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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 consider discrete-time infinite horizon problems of optimal control to a terminal set of states, and establish the uniqueness of the solution of Bellman's equation, and provide convergence results for value and policy iterations.
Abstract: In this paper, we consider discrete-time infinite horizon problems of optimal control to a terminal set of states. These are the problems that are often taken as the starting point for adaptive dynamic programming. Under very general assumptions, we establish the uniqueness of the solution of Bellman’s equation, and we provide convergence results for value and policy iterations.
152 citations
TL;DR: For particular classes of systems, asymptotics for vanishing risk factor is investigated, showing that in the limit the optimal value for an average cost per unit time is obtained.
Abstract: In this paper we study existence of solutions to the Bellman equation corresponding to risk-sensitive ergodic control of discrete-time Markov processes using three different approaches. Also, for particular classes of systems, asymptotics for vanishing risk factor is investigated, showing that in the limit the optimal value for an average cost per unit time is obtained.
152 citations
TL;DR: In this article, the authors use dynamic programming techniques to describe reach sets and related problems of forward and backward reachability, which are reformulated in terms of optimization problems solved through the Hamilton-Jacobi-Bellman Equations.
Abstract: This paper uses dynamic programming techniques to describe reach sets andrelated problems of forward and backward reachability The original problemsdo not involve optimization criteria and are reformulated in terms ofoptimization problems solved through the Hamilton–Jacobi–Bellmanequations The reach sets are the level sets of the value function solutionsto these equations Explicit solutions for linear systems with hard boundsare obtained Approximate solutions are introduced and illustrated forlinear systems and for a nonlinear system similar to that of theLotka–Volterra type
152 citations
Book•
31 Oct 1994
TL;DR: In this article, the Ekeland Variational Principle is applied to the problem of optimal control of nonlinear Parameter Distributed Systems (PDS) and the HINFINITY-Control Problem is formulated.
Abstract: Preface. Symbols and Notations. I: Generalized Gradients and Optimality. 1. Fundamentals of Convex Analysis. 2. Generalized Gradients. 3. The Ekeland Variational Principle. II: Optimal Control of Ordinary Differential Systems. 1. Formulation of the Problem and Existence. 2. The Maximum Principle. 3. Applications of the Maximum Principle. III: The Dynamic Programming Method. 1. The Dynamic Programming Equation. 2. Variational and Viscosity Solutions to the Equation of Dynamic Programming. 3. Constructive Approaches to Synthesis Problem IV: Optimal Control of Parameter Distributed Systems. 1. General Description of Parameter Distributed Systems. 2. Optimal Convex Control Problems. 3. The HINFINITY-Control Problem. 4. Optimal Control of Nonlinear Parameter Distributed Systems. Subject Index.
150 citations
10 Dec 2002
TL;DR: In this article, a solution to the problem of optimal control of piecewise affine systems with a bounded disturbance is characterised results that allow one to compute the value function, its domain (robustly controllable set) and the optimal control law are presented.
Abstract: The solution to the problem of optimal control of piecewise affine systems with a bounded disturbance is characterised Results that allow one to compute the value function, its domain (robustly controllable set) and the optimal control law are presented The tools that are employed include dynamic programming, polytopic set algebra and parametric programming When the cost is time (robust time-optimal control problem) or the stage cost is piecewise affine (robust optimal and robust receding horizon control problems), the value function and the optimal control law are both piecewise affine and each robustly controllable set is the union of a finite set of polytopes Conditions on the cost and constraints are also proposed in order to ensure that the optimal control laws are robustly stabilising
150 citations