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 Publications about 'stabilization'
 Articles in journal or book chapters
1. S. J. Rahi, J. Larsch, K. Pecani, N. Mansouri, A. Y. Katsov, K. Tsaneva-Atanasova, E. D. Sontag, and F. R. Cross. Oscillatory stimuli differentiate adapting circuit topologies. Nature Methods, 14:1010-1016, 2017. [PDF] Keyword(s): biochemical networks, periodic behaviors, monotone systems, entrainment, oscillations, incoherent feedforward loop, feedforward, IFFL, systems biology.
Abstract:
 Elucidating the structure of biological intracellular networks from experimental data remains a major challenge. This paper studies two types of response signatures'' to identify specific circuit motifs, from the observed response to periodic inputs. In particular, the objective is to distinguish negative feedback loops (NFLs) from incoherent feedforward loops (IFFLs), which are two types of circuits capable of producing exact adaptation. The theory of monotone systems with inputs is used to show that period skipping'' (non-harmonic responses) is ruled out in IFFL's, and a notion called refractory period stabilization'' is also analyzed. The approach is then applied to identify a circuit dominating cell cycle timing in yeast, and to uncover a calcium-mediated NFL circuit in \emph{C.elegans} olfactory sensory neurons.

2. E.D. Sontag. Stability and feedback stabilization. In Robert Meyers, editor, Mathematics of Complexity and Dynamical Systems, pages 1639-1652. Springer-Verlag, Berlin, 2011. [PDF] Keyword(s): stability, nonlinear control, feedback stabilization.
Abstract:
 The problem of stabilization of equilibria is one of the central issues in control. In addition to its intrinsic interest, it represents a first step towards the solution of more complicated problems, such as the stabilization of periodic orbits or general invariant sets, or the attainment of other control objectives, such as tracking, disturbance rejection, or output feedback, all of which may be interpreted as requiring the stabilization of some quantity (typically, some sort of error'' signal). A very special case, when there are no inputs, is that of stability. This short and informal article provides an introduction to the subject.

3. E.D. Sontag. Stability and Feedback Stabilization. In Robert Meyers, editor, Encyclopedia of Complexity and Systems Science. Springer-Verlag, Berlin, 2007. Keyword(s): stability, nonlinear control, feedback stabilization.
Abstract:
 The problem of stabilization of equilibria is one of the central issues in control. In addition to its intrinsic interest, it represents a first step towards the solution of more complicated problems, such as the stabilization of periodic orbits or general invariant sets, or the attainment of other control objectives, such as tracking, disturbance rejection, or output feedback, all of which may be interpreted as requiring the stabilization of some quantity (typically, some sort of error'' signal). A very special case, when there are no inputs, is that of stability. This short and informal article provides an introduction to the subject.

4. M. Malisoff, M. Krichman, and E.D. Sontag. Global stabilization for systems evolving on manifolds. Journal of Dynamical and Control Systems, 12:161-184, 2006. [PDF] Keyword(s): nonlinear stability, nonlinear control, feedback stabilization.
Abstract:
 This paper shows that any globally asymptotically controllable system on any smooth manifold can be globally stabilized by a state feedback. Since discontinuous feedbacks are allowed, solutions are understood in the sample and hold'' sense introduced by Clarke-Ledyaev-Sontag-Subbotin (CLSS). This work generalizes the CLSS Theorem, which is the special case of our result for systems on Euclidean space. We apply our result to the input-to-state stabilization of systems on manifolds relative to actuator errors, under small observation noise.

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.
Abstract:
 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:http://dx.doi.org/10.1137/S0363012903422333] Keyword(s): input to state stability, control-Lyapunov functions, nonlinear control, feedback stabilization.
Abstract:
 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. D. Liberzon, E.D. Sontag, and Y. Wang. Universal construction of feedback laws achieving ISS and integral-ISS disturbance attenuation. Systems Control Lett., 46(2):111-127, 2002. Note: Errata here: http://sontaglab.org/FTPDIR/iiss-clf-errata.pdf. [PDF] Keyword(s): input to state stability, integral input to state stability, ISS, iISS, nonlinear control, feedback stabilization.
Abstract:
 We study nonlinear systems with both control and disturbance inputs. The main problem addressed in the paper is design of state feedback control laws that render the closed-loop system integral-input-to-state stable (iISS) with respect to the disturbances. We introduce an appropriate concept of control Lyapunov function (iISS-CLF), whose existence leads to an explicit construction of such a control law. The same method applies to the problem of input-to-state stabilization. Converse results and techniques for generating iISS-CLFs are also discussed.

