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Bellman equation
About: Bellman equation is a research topic. Over the lifetime, 5884 publications have been published within this topic receiving 135589 citations.
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TL;DR: In this article, the authors compare different solution methods for computing the equilibrium of dynamic stochastic general equilibrium (DSGE) models with recursive preferences such as those in Epstein and Zin (1989, 1991) and Stochastic volatility and conclude that perturbation methods are an attractive approach for computing this class of problems.
107 citations
TL;DR: It is proved that semiglobal uniform ultimate boundedness can be guaranteed for states and NN weight errors with the ADP-based ETOC, and a predetermined upper bound is provided by proving the existence of a lower bound for interexecution time.
Abstract: This paper studies the problem of event-triggered optimal control (ETOC) for continuous-time nonlinear systems and proposes a novel event-triggering condition that enables designing ETOC methods directly based on the solution of the Hamilton–Jacobi–Bellman (HJB) equation. We provide formal performance guarantees by proving a predetermined upper bound. Moreover, we also prove the existence of a lower bound for interexecution time. For implementation purposes, an adaptive dynamic programming (ADP) method is developed to realize the ETOC using a critic neural network (NN) to approximate the value function of the HJB equation. Subsequently, we prove that semiglobal uniform ultimate boundedness can be guaranteed for states and NN weight errors with the ADP-based ETOC. Simulation results demonstrate the effectiveness of the developed ADP-based ETOC method.
107 citations
TL;DR: The value function of a finite horizon stochastic control problem with unbounded controls is characterized as the unique viscosity solution of the corresponding dynamic programming equation.
Abstract: In this paper, we prove a comparison result between semicontinuous viscosity sub- and supersolutions growing at most quadratically of second-order degenerate parabolic Hamilton--Jacobi--Bellman and Isaacs equations. As an application, we characterize the value function of a finite horizon stochastic control problem with unbounded controls as the unique viscosity solution of the corresponding dynamic programming equation.
106 citations
TL;DR: In this article, the authors considered the homogenization of Hamilton-Jacobi equations and degenerate Bellman equations in stationary, ergodic, unbounded environments, and showed that, as the microscopic scale tends to zero, the equation averages to a deterministic Hamilton -Jacobi equation and studied some properties of the effective Hamiltonian.
106 citations
TL;DR: This paper provides a characterization of viability kernels and capture basins of a target viable in a constrained subset as a unique closed subset between the target and the constrained subset satisfying tangential conditions or, by duality, normal conditions.
Abstract: This paper provides a characterization of viability kernels and capture basins of a target viable in a constrained subset as a unique closed subset between the target and the constrained subset satisfying tangential conditions or, by duality, normal conditions. It is based on a method devised by Helene Frankowska for characterizing the value function of an optimal control problem as generalized (contingent or viscosity) solutions to Hamilton--Jacobi equations. These abstract results, interesting by themselves, can be applied to epigraphs of functions or graphs of maps and happen to be very efficient for solving other problems, such as stopping time problems, dynamical games, boundary-value problems for systems of partial differential equations, and impulse and hybrid control systems, which are the topics of other companion papers.
106 citations