Results 1  10
of
22
A regularization of Zubov's equation for robust domains of attraction
, 2000
"... We derive a method for the computation of robust domains of attraction based on a recent generalization of Zubov's theorem on representing robust domains of attraction for perturbed systems via the viscosity solution of a suitable partial differential equation. While a direct discretization of the e ..."
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Cited by 34 (5 self)
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We derive a method for the computation of robust domains of attraction based on a recent generalization of Zubov's theorem on representing robust domains of attraction for perturbed systems via the viscosity solution of a suitable partial differential equation. While a direct discretization of the equation leads to numerical difficulties due to a singularity at the stable equilibrium, a suitable regularization enables us to apply a standard discretization technique for HamiltonJacobiBellman equations. We present the resulting fully discrete scheme and show a numerical example.
Adaptive Choice of Grid and Time in Reinforcement Learning
 IN NIPS ’97: PROCEEDINGS OF THE 1997 CONFERENCE ON ADVANCES IN NEURAL INFORMATION PROCESSING SYSTEMS 10
, 1997
"... We propose local error estimates together with algorithms for adaptive aposteriori grid and time refinement in reinforcement learning. We consider a deterministic system with continuous state and time with infinite horizon discounted cost functional. For grid refinement we follow the procedure of ..."
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Cited by 15 (1 self)
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We propose local error estimates together with algorithms for adaptive aposteriori grid and time refinement in reinforcement learning. We consider a deterministic system with continuous state and time with infinite horizon discounted cost functional. For grid refinement we follow the procedure of numerical methods for the Bellmanequation. For time refinement we propose a new criterion, based on consistency estimates of discrete solutions of the Bellmanequation. We demonstrate, that an optimal ratio of time to space discretization is crucial for optimal learning rates and accuracy of the approximate optimal value function.
Using dynamic programming with adaptive grid scheme for optimal control problems in economics
 JOURNAL OF ECONOMIC DYNAMICS AND CONTROL, ELSEVIER
, 2004
"... The study of the solutions of dynamic models with optimizing agents has often been limited by a lack of available analytical techniques to explicitly find the global solution paths. On the other hand the application of numerical techniques such as dynamic programming (DP) to find the solution in int ..."
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Cited by 14 (8 self)
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The study of the solutions of dynamic models with optimizing agents has often been limited by a lack of available analytical techniques to explicitly find the global solution paths. On the other hand the application of numerical techniques such as dynamic programming (DP) to find the solution in interesting regions of the state state was restricted by the use of fixed grid size techniques. Following Grüne (1997) and (2003) in this paper an adaptive grid scheme is used for finding the global solutions of discrete time HamiltonJacobiBellman (HJB) equations. Local error estimates are established and an adapting iteration for the discretization of the state space is developed. The advantage of the use of adaptive grid scheme is demonstrated by computing the solutions of one and two dimensional economic models which exhibit steep curvature, complicated dynamics due to multiple equilibria, thresholds (Skiba sets) separating domains of attraction and periodic solutions. We consider deterministic and stochastic model variants. The studied examples are from economic growth, investment theory, environmental and resource economics.
A Set Oriented Approach To Global Optimal Control
 ESAIM: CONTROL, OPTIMISATION AND CALCULUS OF VARIATIONS
, 1999
"... We describe an algorithm for computing the value function for "all source, single destination " discretetime nonlinear optimal control problems together with approximations of associated globally optimal control strategies. The method is based on a set oriented approach for the discretization of ..."
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Cited by 8 (4 self)
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We describe an algorithm for computing the value function for "all source, single destination " discretetime nonlinear optimal control problems together with approximations of associated globally optimal control strategies. The method is based on a set oriented approach for the discretization of the problem in combination with graphtheoretic techniques. The central idea is that a discretization of phase space of the given problem leads to an (all source, single destination) shortest path problem on a finite graph. The method is illustrated by two numerical examples, namely a single pendulum on a cart and a parametrically driven inverted double pendulum.
Solving Asset Pricing Models with Stochastic Dynamic Programming
 BIELEFELD UNIVERSITY
, 2004
"... The study of asset price characteristics of stochastic growth models such as the riskfree interest rate, equity premium and the Sharpe ratio has been limited by the lack of global and accurate methods to solve dynamic optimization models. In this paper a stochastic version of a dynamic programming m ..."