8. E.D. Sontag. Structure and stability of certain chemical networks and applications to the kinetic proofreading model of T-cell receptor signal transduction. IEEE Trans. Automat. Control, 46(7):1028-1047, 2001. [PDF] Keyword(s): zero-deficiency networks, systems biology, biochemical networks, nonlinear stability, dynamical systems, kinetic proofreading, T cells, immunology.
Abstract:
 This paper deals with the theory of structure, stability, robustness, and stabilization for an appealing class of nonlinear systems which arises in the analysis of chemical networks. The results given here extend, but are also heavily based upon, certain previous work by Feinberg, Horn, and Jackson, of which a self-contained and streamlined exposition is included. The theoretical conclusions are illustrated through an application to the kinetic proofreading model proposed by McKeithan for T-cell receptor signal transduction.

9. X. Bao, Z. Lin, and E.D. Sontag. Finite gain stabilization of discrete-time linear systems subject to actuator saturation. Automatica, 36(2):269-277, 2000. [PDF] Keyword(s): discrete-time, saturation, input-to-state stability, stabilization, ISS, bounded inputs.
Abstract:
 It is shown that, for neutrally stable discrete-time linear systems subject to actuator saturation, finite gain lp stabilization can be achieved by linear output feedback, for all p>1. An explicit construction of the corresponding feedback laws is given. The feedback laws constructed also result in a closed-loop system that is globally asymptotically stable, and in an input-to-state estimate.

10. 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.
Abstract:
 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.

11. E.D. Sontag. Nonlinear feedback stabilization revisited. In Dynamical systems, control, coding, computer vision (Padova, 1998), volume 25 of Progr. Systems Control Theory, pages 223-262. Birkhäuser, Basel, 1999. Note: This is a short conference proceedings paper. Please consult the full version Stability and stabilization: discontinuities and the effect of disturbances.

12. 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.
Abstract:
 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.

13. 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.
Abstract:
 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.

14. E.D. Sontag. Clocks and insensitivity to small measurement errors. ESAIM Control Optim. Calc. Var., 4:537-557, 1999. [PDF] Keyword(s): nonlinear control, feedback stabilization, hybrid systems, discontinuous feedback, measurement noise.
Abstract:
 This paper provides a precise result which shows that insensitivity to small measurement errors in closed-loop stabilization can be attained provided that the feedback controller ignores observations during small time intervals.

15. D. Nesic and E.D. Sontag. Input-to-state stabilization of linear systems with positive outputs. Systems Control Lett., 35(4):245-255, 1998. [PDF] Keyword(s): input to state stability, ISS, stabilization.
Abstract:
 This paper considers the problem of stabilization of linear systems for which only the magnitudes of outputs are measured. It is shown that, if a system is controllable and observable, then one can find a stabilizing controller, which is robust with respect to observation noise (in the ISS sense).

16. F. H. Clarke, Y.S. Ledyaev, E.D. Sontag, and A.I. Subbotin. Asymptotic controllability implies feedback stabilization. IEEE Trans. Automat. Control, 42(10):1394-1407, 1997. [PDF]
Abstract:
 It is shown that every asymptotically controllable system can be stabilized by means of some (discontinuous) feedback law. One of the contributions of the paper is in defining precisely the meaning of stabilization when the feedback rule is not continuous. The main ingredients in our construction are: (a) the notion of control-Lyapunov function, (b) methods of nonsmooth analysis, and (c) techniques from positional differential games.

17. Y. Yang, E.D. Sontag, and H.J. Sussmann. Global stabilization of linear discrete-time systems with bounded feedback. Systems Control Lett., 30(5):273-281, 1997. [PDF] [doi:http://dx.doi.org/10.1016/S0167-6911(97)00021-2] Keyword(s): discrete-time, saturation, bounded inputs.
Abstract:
 This paper deals with the problem of global stabilization of linear discrete time systems by means of bounded feedback laws. The main result proved is an analog of one proved for the continuous time case by the authors, and shows that such stabilization is possible if and only if the system is stabilizable with arbitrary controls and the transition matrix has spectral radius less or equal to one. The proof provides in principle an algorithm for the construction of such feedback laws, which can be implemented either as cascades or as parallel connections (single hidden layer neural networks'') of simple saturation functions.

