Book•
Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations
18 Dec 1997-
TL;DR: In this paper, the main ideas on a model problem with continuous viscosity solutions of Hamilton-Jacobi equations are discussed. But the main idea of the main solutions is not discussed.
Abstract: Preface.- Basic notations.- Outline of the main ideas on a model problem.- Continuous viscosity solutions of Hamilton-Jacobi equations.- Optimal control problems with continuous value functions: unrestricted state space.- Optimal control problems with continuous value functions: restricted state space.- Discontinuous viscosity solutions and applications.- Approximation and perturbation problems.- Asymptotic problems.- Differential Games.- Numerical solution of Dynamic Programming.- Nonlinear H-infinity control by Pierpaolo Soravia.- Bibliography.- Index
Citations
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TL;DR: In this paper, the authors present three examples of the mean-field approach to modelling in economics and finance (or other related subjects) and show that these nonlinear problems are essentially well-posed problems with unique solutions.
Abstract: We survey here some recent studies concerning what we call mean-field models by analogy with Statistical Mechanics and Physics. More precisely, we present three examples of our mean-field approach to modelling in Economics and Finance (or other related subjects...). Roughly speaking, we are concerned with situations that involve a very large number of “rational players” with a limited information (or visibility) on the “game”. Each player chooses his optimal strategy in view of the global (or macroscopic) informations that are available to him and that result from the actions of all players. In the three examples we mention here, we derive a mean-field problem which consists in nonlinear differential equations. These equations are of a new type and our main goal here is to study them and establish their links with various fields of Analysis. We show in particular that these nonlinear problems are essentially well-posed problems i.e., have unique solutions. In addition, we give various limiting cases, examples and possible extensions. And we mention many open problems.
2,385 citations
TL;DR: An algorithm for computing the set of reachable states of a continuous dynamic game based on a proof that the reachable set is the zero sublevel set of the viscosity solution of a particular time-dependent Hamilton-Jacobi-Isaacs partial differential equation.
Abstract: We describe and implement an algorithm for computing the set of reachable states of a continuous dynamic game. The algorithm is based on a proof that the reachable set is the zero sublevel set of the viscosity solution of a particular time-dependent Hamilton-Jacobi-Isaacs partial differential equation. While alternative techniques for computing the reachable set have been proposed, the differential game formulation allows treatment of nonlinear systems with inputs and uncertain parameters. Because the time-dependent equation's solution is continuous and defined throughout the state space, methods from the level set literature can be used to generate more accurate approximations than are possible for formulations with potentially discontinuous solutions. A numerical implementation of our formulation is described and has been released on the web. Its correctness is verified through a two vehicle, three dimensional collision avoidance example for which an analytic solution is available.
1,107 citations
TL;DR: It is shown that the constrained optimal control law has the largest region of asymptotic stability (RAS) and the result is a nearly optimal constrained state feedback controller that has been tuned a priori off-line.
1,045 citations
Cites background from "Optimal Control and Viscosity Solut..."
...Optimal control problems do not necessarily have smooth or even continuous value functions, (Huang et al., 2000; Bardi & Capuzzo-Dolcetta, 1997)....
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...Bardi and Capuzzo-Dolcetta (1997)showed that if the Hamiltonian is strictly convex and if the continuous viscosity solution is semiconcave, thenV ∗(x) ∈ C1( ) satisfying the HJB equation everywhere....
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...Bardi and Capuzzo-Dolcetta (1997) showed that if the Hamiltonian is strictly convex and if the continuous viscosity solution is semiconcave, then V ∗(x) ∈ C1( ) satisfying the HJB equation everywhere....
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TL;DR: The near-optimal control problem for a class of nonlinear discrete-time systems with control constraints is solved by iterative adaptive dynamic programming algorithm.
Abstract: In this paper, the near-optimal control problem for a class of nonlinear discrete-time systems with control constraints is solved by iterative adaptive dynamic programming algorithm. First, a novel nonquadratic performance functional is introduced to overcome the control constraints, and then an iterative adaptive dynamic programming algorithm is developed to solve the optimal feedback control problem of the original constrained system with convergence analysis. In the present control scheme, there are three neural networks used as parametric structures for facilitating the implementation of the iterative algorithm. Two examples are given to demonstrate the convergence and feasibility of the proposed optimal control scheme.
574 citations
Cites background from "Optimal Control and Viscosity Solut..."
...Digital Object Identifier 10.1109/TNN.2009.2027233 for general nonlinear discrete-time systems....
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Book•
01 Jan 2005
TL;DR: In this article, the authors present a set of principles for using variational techniques in nonlinear functional analysis in the presence of symmetry and symmetry in the context of convex analysis.
Abstract: and Notation.- Variational Principles.- Variational Techniques in Subdifferential Theory.- Variational Techniques in Convex Analysis.- Variational Techniques and Multifunctions.- Variational Principles in Nonlinear Functional Analysis.- Variational Techniques In the Presence of Symmetry.
551 citations