Publications about 'differential inclusions'
Articles in journal or book chapters
  1. D. Angeli, P. de Leenheer, and E.D. Sontag. Chemical networks with inflows and outflows: A positive linear differential inclusions approach. Biotechnology Progress, 25:632-642, 2009. [PDF] Keyword(s): biochemical networks, fluxes, differential inclusions, positive systems, Petri nets, persistence, switched systems.
    Certain mass-action kinetics models of biochemical reaction networks, although described by nonlinear differential equations, may be partially viewed as state-dependent linear time-varying systems, which in turn may be modeled by convex compact valued positive linear differential inclusions. A result is provided on asymptotic stability of such inclusions, and applied to biochemical reaction networks with inflows and outflows. Included is also a characterization of exponential stability of general homogeneous switched systems

  2. J. L. Mancilla-Aguilar, R. Garcža, E.D. Sontag, and Y. Wang. On the representation of switched systems with inputs by perturbed control systems. Nonlinear Anal., 60(6):1111-1150, 2005. [PDF]
    This paper provides representations of switched systems described by controlled differential inclusions, in terms of perturbed control systems. The control systems have dynamics given by differential equations, and their inputs consist of the original controls together with disturbances that evolve in compact sets; their sets of maximal trajectories contain, as a dense subset, the set of maximal trajectories of the original system. Several applications to control theory, dealing with properties of stability with respect to inputs and of detectability, are derived as a consequence of the representation theorem.

  3. J. L. Mancilla-Aguilar, R. Garcža, 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.

  4. D. Angeli, B.P. Ingalls, E.D. Sontag, and Y. Wang. Uniform global asymptotic stability of differential inclusions. J. Dynam. Control Systems, 10(3):391-412, 2004. [PDF] [doi:] Keyword(s): differential inclusions.
    The stability of differential inclusions defined by locally Lipschitz compact valued maps is addressed. It is shown that if such a differential inclusion is globally asymptotically stable, then in fact it is uniformly globally asymptotically stable (with respect to initial states in compacts). This statement is trivial for differential equations, but here we provide the extension to compact (not necessarily convex) valued differential inclusions. The main result is presented in a context which is useful for control-theoretic applications: a differential inclusion with two outputs is considered, and the result applies to the property of global error detectability.

  5. B.P. Ingalls, E.D. Sontag, and Y. Wang. An infinite-time relaxation theorem for differential inclusions. Proc. Amer. Math. Soc., 131(2):487-499, 2003. [PDF]
    The fundamental relaxation result for Lipschitz differential inclusions is the Filippov-Wazewski Relaxation Theorem, which provides approximations of trajectories of a relaxed inclusion on finite intervals. A complementary result is presented, which provides approximations on infinite intervals, but does not guarantee that the approximation and the reference trajectory satisfy the same initial condition.

Conference articles
  1. B.P. Ingalls, E.D. Sontag, and Y. Wang. A relaxation theorem for differential inclusions with applications to stability properties. In D. Gilliam and J. Rosenthal, editors, Mathematical Theory of Networks and Systems, Electronic Proceedings of MTNS-2002 Symposium held at the University of Notre Dame, August 2002, 2002. Note: (12 pages). [PDF]
    The fundamental Filippov--Wazwski Relaxation Theorem states that the solution set of an initial value problem for a locally Lipschitz inclusion is dense in the solution set of the same initial value problem for the corresponding relaxation inclusion on compact intervals. In a recent paper of ours, a complementary result was provided for inclusions with finite dimensional state spaces which says that the approximation can be carried out over non-compact or infinite intervals provided one does not insist on the same initial values. This note extends the infinite-time relaxation theorem to the inclusions whose state spaces are Banach spaces. To illustrate the motivations for studying such approximation results, we briefly discuss a quick application of the result to output stability and uniform output stability properties.

  2. B.P. Ingalls, E.D. Sontag, and Y. Wang. Measurement to error stability: a notion of partial detectability for nonlinear systems. In Proc. IEEE Conf. Decision and Control, Las Vegas, Dec. 2002, IEEE Publications, pages 3946-3951, 2002. [PDF] Keyword(s): input to state stability.
    For systems whose output is to be kept small (thought of as an error output), the notion of input to output stability (IOS) arises. Alternatively, when considering a system whose output is meant to provide information about the state (i.e. a measurement output), one arrives at the detectability notion of output to state stability (OSS). Combining these concepts, one may consider a system with two types of outputs, an error and a measurement. This leads naturally to a notion of partial detectability which we call measurement to error stability (MES). This property characterizes systems in which the error signal is detectable through the measurement signal. This paper provides a partial Lyapunov characterization of the MES property. A closely related property of stability in three measures (SIT) is introduced, which characterizes systems for which the error decays whenever it dominates the measurement. The SIT property is shown to imply MES, and the two are shown to be equivalent under an additional boundedness assumption. A nonsmooth Lyapunov characterization of the SIT property is provided, which yields the partial characterization of MES. The analysis is carried out on systems described by differential inclusions -- implicitly incorporating a disturbance input with compact value-set.



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