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We consider a class of bilinear weakly contractive systems that entrain to periodic excitations. Entrainment is important in many natural and artificial processes. For example, in order to function properly synchronous generators must entrain to the frequency of the electrical grid, and biological organisms must entrain to the 24h solar day. A dynamical system has a positive gain of entrainment (GOE) if entrainment also yields a larger output, on average. This property is important in many applications from the periodic operation of bioreactors to the periodic production of proteins during the cell cycle division process. We derive a closed-form formula for the GOE to first-order in the control perturbation. This is used to show that in the class of systems that we consider the GOE is always a higher-order phenomenon. We demonstrate the theoretical results using two applications: the master equation and a model from systems biology called the ribosome flow model, both with time-varying and periodic transition rates. [ABSTRACT FROM AUTHOR]

To link to full-text access for this article, visit this link: http://dx.doi.org/10.1016/j.jedc.2008.01.002 Byline: Lars Grune (a), Willi Semmler (b)(c) Abstract: The use of standard preferences for asset pricing has not been very successful in matching asset price characteristics, such as the risk-free interest rate, equity premium and the Sharpe ratio, to time series data. Behavioral finance has recently proposed more realistic preferences such as those with loss aversion. Research is starting to explore the implications of behaviorally founded preferences for asset price characteristics. Encouraged by some studies of Benartzi and Thaler [1995. Myopic loss aversion and the equity premium puzzle. The Quarterly Journal of Economics 110 (1), 73-92] and Barberis et al. [2001. Prospect theory and asset prices. Quarterly Journal of Economics CXVI (1), 1-53] we study asset pricing with loss aversion in a production economy. Here, we employ a stochastic growth model and use a stochastic version of a dynamic programming method with an adaptive grid scheme to compute the above mentioned asset price characteristics of a model with loss aversion in preferences. As our results show using loss aversion we get considerably better results than one usually obtains from pure consumption-based asset pricing models including the habit formation variant. Author Affiliation: (a) Mathematisches Institut, Fakultat fur Mathematik und Physik, Universitat Bayreuth, 95440 Bayreuth, Germany (b) Department of Economics, New School, 65 Fifth Avenue, New York, NY 10003, USA (c) Center for Empirical Macroeconomics, University of Bielefeld, Germany Article History: Received 11 August 2005; Accepted 9 January 2008

Receding horizon control is a well established approach for control of systems with constraints and nonlinearities. Optimization over an infinite time-horizon, which is often computationally intractable, is therein replaced by a sequence of finite horizon problems. This paper provides a method to quantify the performance degradation that comes with this approximation. Results are provided for problems both with and without terminal costs and constraints and for both exactly and practically asymptotically stabilizable systems. Index Terms--Dynamic programming, model predictive control (MPC), receding horizon control (RHC), suboptimality.

To link to full-text access for this article, visit this link: http://dx.doi.org/10.1016/j.jebo.2005.04.002 Byline: Lars Grune (a), Mika Kato (b), Willi Semmler (c) Keywords: Renewable resources; Ecological management; Optimal taxation; Bell equation; Dynamic programming Abstract: We study an ecological management problem where the interaction of the maximizing-welfare dynamic decision and the dynamics of the ecosystem admits multiple equilibria. We follow the example of a shallow lake by Brock, etc. Low loading preserves resilience of the ecosystem while high loading may lead to the deterioration of the ecosystem. We consider instruments of a regulatory agency that may help to maintain and enhance resilience by enlarging the domain of attraction of the low-pollution equilibrium. The global dynamics of all our model variants, without and with tax rates, are analytically studied by the Hamilton-Jacobi-Bellman method and numerically solved through dynamic programming. Author Affiliation: (a) Faculty of Mathematics, University of Bayreuth, Bayreuth, 95440 Bayreuth, Germany (b) Department of Economics, Howard University, ASB-B Bldg. Room 319, Washington, DC 20059, USA (c) Center for Empirical Macroeconomics, Bielefeld and New School University, New York, 65 Fifth Ave., New York, NY 10003, USA

An adaptive grid scheme that follows Grune is used to find the global solutions of discrete time Hamilton-Jacobi-Bellman equations, establishing local error estimates and developing an adapting iteration for the discretization of the grid space. The advantage of the use of the adaptive grid scheme is evident after computing the solutions of one- and two-dimensional economic models exhibiting steep curvature, complicated dynamics thresholds separating domains of attraction and periodic solutions. Deterministic and stochastic model variants are considered.

We present a new variant of the input-to-state stability (ISS) property which is based on using a one-dimensional dynamical system for building the class KL function for the decay estimate and for describing the influence of the perturbation. We show the relation to the original ISS formulation and describe characterizations by means of suitable Lyapunov functions. As applications, we derive quantitative results on stability margins for nonlinear systems and a quantitative version of a small gain theorem for nonlinear systems. Index Terms--Input-to-state dynamical stability (ISDS), Lyapunov function, nonlinear stability margin, small gain theorem.

A generalization of Zubov's theorem on representing the domain of attraction via the solution of a suitable partial differential equation is presented for the case of perturbed systems with a singular fixed point. For the construction it is necessary to consider solutions in the viscosity sense. As a consequence, maximal robust Lyapunov functions can be characterized as viscosity solutions. [ABSTRACT FROM AUTHOR]