, we review constructions of feedbacks that render the system input-to-state stable with respect to actuator errors. Then, we discuss a recent paper which provides a new feedback design that makes globally asymptotically controllable systems input-to-state stable to actuator errors and small observation noise. We illustrate our constructions using the nonholonomic integrator, and discuss a related feedback design for systems with disturbances.

  6. M. Malisoff, L. Rifford, and E.D. Sontag. Global Asymptotic Controllability Implies Input-to-State Stabilization. SIAM J. Control Optim., 42(6):2221-2238, 2004. [PDF] [doi:] Keyword(s): input to state stability, control-Lyapunov functions, nonlinear control, feedback stabilization.
    The main problem addressed in this paper is the design of feedbacks for globally asymptotically controllable (GAC) control affine systems that render the closed loop systems input to state stable with respect to actuator errors. Extensions for fully nonlinear GAC systems with actuator errors are also discussed. Our controllers have the property that they tolerate small observation noise as well.

  7. E.D. Sontag and M. Krichman. An example of a GAS system which can be destabilized by an integrable perturbation. IEEE Trans. Automat. Control, 48(6):1046-1049, 2003. [PDF] Keyword(s): nonlinear stability.
    A construction is given of a globally asymptotically stable time-invariant system which can be destabilized by some integrable perturbation. Besides its intrinsic interest, this serves to provide counterexamples to an open question regarding Lyapunov functions.

  8. M. Malisoff and E.D. Sontag. Universal formulas for feedback stabilization with respect to Minkowski balls. Systems Control Lett., 40(4):247-260, 2000. [PDF] Keyword(s): nonlinear control, feedback stabilization, saturation, control-Lyapunov functions, bounded inputs.
    This note provides explicit algebraic stabilizing formulas for clf's when controls are restricted to certain Minkowski balls in Euclidean space. Feedbacks of this kind are known to exist by a theorem of Artstein, but the proof of Artstein's theorem is nonconstructive. The formulas are obtained from a general feedback stabilization technique and are used to construct approximation solutions to some stabilization problems.

  9. E.D. Sontag. Control-Lyapunov functions. In Open problems in mathematical systems and control theory, Comm. Control Engrg. Ser., pages 211-216. Springer, London, 1999. Keyword(s): control-Lyapunov functions.

  10. E.D. Sontag. Stability and stabilization: discontinuities and the effect of disturbances. In Nonlinear analysis, differential equations and control (Montreal, QC, 1998), volume 528 of NATO Sci. Ser. C Math. Phys. Sci., pages 551-598. Kluwer Acad. Publ., Dordrecht, 1999. [PDF] Keyword(s): feedback stabilization, nonlinear control, input to state stability.
    In this expository paper, we deal with several questions related to stability and stabilization of nonlinear finite-dimensional continuous-time systems. We review the basic problem of feedback stabilization, placing an emphasis upon relatively new areas of research which concern stability with respect to "noise" (such as errors introduced by actuators or sensors). The table of contents is as follows: Review of Stability and Asymptotic Controllability, The Problem of Stabilization, Obstructions to Continuous Stabilization, Control-Lyapunov Functions and Artstein's Theorem, Discontinuous Feedback, Nonsmooth CLF's, Insensitivity to Small Measurement and Actuator Errors, Effect of Large Disturbances: Input-to-State Stability, Comments on Notions Related to ISS.

  11. F. Albertini and E.D. Sontag. Continuous control-Lyapunov functions for asymptotically controllable time-varying systems. Internat. J. Control, 72(18):1630-1641, 1999. [PDF] Keyword(s): control-Lyapunov functions.
    This paper shows that, for time varying systems, global asymptotic controllability to a given closed subset of the state space is equivalent to the existence of a continuous control-Lyapunov function with respect to the set.

  12. Y.S. Ledyaev and E.D. Sontag. A Lyapunov characterization of robust stabilization. Nonlinear Anal., 37(7, Ser. A: Theory Methods):813-840, 1999. [PDF] Keyword(s): nonlinear control, feedback stabilization.
    One of the fundamental facts in control theory (Artstein's theorem) is the equivalence, for systems affine in controls, between continuous feedback stabilizability to an equilibrium and the existence of smooth control Lyapunov functions. This equivalence breaks down for general nonlinear systems, not affine in controls. One of the main results in this paper establishes that the existence of smooth Lyapunov functions implies the existence of (in general, discontinuous) feedback stabilizers which are insensitive to small errors in state measurements. Conversely, it is shown that the existence of such stabilizers in turn implies the existence of smooth control Lyapunov functions. Moreover, it is established that, for general nonlinear control systems under persistently acting disturbances, the existence of smooth Lyapunov functions is equivalent to the existence of (possibly) discontinuous) feedback stabilizers which are robust with respect to small measurement errors and small additive external disturbances.