18. W. Liu, Y. Chitour, and E.D. Sontag. On finite-gain stabilizability of linear systems subject to input saturation. SIAM J. Control Optim., 34(4):1190-1219, 1996. [PDF] [doi:http://dx.doi.org/10.1137/S0363012994263469] Keyword(s): saturation, bounded inputs.
Abstract:
 This paper deals with (global) finite-gain input/output stabilization of linear systems with saturated controls. For neutrally stable systems, it is shown that the linear feedback law suggested by the passivity approach indeed provides stability, with respect to every Lp-norm. Explicit bounds on closed-loop gains are obtained, and they are related to the norms for the respective systems without saturation. These results do not extend to the class of systems for which the state matrix has eigenvalues on the imaginary axis with nonsimple (size >1) Jordan blocks, contradicting what may be expected from the fact that such systems are globally asymptotically stabilizable in the state-space sense; this is shown in particular for the double integrator.

19. Y. Lin, E.D. Sontag, and Y. Wang. Input to state stabilizability for parametrized families of systems. Internat. J. Robust Nonlinear Control, 5(3):187-205, 1995. [PDF] Keyword(s): ISS, stabilization.
Abstract:
 This paper studies various stability issues for parameterized families of systems, including problems of stabilization with respect to sets. The study of such families is motivated by robust control applications. A Lyapunov-theoretic necessary and sufficient characterization is obtained for a natural notion of robust uniform set stability; this characterization allows replacing ad hoc conditions found in the literature by more conceptual stability notions. We then use these techniques to establish a result linking state space stability to input to state'' (bounded-input bounded-state) stability. In addition, the preservation of stabilizability under certain types of cascade interconnections is analyzed.

20. E.D. Sontag. Control of systems without drift via generic loops. IEEE Trans. Automat. Control, 40(7):1210-1219, 1995. [PDF] Keyword(s): stabilization, non-holonomic systems, path-planning, systems without drift, nonlinear control, controllability.
Abstract:
 This paper proposes a simple numerical technique for the steering of arbitrary analytic systems with no drift. It is based on the generation of "nonsingular loops" which allow linearized controllability along suitable trajetories. Once such loops are available, it is possible to employ standard Newton or steepest descent methods, as classically done in numerical control. The theoretical justification of the approach relies on recent results establishing the genericity of nonsingular controls, as well as a simple convergence lemma.

21. E.D. Sontag. On the input-to-state stability property. European J. Control, 1:24-36, 1995. [PDF] Keyword(s): input to state stability, ISS.
Abstract:
 The "input to state stability" (ISS) property provides a natural framework in which to formulate notions of stability with respect to input perturbations. In this expository paper, we review various equivalent definitions expressed in stability, Lyapunov-theoretic, and dissipation terms. We sketch some applications to the stabilization of cascades of systems and of linear systems subject to control saturation.

22. H.J. Sussmann, E.D. Sontag, and Y. Yang. A general result on the stabilization of linear systems using bounded controls. IEEE Trans. Automat. Control, 39(12):2411-2425, 1994. [PDF] Keyword(s): saturation, neural networks, global stability, nonlinear stability, bounded inputs.
Abstract:
 We present two constructions of controllers that globally stabilize linear systems subject to control saturation. We allow essentially arbitrary saturation functions. The only conditions imposed on the system are the obvious necessary ones, namely that no eigenvalues of the uncontrolled system have positive real part and that the standard stabilizability rank condition hold. One of the constructions is in terms of a "neural-network type" one-hidden layer architecture, while the other one is in terms of cascades of linear maps and saturations.

