Journal ArticleDOI
A Viscosity Approach to Infinite-Dimensional Hamilton--Jacobi Equations Arising in Optimal Control with State Constraints
Maciej Kocan,Pierpaolo Soravia +1 more
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In this article, the authors consider nonlinear optimal control problems with state constraints and nonnegative cost in infinite dimensions, where the constraint is a closed set possibly with empty interior for a class of systems with a maximal monotone operator and satisfying certain stability properties.Abstract:
We consider nonlinear optimal control problems with state constraints and nonnegative cost in infinite dimensions, where the constraint is a closed set possibly with empty interior for a class of systems with a maximal monotone operator and satisfying certain stability properties of the set of trajectories that allow the value function to be lower semicontinuous. We prove that the value function is a viscosity solution of the Bellman equation and is in fact the minimal nonnegative supersolution.read more
Citations
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Journal ArticleDOI
A stochastic control problem with delay arising in a pension fund model
TL;DR: The equivalence between the one-dimensional delay problem and the associated infinite-dimensional problem without delay is shown and it is proved that the value function is continuous in this infinite- dimensional setting.
Journal ArticleDOI
Numerical solution to the optimal feedback control of continuous casting process
Bao-Zhu Guo,Bing Sun +1 more
TL;DR: It is shown that the value function is the viscosity solution of the Hamilton–Jacobi–Bellman equation that leads to the optimal feedback control of the continuous casting process in the secondary cooling zone with water spray control.
Journal ArticleDOI
Optimality principles and uniqueness for Bellman equations of unbounded control problems with discontinuous running cost
TL;DR: In this paper, the authors study viscosity solutions of Hamilton-Jacobi equations that arise in optimal control problems with unbounded controls and discontinuous Lagrangian and prove optimality principles that extend the scope of the results of [23] under very general assumptions, allowing unbounded control.
Journal ArticleDOI
Viscosity Solutions of Dynamic-Programming Equations for the Optimal Control of the Two-Dimensional Navier-Stokes Equations
TL;DR: In this paper, the authors study the infinite-dimensional Hamilton-Jacobi equation associated with the optimal feedback control of viscous hydrodynamics and resolve the global unique solvability problem of this equation by showing that the value function is the unique viscosity solution.
Journal ArticleDOI
On Differential Games for Infinite-Dimensional Systems with Nonlinear, Unbounded Operators
TL;DR: In this paper, the authors consider a two-player zero-sum game governed by an abstract nonlinear differential equation of accretive type in an infinite-dimensional space and prove that the value function of the game is the unique viscosity solution of the corresponding Hamilton-Jacobi-Isaacs equation.
References
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Journal ArticleDOI
User’s guide to viscosity solutions of second order partial differential equations
TL;DR: The notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems, existence theorem, and continuous dependence may now be proved by very efficient and striking arguments as discussed by the authors.
Book
Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert
TL;DR: In this article, Operateurs Maximaux Monotones: Et Semi-Groupes De Contractions Dans Les Espaces De Hllbert are described and discussed. But the focus is not on the performance of the operators.
Book
Nonlinear semigroups and differential equations in Banach spaces
TL;DR: In this article, the authors consider boundary value problems for second order nonlinear differential equations in Banach spaces, and show that the boundary value problem can be expressed as a boundary value maximization problem.
Journal ArticleDOI
Optimal control with state-space constraint I
TL;DR: In this paper, the optimal control of piecewise deterministic processes with state space constraint is studied under appropriate assumptions, and it is shown that the optimal value function is the only viscosity solution on the open domain which is also a supersolution on the closed domain.
Book
Compactness methods for nonlinear evolutions
TL;DR: In this article, the Second Edition Notation and Conventions Elements of Nonlinear Functional Analysis Fundamental Compactness Results Nonlinear Perturbations of Accretive Operators Demiclosed Perturations of Subdifferentials Functional and Integrodifferential Equations Bibliographical Notes, Comments and Open Problems References Index