  13. E.D. Sontag and H.J. Sussmann. General classes of control-Lyapunov functions. In Stability theory (Ascona, 1995), volume 121 of Internat. Ser. Numer. Math., pages 87-96. Birkhäuser, Basel, 1996. [PDF] Keyword(s): control-Lyapunov functions.
    Shorter and more expository version of "Nonsmooth control-Lyapunov functions"

  14. Y. Lin, E.D. Sontag, and Y. Wang. A smooth converse Lyapunov theorem for robust stability. SIAM J. Control Optim., 34(1):124-160, 1996. [PDF] [doi:] Keyword(s): input to state stability.
    This paper presents a Converse Lyapunov Function Theorem motivated by robust control analysis and design. Our result is based upon, but generalizes, various aspects of well-known classical theorems. In a unified and natural manner, it (1) allows arbitrary bounded time-varying parameters in the system description, (2) deals with global asymptotic stability, (3) results in smooth (infinitely differentiable) Lyapunov functions, and (4) applies to stability with respect to not necessarily compact invariant sets.

  15. Y. Lin and E.D. Sontag. Control-Lyapunov universal formulas for restricted inputs. Control Theory Adv. Tech., 10(4, part 5):1981-2004, 1995. [PDF] Keyword(s): control-Lyapunov functions, saturation, bounded inputs.
    We deal with the question of obtaining explicit feedback control laws that stabilize a nonlinear system, under the assumption that a "control Lyapunov function" is known. In previous work, the case of unbounded controls was considered. Here we obtain results for bounded and/or positive controls. We also provide some simple preliminary remarks regarding a set stability version of the problem and a version for systems subject to disturbances.

  16. E.D. Sontag and A.R. Teel. Changing supply functions in input/state stable systems. IEEE Trans. Automat. Control, 40(8):1476-1478, 1995. [PDF] Keyword(s): input to state stability, ISS, input to state stability, Lyapunov functions.
    We consider the problem of characterizing possible supply functions for a given dissipative nonlinear system, and provide a result that allows some freedom in the modification of such functions.

  17. Y. Lin and E.D. Sontag. A universal formula for stabilization with bounded controls. Systems Control Lett., 16(6):393-397, 1991. [PDF] [doi:] Keyword(s): stabilization, nonlinear systems, saturation, bounded inputs, control-Lyapunov functions, real-analytic functions.
    We provide a formula for a stabilizing feedback law using a bounded control, under the assumption that an appropriate control-Lyapunov function is known. Such a feedback, smooth away from the origin and continuous everywhere, is known to exist via Artstein's Theorem. As in the unbounded-control case treated in a previous note, we provide an explicit and ``universal'' formula given by an algebraic function of Lie derivatives. In particular, we extend to the bounded case the result that the feedback can be chosen analytic if the Lyapunov function and the vector fields defining the system are analytic.

  18. E.D. Sontag. A ``universal'' construction of Artstein's theorem on nonlinear stabilization. Systems Control Lett., 13(2):117-123, 1989. [PDF] Keyword(s): control-Lyapunov functions, stabilization, real-analytic functions.
    This note presents an explicit proof of the theorem - due to Artstein - which states that the existence of a smooth control-Lyapunov function implies smooth stabilizability. Moreover, the result is extended to the real-analytic and rational cases as well. The proof uses a "universal" formula given by an algebraic function of Lie derivatives; this formula originates in the solution of a simple Riccati equation.

  19. E.D. Sontag. A Lyapunov-like characterization of asymptotic controllability. SIAM J. Control Optim., 21(3):462-471, 1983. [PDF] Keyword(s): control-Lyapunov functions.
    It is shown that a control system in Rn is asymptotically controllable to the origin if and only if there exists a positive definite continuous functional of the states whose derivative can be made negative by appropriate choices of controls.

  20. E.D. Sontag. A characterization of asymptotic controllability. In A. Bednarek and L. Cesari, editors, Dynamical Systems II, pages 645-648. Academic Press, NY, 1982. [PDF] Keyword(s): control-Lyapunov functions.
    This paper was a conference version of the SIAM paper that introduced the idea of control-Lyapunov functions for arbitrary nonlinear systems. (The journal paper was submitted in 1981 but only published in 1983.)