23. E.D. Sontag. Feedback stabilization using two-hidden-layer nets. IEEE Trans. Neural Networks, 3:981-990, 1992. [PDF] Keyword(s): machine learning, neural networks, feedback stabilization.
Abstract:
 This paper compares the representational capabilities of one hidden layer and two hidden layer nets consisting of feedforward interconnections of linear threshold units. It is remarked that for certain problems two hidden layers are required, contrary to what might be in principle expected from the known approximation theorems. The differences are not based on numerical accuracy or number of units needed, nor on capabilities for feature extraction, but rather on a much more basic classification into "direct" and "inverse" problems. The former correspond to the approximation of continuous functions, while the latter are concerned with approximating one-sided inverses of continuous functions - and are often encountered in the context of inverse kinematics determination or in control questions. A general result is given showing that nonlinear control systems can be stabilized using two hidden layers, but not in general using just one.

24. E.D. Sontag. Input/output and state-space stability. In New trends in systems theory (Genoa, 1990), volume 7 of Progr. Systems Control Theory, pages 684-691. Birkhäuser Boston, Boston, MA, 1991. [PDF] Keyword(s): input to state stability, input to state stability.
Abstract:
 This conference paper reviews various results relating state-space (Lyapunov) stabilization and exponential stabilization to several notions of input/output or bounded-input bounded-output stabilization. It also provides generalizations of some of these results to systems with saturating controls. Some of these latter results were not included in journal papers.

25. Y. Lin and E.D. Sontag. A universal formula for stabilization with bounded controls. Systems Control Lett., 16(6):393-397, 1991. [PDF] [doi:http://dx.doi.org/10.1016/0167-6911(91)90111-Q] Keyword(s): saturation, bounded inputs.
Abstract:
 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.

26. E.D. Sontag. Constant McMillan degree and the continuous stabilization of families of transfer matrices. In Control of uncertain systems (Bremen, 1989), volume 6 of Progr. Systems Control Theory, pages 289-295. Birkhäuser Boston, Boston, MA, 1990. [PDF] Keyword(s): systems over rings.

27. E.D. Sontag. Feedback stabilization of nonlinear systems. In Robust control of linear systems and nonlinear control (Amsterdam, 1989), volume 4 of Progr. Systems Control Theory, pages 61-81. Birkhäuser Boston, Boston, MA, 1990. [PDF]
Abstract:
 This paper surveys some well-known facts as well as some recent developments on the topic of stabilization of nonlinear systems. (NOTE: figures are not included in file; they were pasted-in.)

28. E.D. Sontag. Further facts about input to state stabilization. IEEE Trans. Automat. Control, 35(4):473-476, 1990. [PDF] Keyword(s): input to state stability, ISS, stabilization.
Abstract:
 Previous results about input to state stabilizability are shown to hold even for systems which are not linear in controls, provided that a more general type of feedback be allowed. Applications to certain stabilization problems and coprime factorizations, as well as comparisons to other results on input to state stability, are also briefly discussed.d local minima may occur, if the data are not separable and sigmoids are used.

29. 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.
Abstract:
 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.

30. E.D. Sontag. Smooth stabilization implies coprime factorization. IEEE Trans. Automat. Control, 34(4):435-443, 1989. [PDF] Keyword(s): input to state stability, ISS, input to state stability.
Abstract:
 This paper shows that coprime right factorizations exist for the input to state mapping of a continuous time nonlinear system provided that the smooth feedback stabilization problem be solvable for this system. In particular, it follows that feedback linearizable systems admit such factorizations. In order to establish the result a Lyapunov-theoretic definition is proposed for bounded input bounded output stability. The main technical fact proved relates the notion of stabilizability studied in the state space nonlinear control literature to a notion of stability under bounded control perturbations analogous to those studied in operator theoretic approaches to systems; it states that smooth stabilization implies smooth input-to-state stabilization. (Note: This is the original ISS paper, but the ISS results have been much improved in later papers. The material on coprime factorizations is still of interest, but the 89 CDC paper has some improvements and should be read too.)

31. E.D. Sontag and H.J. Sussmann. Further comments on the stabilizability of the angular velocity of a rigid body. Systems Control Lett., 12(3):213-217, 1989. [PDF] [doi:http://dx.doi.org/10.1016/0167-6911(89)90052-2] Keyword(s): satellite control, feedback stabilization.
Abstract:
 We prove that the angular velocity equations can be smoothly stabilized with a single torque controller for bodies having an axis of symmetry. This complements a recent result of Aeyels and Szafranski.

