Publications about 'optimal control'
Articles in journal or book chapters
  1. A.M. Weinstein and E.D. Sontag. Modeling proximal tubule cell homeostasis: Tracking changes in luminal flow. Bulletin of Mathematical Biology, 71:1285-1322, 2009. [PDF]
    During normal kidney function, there are are routinely wide swings in proximal tubule fluid flow and proportional changes in Na+ reabsorption across tubule epithelial cells. This "glomerulotubular balance" occurs in the absence of any substantial change in cell volume, and is thus a challenge to coordinate luminal membrane solute entry with peritubular membrane solute exit. In this work, linear optimal control theory is applied to generate a configuration of regulated transporters that could achieve this result. A previously developed model of rat proximal tubule epithelium is linearized about a physiologic reference condition; the approximate linear system is recast as a dynamical system; and a Riccati equation is solved to yield optimal linear feedback that stabilizes Na+ flux, cell volume, and cell pH. This optimal feedback control is largely consigned to three physiologic variables, cell volume, cell electrical potential, and lateral intercellular hydrostatic pressure. Transport modulation by cell volume stabilizes cell volume; transport modulation by electrical potential or interspace pressure act to stabilize Na+ flux and cell pH. This feedback control is utilized in a tracking problem, in which reabsorptive Na+ flux varies over a factor of two. The resulting control parameters consist of two terms, an autonomous term and a feedback term, and both terms include transporters on both luminal and peritubular cell membranes. Overall, the increase in Na+ flux is achieved with upregulation of luminal Na+/H+ exchange and Na+-glucose cotransport, with increased peritubular Na+-3HCO_3- and K+-Cl- cotransport, and with increased Na+,K+-ATPase activity. The configuration of activated transporters emerges as testable hypothesis of the molecular basis for glomerulotubular balance. It is suggested that the autonomous control component at each cell membrane could represent the cytoskeletal effects of luminal flow.

  2. M. Chyba, N. E. Leonard, and E.D. Sontag. Singular trajectories in multi-input time-optimal problems: Application to controlled mechanical systems. Journal of Dynamical and Control Systems, 9(1):103-129, 2003. [PDF] [doi:] Keyword(s): optimal control.
    This paper addresses the time-optimal control problem for a class of control systems which includes controlled mechanical systems with possible dissipation terms. The Lie algebras associated with such mechanical systems enjoy certain special properties. These properties are explored and are used in conjunction with the Pontryagin maximum principle to determine the structure of singular extremals and, in particular, time-optimal trajectories. The theory is illustrated with an application to a time-optimal problem for a class of underwater vehicles.

  3. E.D. Sontag and H.J. Sussmann. Time-optimal control of manipulators (reprint of 1986 IEEE Int Conf on Robotics and Automation paper. In M.W. Spong, F.L. Lewis, and C.T. Abdallah, editors, Robot Control, pages 266-271. IEEE Press, New York, 1993. Keyword(s): robotics, optimal control.

Conference articles
  1. J.M. Greene, C. Sanchez-Tapia, and E.D. Sontag. Control structures of drug resistance in cancer chemotherapy. In Proc. 2018 IEEE Conf. Decision and Control, pages 5195-5200, 2018. [PDF]
    The primary factor limiting the success of chemotherapy in cancer treatment is the phenomenon of drug resistance. This work extends the work reported in "A mathematical approach to distinguish spontaneous from induced evolution of drug resistance during cancer treatment" by introducing a time-optimal control problem that is analyzed utilizing differential-geometric techniques: we seek a treatment protocol which maximizes the time of treatment until a critical tumor size is reached. The general optimal control structure is determined as a combination of both bang-bang and path-constrained arcs. Numerical results are presented which demonstrate decreasing treatment efficacy as a function of the ability of the drug to induce resistance. Thus, drug-induced resistance may dramatically effect the outcome of chemotherapy, implying that factors besides cytotoxicity should be considered when designing treatment regimens.