Conference articles
  1. D. K. Agrawal, R. Marshall, M.A. Al-Radhawi, V. Noireaux, and E. D. Sontag. Some remarks on robust gene regulation in a biomolecular integral controller. In Proc. 2019 IEEE Conf. Decision and Control, pages 2820-2825, 2019. [PDF] Keyword(s): adaptation, biological adaptation, perfect adaptation, tracking, synthetic biology, integral feedback, TX/TL, systems biology, dynamical systems, adaptation, internal model principle, systems biology.
    Integral feedback can help achieve robust tracking independently of external disturbances. Motivated by this knowledge, biological engineers have proposed various designs of biomolecular integral feedback controllers to regulate biological processes. In this paper, we theoretically analyze the operation of a particular synthetic biomolecular integral controller, which we have recently proposed and implemented experimentally. Using a combination of methods, ranging from linearized analysis to sum-of-squares (SOS) Lyapunov functions, we demonstrate that, when the controller is operated in closed-loop, it is capable of providing integral corrections to the concentration of an output species in such a manner that the output tracks a reference signal linearly over a large dynamic range. We investigate the output dependency on the reaction parameters through sensitivity analysis, and quantify performance using control theory metrics to characterize response properties, thus providing clear selection guidelines for practical applications. We then demonstrate the stable operation of the closed-loop control system by constructing quartic Lyapunov functions using SOS optimization techniques, and establish global stability for a unique equilibrium. Our analysis suggests that by incorporating effective molecular sequestration, a biomolecular closed-loop integral controller that is capable of robustly regulating gene expression is feasible.

  2. D. Liberzon, E.D. Sontag, and Y. Wang. On integral-input-to-state stabilization. In Proc. American Control Conf., San Diego, June 1999, pages 1598-1602, 1999. [PDF] Keyword(s): input to state stability, integral input to state stability, iISS, ISS, control-Lyapunov functions.
    This paper continues the investigation of the recently introduced integral version of input-to-state stability (iISS). We study the problem of designing control laws that achieve iISS disturbance attenuation. The main contribution is an appropriate concept of control Lyapunov function (iISS-CLF), whose existence leads to an explicit construction of such a control law. The results are compared and contrasted with the ones available for the ISS case.

  3. E.D. Sontag. Recent results on discontinuous stabilization and control-Lyapunov functions. In Proc. Workshop on Control of Nonlinear and Uncertain Systems, London, Feb. 1998, 1998. Keyword(s): control-Lyapunov functions.

  4. F. Albertini and E.D. Sontag. Control-Lyapunov functions for time-varying set stabilization. In Proc. European Control Conf., Brussels, July 1997, 1997. Note: (Paper WE-E A5, CD-ROM file ECC515.pdf, 6 pages). Keyword(s): control-Lyapunov functions.

  5. F.H. Clarke, Y.S. Ledyaev, E.D. Sontag, and A.I. Subbotin. Asymptotic controllability and feedback stabilization. In Proc. Conf. on Information Sciences and Systems (CISS 96)Princeton, NJ, pages 1232-1237, 1996. Keyword(s): control-Lyapunov functions, feedback stabilization.

  6. E.D. Sontag and H.J. Sussmann. Nonsmooth control-Lyapunov functions. In Proc. IEEE Conf. Decision and Control, New Orleans, Dec. 1995, IEEE Publications, 1995, pages 2799-2805, 1995. [PDF] Keyword(s): control-Lyapunov functions.
    It is shown that the existence of a continuous control-Lyapunov function (CLF) is necessary and sufficient for null asymptotic controllability of nonlinear finite-dimensional control systems. The CLF condition is expressed in terms of a concept of generalized derivative (upper contingent derivative). This result generalizes to the non-smooth case the theorem of Artstein relating closed-loop feedback stabilization to smooth CLF's. It relies on viability theory as well as optimal control techniques. A "non-strict" version of the results, analogous to the LaSalle Invariance Principle, is also provided.

  7. Y. Lin and E.D. Sontag. On control-Lyapunov functions under input constraints. In Proc. IEEE Conf. Decision and Control, Orlando, Dec. 1994, IEEE Publications, 1994, pages 640-645, 1994. Keyword(s): control-Lyapunov functions.

  8. G.A. Lafferriere and E.D. Sontag. Remarks on control Lyapunov functions for discontinuous stabilizing feedback. In Proc. IEEE Conf. Decision and Control, San Antonio, Dec. 1993, IEEE Publications, 1993, pages 306-308, 1993. [PDF] Keyword(s): feedback stabilization.
    We present a formula for a stabilizing feedback law under the assumption that a piecewise smooth control-Lyapunov function exists. The resulting feedback is continuous at the origin and smooth everywhere except on a hypersurface of codimension 1, assuming that certain transversality conditions are imposed there.



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