32. E.D. Sontag. An explicit construction of the equilinearization controller. In C.I. Byrnes, C.F. Martin, and R. Saek, editors, Analysis and Control of Nonlinear Systems, pages 483-492. North Holland, Amsterdam, 1988. [PDF]
Abstract:
 This paper provides further results about the equilinearization method of control design recently introduced by the author. A simplified derivation of the controller is provided, as well as a theorem on local stabilization along reference trajectories.

33. E.D. Sontag. Continuous stabilizers and high-gain feedback. IMA Journal of Mathematical Control and Information, 3:237-253, 1986. [PDF] Keyword(s): adaptive control, systems over rings.
Abstract:
 A controller is shown to exist, universal for the family of all systems of fixed dimension n, and m controls, which stabilizes those systems that are stabilizable, if certain gains are large enough. The controller parameters are continuous, in fact polynomial, functions of the entries of the plant. As a consequence, a result is proved on polynomial stabilization of families of systems.

34. E.D. Sontag. An introduction to the stabilization problem for parametrized families of linear systems. In Linear algebra and its role in systems theory (Brunswick, Maine, 1984), volume 47 of Contemp. Math., pages 369-400. Amer. Math. Soc., Providence, RI, 1985. [PDF] Keyword(s): systems over rings.
Abstract:
 This paper provides an introduction to definitions and known facts relating to the stabilization of parametrized families of linear systems using static and dynamic controllers. New results are given in the rational and polynomial cases.

35. E.D. Sontag. Parametric stabilization is easy. Systems Control Lett., 4(4):181-188, 1984. [PDF] Keyword(s): systems over rings.
Abstract:
 A polynomially parametrized family of continuous-time controllable linear systems is always stabilizable by polynomially parametrized feedback. (Note: appendix had a MACSYMA computation. I cannot find the source file for that. Please look at journal if interested, but this is not very important. Also, two figures involving root loci are not in the web version.)

36. R.T. Bumby and E.D. Sontag. Stabilization of polynomially parametrized families of linear systems. The single-input case. Systems Control Lett., 3(5):251-254, 1983. [PDF] Keyword(s): systems over rings.
Abstract:
 Given a continuous-time family of finite dimensional single input linear systems, parametrized polynomially, such that each of the systems in the family is controllable, there exists a polynomially parametrized control law making each of the systems in the family stable.

37. E.D. Sontag. Abstract regulation of nonlinear systems: stabilization. In Feedback control of linear and nonlinear systems (Bielefeld/Rome, 1981), volume 39 of Lecture Notes in Control and Inform. Sci., pages 227-243. Springer, Berlin, 1982.

38. E.D. Sontag. Conditions for abstract nonlinear regulation. Inform. and Control, 51(2):105-127, 1981. [PDF] Keyword(s): feedback stabilization.
Abstract:
 A paper that introduces a separation principle for general finite dimensional analytic continuous-time systems, proving the equivalence between existence of an output regulator (which is an abstract dynamical system) and certain "0-detectability" and asymptotic controllability assumptions.

 Conference articles
1. M. Sznaier, A. Olshevsky, and E.D. Sontag. The role of systems theory in control oriented learning. In Proc. 25th Int. Symp. Mathematical Theory of Networks and Systems (MTNS 2022), 2022. Note: To appear.[PDF] Keyword(s): control oriented learning, neural networks, reinforcement learning, feedback control, machine learning.
Abstract:
 Systems theory can play an important in unveiling fundamental limitations of learning algorithms and architectures when used to control a dynamical system, and in suggesting strategies for overcoming these limitations. As an example, a feedforward neural network cannot stabilize a double integrator using output feedback. Similarly, a recurrent NN with differentiable activation functions that stabilizes a non-strongly stabilizable system must be itself open loop unstable, a fact that has profound implications for training with noisy, finite data. A potential solution to this problem, motivated by results on stabilization with periodic control, is the use of neural nets with periodic resets, showing that indeed systems theoretic analysis is instrumental in developing architectures capable of controlling certain classes of unstable systems. This short conference paper also argues that when the goal is to learn control oriented models, the loss function should reflect closed loop, rather than open loop model performance, a fact that can be accomplished by using gap-metric motivated loss functions.