  2. M. Chyba, N.E. Leonard, and E.D. Sontag. Optimality for underwater vehicles. In Proc. IEEE Conf. Decision and Control, Orlando, Dec. 2001,IEEE Publications, 2001, pages 4204-4209, 2001. [PDF] Keyword(s): optimal control.

  3. M. Chyba, N.E. Leonard, and E.D. Sontag. Time-optimal control for underwater vehicles. In N.E. Leonard and R. Ortega, editors, Lagrangian and Hamiltonian Methods for Nonlinear Control, pages 117-122, 2000. Pergamon Press, Oxford. [PDF]

  4. E.D. Sontag. An abstract approach to dissipation. In Proc. IEEE Conf. Decision and Control, New Orleans, Dec. 1995, IEEE Publications, 1995, pages 2702-2703, 1995. Note: Full version, never submitted, is here: [PDF]
    We suggest that a very natural mathematical framework for the study of dissipation -in the sense of Willems, Moylan and Hill, and others- is that of indefinite quasimetric spaces. Several basic facts about dissipative systems are seen to be simple consequences of the properties of such spaces. Quasimetric spaces provide also one natural context for optimal control problems, and even for "gap" formulations of robustness.

  5. 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.

  6. E.D. Sontag. Remarks on the time-optimal control of a class of Hamiltonian systems. In Proceedings of the 28th IEEE Conference on Decision and Control, Vol. 1--3 (Tampa, FL, 1989), New York, pages 217-221, 1989. IEEE. [PDF] Keyword(s): robotics, optimal control.
    This paper introduces a subclass of Hamiltonian control systems motivated by mechanical models. It deals with time-optimal control problems. The main results characterize regions of the state space where singular trajectories cannot exist, and provide high-order conditions for optimality.

  7. E.D. Sontag and H.J. Sussmann. Time-optimal control of manipulators. In Proc. IEEE Int.Conf.on Robotics and Automation, San Francisco, April 1986, pages 1692-1697, 1986. [PDF] Keyword(s): robotics, optimal control.
    This paper studies time-optimal control questions for a certain class of nonlinear systems. This class includes a large number of mechanical systems, in particular, rigid robotic manipulators with torque constraints. As nonlinear systems, these systems have many properties that are false for generic systems of the same dimensions.

  8. E.D. Sontag and H.J. Sussmann. Remarks on the time-optimal control of two-link manipulators. In Proc. IEEE Conf. Dec. and Control, 1985, pages 1646-1652, 1985. [PDF] Keyword(s): optimal control, robotics.

Internal reports
  1. J.L. Gevertz, J.M. Greene, and E.D. Sontag. Validation of a mathematical model of cancer incorporating spontaneous and induced evolution to drug resistance. Technical report, Cold Spring Harbor Laboratory, 2019. Note: BioRxiv preprint 10.1101/2019.12.27.889444. Keyword(s): cancer heterogeneity, phenotypic variation, nonlinear systems, epigenetics, optimal control theory.
    This paper continues the study of a model which was introduced in earlier work by the authors to study spontaneous and induced evolution to drug resistance under chemotherapy. The model is fit to existing experimental data, and is then validated on additional data that had not been used when fitting. In addition, an optimal control problem is studied numerically.

  2. J.M. Greene, C. Sanchez-Tapia, and E.D. Sontag. Mathematical details on a cancer resistance model. Technical report, bioRxiv 2018/475533, 2018. [PDF] Keyword(s): identifiability, drug resistance, chemotherapy, optimal control theory, singular controls.
    The primary factor limiting the success of chemotherapy in cancer treatment is the phenomenon of drug resistance. We have recently introduced a framework for quantifying the effects of induced and non-induced resistance to cancer chemotherapy . In this work, the control structure is precisely characterized as a concatenation of bang-bang and path-constrained arcs via the Pontryagin Maximum Principle and differential Lie techniques. A structural identfiability analysis is also presented, demonstrating that patient-specfic parameters may be measured and thus utilized in the design of optimal therapies prior to the commencement of therapy.



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