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Cited by 4 (4 self)
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The study of asset price characteristics of stochastic growth models such as the riskfree interest rate, equity premium and the Sharpe ratio has been limited by the lack of global and accurate methods to solve dynamic optimization models. In this paper a stochastic version of a dynamic programming method with adaptive grid scheme is applied to compute the above mentioned asset price characteristics of a stochastic growth model. The stochastic growth model is of the type as developed by Brock and Mirman (1972) and Brock (1979, 1982). In order to test our method it is applied to a basic stochastic growth model for which the optimal consumption and asset prices can analytical be computed. Since, as shown, our method produces only negligible errors as compared to the analytical solution it is recommended to be used for more elaborate stochastic growth models with different preferences and technology shocks, adjustment costs, and heterogenous agents.
Global optimal control of perturbed systems
 J. Optim. Theory Appl
, 2008
"... We propose a new numerical method for the computation of the optimal value function of perturbed control systems and associated globally stabilizing optimal feedback controllers. The method is based on a set oriented discretization of state space in combination with a new algorithm for the computati ..."
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Cited by 4 (3 self)
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We propose a new numerical method for the computation of the optimal value function of perturbed control systems and associated globally stabilizing optimal feedback controllers. The method is based on a set oriented discretization of state space in combination with a new algorithm for the computation of shortest paths in weighted directed hypergraphs. Using the concept of a multivalued game, we prove convergence of the scheme as the discretization parameter goes to zero.
Default Risk, Asset Pricing, and Debt Control
 Journal of Financial Econometrics
, 2005
"... The pricing and control of firms ’ debt has become a major issue since Merton’s (1974) seminal paper. Yet, Merton as well as other recent theories presume that the asset value of the firm is independent of the debt of the firm. However, when using debt finance firms may have to pay a premium for an ..."
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Cited by 3 (3 self)
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The pricing and control of firms ’ debt has become a major issue since Merton’s (1974) seminal paper. Yet, Merton as well as other recent theories presume that the asset value of the firm is independent of the debt of the firm. However, when using debt finance firms may have to pay a premium for an idiosyncratic default risk and may face debt constraints. We demonstrate that firm specific debt constraints and endogenous risk premia, based on collateralized borrowing, affect the asset value of the firm and, in turn, the collateral value of the firm. In order to explore the interdependence of debt finance and asset pricing of firms we endogenize default premia and borrowing constraints in a production based asset pricing model. In this context then the dynamic decision problem of maximizing the present value of the firm faces an additional constraint giving rise to the debt dependent firm value. We solve for the asset value of the firm with debt finance by the use of numerical dynamic programming. This allows us to solve the debt control problem and to compute sustainable debt as well as firm’s debt value. We want to thank John Donaldson, Martin Lettau and Buz Brock for helpful suggestions and discussions. We also want to thank participants
A Maximum Time Approach to the Computation of Robust Domains of Attraction
"... We present an optimal control based algorithm for the computation of robust domains of attraction for perturbed systems. We give a sufficient condition for the continuity of the optimal value function and a characterization by HamiltonJacobi equations. A numerical scheme is presented and illustrate ..."
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Cited by 3 (1 self)
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We present an optimal control based algorithm for the computation of robust domains of attraction for perturbed systems. We give a sufficient condition for the continuity of the optimal value function and a characterization by HamiltonJacobi equations. A numerical scheme is presented and illustrated by an example.
Discrete Feedback Stabilization Of Nonlinear Control Systems At A Singular Point
"... For continuous time nonlinear control systems with constrained control values stabilizing discrete feedback controls are discussed. It is shown that under an accessibility condition exponential discrete feedback stabilizability is equivalent to open loop uniform exponential asymptotic controllabilit ..."
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Cited by 2 (1 self)
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For continuous time nonlinear control systems with constrained control values stabilizing discrete feedback controls are discussed. It is shown that under an accessibility condition exponential discrete feedback stabilizability is equivalent to open loop uniform exponential asymptotic controllability. A numerical algorithm for the computation of discrete feedback controls is presented and a numerical example is discussed. 1 Introduction In this paper we consider nonlinear control systems of the form y(t) = f(y(t); u(t)); y(0) = y 0 2 R d n f0g u(\Delta) 2 U := fu : R! U; measurableg U ae R m compact (1) where f : R d \Theta R m ! R d is C 2 in y and Lipschitz in u. We assume that x is a singular point of f , i.e. that f(x ; u) = 0 for all u 2 U . Our goal is now to obtain a feedback control strategy such that x becomes an asymptotically stable equilibrium point for the closed loop system. The problem of stabilization of nonlinear control systems has been cons...