2. M. Malisoff, L. Rifford, and E.D. Sontag. Remarks on input to state stabilization. In Proc. IEEE Conf. Decision and Control, Maui, Dec. 2003, IEEE Publications, 2003, pages 1053-1058, 2003. [PDF] Keyword(s): nonlinear control, feedback stabilization.

3. M. Arcak, D. Angeli, and E.D. Sontag. Stabilization of cascades using integral input-to-state stability. In Proc. IEEE Conf. Decision and Control, Orlando, Dec. 2001, IEEE Publications, 2001, pages 3814-3819, 2001. Keyword(s): nonlinear control, feedback stabilization, input to state stability.

4. B.P. Ingalls, D. Angeli, E.D. Sontag, and Y. Wang. Asymptotic characterizations of IOSS. In Proc. IEEE Conf. Decision and Control, Orlando, Dec. 2001, IEEE Publications, 2001, pages 881-886, 2001. Keyword(s): nonlinear control, feedback stabilization, input to state stability.

5. 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.
Abstract:
 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.

6. X. Bao, Z. Lin, and E.D. Sontag. Some new results on finite gain $l_p$ stabilization of discrete-time linear systems subject to actuator saturation. In Proc. IEEE Conf. Decision and Control, Tampa, Dec. 1998, IEEE Publications, 1998, pages 4628-4629, 1998. Keyword(s): saturation, bounded inputs.

7. Y.S. Ledyaev and E.D. Sontag. Stabilization under measurement noise: Lyapunov characterization. In Proc. American Control Conf., Philadelphia, June 1998, pages 1658-166, 1998.

8. D. Nesic and E.D. Sontag. Output stabilization of nonlinear systems: Linear systems with positive outputs as a case study. In Proc. IEEE Conf. Decision and Control, Tampa, Dec. 1998, IEEE Publications, 1998, pages 885-890, 1998.

9. 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.

10. 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.

11. Y.S. Ledyaev and E.D. Sontag. A remark on robust stabilization of general asymptotically controllable systems. In Proc. Conf. on Information Sciences and Systems (CISS 97), Johns Hopkins, Baltimore, MD, March 1997, pages 246-251, 1997. [PDF]
Abstract:
 We showned in another recent paper that any asymptotically controllable system can be stabilized by means of a certain type of discontinuous feedback. The feedback laws constructed in that work are robust with respect to actuator errors as well as to perturbations of the system dynamics. A drawback, however, is that they may be highly sensitive to errors in the measurement of the state vector. This paper addresses this shortcoming, and shows how to design a dynamic hybrid stabilizing controller which, while preserving robustness to external perturbations and actuator error, is also robust with respect to measurement error. This new design relies upon a controller which incorporates an internal model of the system driven by the previously constructed feedback.

12. 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.

13. 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.
Abstract:
 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.

14. 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.
Abstract:
 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.

15. Y. Lin, E.D. Sontag, and Y. Wang. Lyapunov-function characterizations of stability and stabilization for parameterized families of systems. In Proc. IEEE Conf. Decision and Control, San Antonio, Dec. 1993, IEEE Publications, 1993, pages 1978-1983, 1993.

16. H.J. Sussmann, E.D. Sontag, and Y. Yang. A general result on the stabilization of linear systems using bounded controls. In Proc. IEEE Conf. Decision and Control, San Antonio, Dec. 1993, IEEE Publications, 1993, pages 1802-1807, 1993. Keyword(s): saturation, bounded inputs.

17. Y. Yang and E.D. Sontag. Stabilization with saturated actuators, a worked example: F-8 longitudinal flight control. In Proc. 1993 IEEE Conf. on Aerospace Control Systems, Thousand Oaks, CA, May 1993, pages 289-293, 1993. [PDF] Keyword(s): saturation, bounded inputs, aircraft, airplanes.
Abstract:
 This paper develops in detail an explicit design for control under saturation limits for the linearized equations of longitudinal flight control for an F-8 aircraft, and tests the obtained controller on the original nonlinear model.

18. E.D. Sontag and Y. Lin. Stabilization with respect to noncompact sets: Lyapunov characterizations and effect of bounded inputs. In Nonlinear Control Systems Design 1992, IFAC Symposia Series, M. Fliess Ed., Pergamon Press, Oxford, 1993, pages 43-49, 1992. Note: Also in Proc. Nonlinear Control Systems Design Symp., Bordeaux, June 1992,(M. Fliess, Ed.), IFAC Publications, pp. 9--14. [PDF] Keyword(s): saturation, bounded inputs.

