1. R. Alur, T.A. Henzinger, and E.D. Sontag. Hybrid Systems III. Verification and Control (edited book). Springer Verlag, Berlin, 1996. Note: (LNCS 1066).

1. M. Chaves, E.D. Sontag, and R. Albert. Methods of robustness analysis for Boolean models of gene control networks. IET Systems Biology, 153:154-167, 2006. [PDF] Keyword(s): systems biology, biochemical networks, boolean systems, identifiability, robust, robustness, geometry, Boolean, segment polarity network, gene and protein networks, hybrid systems. Abstract:

   As a discrete approach to genetic regulatory networks, Boolean models provide an essential qualitative description of the structure of interactions among genes and proteins. Boolean models generally assume only two possible states (expressed or not expressed) for each gene or protein in the network as well as a high level of synchronization among the various regulatory processes. In this paper, we discuss and compare two possible methods of adapting qualitative models to incorporate the continuous-time character of regulatory networks. The first method consists of introducing asynchronous updates in the Boolean model. In the second method, we adopt the approach introduced by L. Glass to obtain a set of piecewise linear differential equations which continuously describe the states of each gene or protein in the network. We apply both methods to a particular example: a Boolean model of the segment polarity gene network of Drosophila melanogaster. We analyze the dynamics of the model, and provide a theoretical characterization of the model's gene pattern prediction as a function of the timescales of the various processes.




2. B. DasGupta and E.D. Sontag. A polynomial-time algorithm for checking equivalence under certain semiring congruences motivated by the state-space isomorphism problem for hybrid systems. Theor. Comput. Sci., 262(1-2):161-189, 2001. [PDF] [doi:http://dx.doi.org/10.1016/S0304-3975(00)00188-2] Keyword(s): hybrid systems, computational complexity. Abstract:

   The area of hybrid systems concerns issues of modeling, computation, and control for systems which combine discrete and continuous components. The subclass of piecewise linear (PL) systems provides one systematic approach to discrete-time hybrid systems, naturally blending switching mechanisms with classical linear components. PL systems model arbitrary interconnections of finite automata and linear systems. Tools from automata theory, logic, and related areas of computer science and finite mathematics are used in the study of PL systems, in conjunction with linear algebra techniques, all in the context of a "PL algebra" formalism. PL systems are of interest as controllers as well as identification models. Basic questions for any class of systems are those of equivalence, and, in particular, if state spaces are equivalent under a change of variables. This paper studies this state-space equivalence problem for PL systems. The problem was known to be decidable, but its computational complexity was potentially exponential; here it is shown to be solvable in polynomial-time.




3. 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.




4. E.D. Sontag. Interconnected automata and linear systems: a theoretical framework in discrete-time. In R. Alur, T.A. Henzinger, and E.D. Sontag, editors, Proceedings of the DIMACS/SYCON workshop on Hybrid systems III : verification and control, pages 436-448. Springer-Verlag New York, Inc., Secaucus, NJ, USA, 1996. [PDF] Keyword(s): hybrid systems. Abstract:

   This paper summarizes the definitions and several of the main results of an approach to hybrid systems, which combines finite automata and linear systems, developed by the author in the early 1980s. Some related more recent results are briefly mentioned as well.




5. E.D. Sontag. Remarks on piecewise-linear algebra. Pacific J. Math., 98(1):183-201, 1982. [PDF] Keyword(s): hybrid systems, piecewise linear systems. Abstract:

   Algebraic study of functions defined by piecewise linear (generally discontinuous) equations. File obtained by scanning a reprint.




6. E.D. Sontag. Nonlinear regulation: the piecewise linear approach. IEEE Trans. Automat. Control, 26(2):346-358, 1981. [PDF] Keyword(s): hybrid systems. Abstract:

   Development of an approach to nonlinear control based on mixtures of linear systems and finite automata. File obtained by scanning.

1. M. Chaves, E.D. Sontag, and R. Albert. Structure and timescale analysis in genetic regulatory networks. In Proc. IEEE Conf. Decision and Control, San Diego, Dec. 2006, pages 2358-2363, 2006. IEEE. [PDF] Keyword(s): genetic regulatory networks, Boolean systems, hybrid systems. Abstract:

   This work is concerned with the study of the robustness and fragility of gene regulation networks to variability in the timescales of the distinct biological processes involved. It explores and compares two methods: introducing asynchronous updates in a Boolean model, or integrating the Boolean rules in a continuous, piecewise linear model. As an example, the segment polarity network of the fruit fly is analyzed. A theoretical characterization is given of the model's ability to predict the correct development of the segmented embryo, in terms of the specific timescales of the various regulation interactions.




2. B. Dasgupta and E.D. Sontag. A polynomial-time algorithm for an equivalence problem which arises in hybrid systems theory. In Proc. IEEE Conf. Decision and Control, Tampa, Dec. 1998, IEEE Publications, 1998, pages 1629-1634, 1998.

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