19. Y. Yang, H.J. Sussmann, and E.D. Sontag. Stabilization of linear systems with bounded controls. In Nonlinear Control Systems Design 1992, IFAC Symposia Series, 1993, M. Fliess Ed., Pergamon Press, Oxford, 1993, pages 51-56, 1992. Note: Also in Proc. Nonlinear Control Systems Design Symp., Bordeaux, June 1992,(M. Fliess, Ed.), IFAC Publications, pp. 15-20.Keyword(s): saturation, bounded inputs.

20. Y. Lin and E.D. Sontag. Further universal formulas for Lyapunov approaches to nonlinear stabilization. In Proc. Conf. Inform. Sci. and Systems, John Hopkins University, March 1991, pages 541-546, 1991.

21. E.D. Sontag. Feedback Stabilization Using Two-Hidden-Layer Nets. In Proc. Amer. Automatic Control Conf. , Boston, June 1991, pages 815-820, 1991.

22. E.D. Sontag. Remarks on stabilization and input-to-state stability. In Proceedings of the 28th IEEE Conference on Decision and Control, Vol. 1--3 (Tampa, FL, 1989), New York, pages 1376-1378, 1989. IEEE. [PDF] Keyword(s): input to state stability, ISS, stabilization.
Abstract:
 This paper describes how notions of input-to-state stabilization are useful when stabilizing cascades of systems. The simplest result along these lines is local, and it states that a cascade of two locally asymptotically stable systems is again asystable. A global result is obtained if both systems have the origin as a globally asymptotically stable state and the "converging input bounded state" property holds for the second system. Relations to input to state stability and the "bounded input bounded state" property as mentioned as well.

23. E.D. Sontag. Some connections between stabilization and factorization. In Proceedings of the 28th IEEE Conference on Decision and Control, Vol. 1--3 (Tampa, FL, 1989), New York, pages 990-995, 1989. IEEE. [PDF]
Abstract:
 Coprime right fraction representations are obtained for nonlinear systems defined by differential equations, under assumptions of stabilizability and detectability. A result is also given on left (not necessarily coprime) factorizations.

24. E.D. Sontag. Stabilizability, i/o stability, and coprime factorizations. In Proc. IEEE Conf. Decision and Control, Austin, Dec. 1988, pages 457-458, 1988. Keyword(s): input to state stability, coprime factorizations, stabilization.

25. E.D. Sontag. Abstract regulation of nonlinear systems: Stabilization, Part II. In Proc.Princeton Conf.on Information Sciences and Systems, Princeton, March 1982, pages 431-435, 1982. Keyword(s): feedback stabilization.

26. E.D. Sontag and H.J. Sussmann. Remarks on continuous feedback. In Proc. IEEE Conf. Decision and Control, Albuquerque, Dec.1980, pages 916-921, 1980. [PDF] Keyword(s): feedback stabilization.
Abstract:
 We show that, in general, it is impossible to stabilize a controllable system by means of a continuous feedback, even if memory is allowed. No optimality considerations are involved. All state spaces are Euclidean spaces, so no obstructions arising from the state space topology are involved either. For one dimensional state and input, we prove that continuous stabilization with memory is always possible. (This is an old conference paper, never published in journal form but widely cited nonetheless. Warning: file is very large, since it was scanned.)

 Internal reports
1. E.D. Sontag. A remark on incoherent feedforward circuits as change detectors and feedback controllers. Technical report, arXiv:1602.00162, 2016. [PDF] Keyword(s): scale invariance, fold change detection, T cells, incoherent feedforward loops, immunology, incoherent feedforward loop, feedforward, IFFL.
Abstract:
 This note analyzes incoherent feedforward loops in signal processing and control. It studies the response properties of IFFL's to exponentially growing inputs, both for a standard version of the IFFL and for a variation in which the output variable has a positive self-feedback term. It also considers a negative feedback configuration, using such a device as a controller. It uncovers a somewhat surprising phenomenon in which stabilization is only possible in disconnected regions of parameter space, as the controlled system's growth rate is varied.

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