SIAM Journal on Numerical Analysis, ISSN 0036-1429, 2017, Volume 55, Issue 2, pp. 691 - 712

This paper is concerned with developing and analyzing convergent semi-Lagrangian methods for the fully nonlinear elliptic Monge-Ampere equation on general...

Howard's algorithm | Hamilton-Jacobi-Bellman equation | Viscosity solution | Monge-Ampfiere equation | Semi-Lagrangian method | Wde stencil | Monotone scheme | Convergence | NUMERICAL-METHODS | JACOBI-BELLMAN EQUATIONS | VISCOSITY SOLUTIONS | MATHEMATICS, APPLIED | Monge-Ampere equation | APPROXIMATION | convergence | FINITE-ELEMENT METHODS | SCHEME | viscosity solution | semi-Lagrangian method | PARTIAL-DIFFERENTIAL-EQUATIONS | wide stencil | monotone scheme

Howard's algorithm | Hamilton-Jacobi-Bellman equation | Viscosity solution | Monge-Ampfiere equation | Semi-Lagrangian method | Wde stencil | Monotone scheme | Convergence | NUMERICAL-METHODS | JACOBI-BELLMAN EQUATIONS | VISCOSITY SOLUTIONS | MATHEMATICS, APPLIED | Monge-Ampere equation | APPROXIMATION | convergence | FINITE-ELEMENT METHODS | SCHEME | viscosity solution | semi-Lagrangian method | PARTIAL-DIFFERENTIAL-EQUATIONS | wide stencil | monotone scheme

Journal Article

European Journal of Control, ISSN 0947-3580, 09/2017, Volume 37, pp. 70 - 74

This paper presents a computational method to deal with the Hamilton–Jacobi–Bellman equation with respect to a nonlinear optimal control problem. With Bellman...

Hamilton–Jacobi–Bellman equation | Global minimizer flow | Nonlinear minimization | Feedback optimal control | Difference equation | Hamilton-Jacobi-Bellman equation | AUTOMATION & CONTROL SYSTEMS | Hamilton-Jacobi equations | Research | Dynamic programming | Feedback control systems | Methods | Mathematical optimization | Economic models | Nonlinear equations | Control systems | Optimization | Problems | Feedback | Optimal control | Ordinary differential equations | Queuing theory | Control theory | Iterative methods | Nonlinear control

Hamilton–Jacobi–Bellman equation | Global minimizer flow | Nonlinear minimization | Feedback optimal control | Difference equation | Hamilton-Jacobi-Bellman equation | AUTOMATION & CONTROL SYSTEMS | Hamilton-Jacobi equations | Research | Dynamic programming | Feedback control systems | Methods | Mathematical optimization | Economic models | Nonlinear equations | Control systems | Optimization | Problems | Feedback | Optimal control | Ordinary differential equations | Queuing theory | Control theory | Iterative methods | Nonlinear control

Journal Article

Stochastic Processes and their Applications, ISSN 0304-4149, 07/2017, Volume 127, Issue 7, pp. 2093 - 2137

Since its introduction by P.L. Lions in his lectures and seminars at the College de France, see Lions [6], and also the very helpful notes of Cardialaguet...

Master equation | Hamilton–Jacobi–Bellman equation | Functionals of probability measures | Mean field type control | Fokker–Planck equation | Mean field games | Hamilton-Jacobi-Bellman equation | STATISTICS & PROBABILITY | Fokker-Planck equation | Seminars | Management science | Business schools | Analysis

Master equation | Hamilton–Jacobi–Bellman equation | Functionals of probability measures | Mean field type control | Fokker–Planck equation | Mean field games | Hamilton-Jacobi-Bellman equation | STATISTICS & PROBABILITY | Fokker-Planck equation | Seminars | Management science | Business schools | Analysis

Journal Article

Neural Networks, ISSN 0893-6080, 03/2018, Volume 99, pp. 166 - 177

This paper presents a new theoretical design of nonlinear optimal control on achieving chaotic synchronization for coupled stochastic neural networks. To...

Lyapunov technique | Coupled stochastic neural networks | Chaotic synchronization | Noise attenuation | Nonlinear optimal control | Hamilton–Jacobi–Bellman (HJB) equation | CRITERIA | FEEDBACK-CONTROL | SLIDING MODE CONTROL | TIME-VARYING DELAYS | NEUROSCIENCES | COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE | EXPONENTIAL LAG SYNCHRONIZATION | Hamilton-Jacobi-Bellman (HJB) equation | SYSTEMS | UNKNOWN-PARAMETERS | Neural networks

Lyapunov technique | Coupled stochastic neural networks | Chaotic synchronization | Noise attenuation | Nonlinear optimal control | Hamilton–Jacobi–Bellman (HJB) equation | CRITERIA | FEEDBACK-CONTROL | SLIDING MODE CONTROL | TIME-VARYING DELAYS | NEUROSCIENCES | COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE | EXPONENTIAL LAG SYNCHRONIZATION | Hamilton-Jacobi-Bellman (HJB) equation | SYSTEMS | UNKNOWN-PARAMETERS | Neural networks

Journal Article

Calculus of Variations and Partial Differential Equations, ISSN 0944-2669, 10/2015, Volume 54, Issue 2, pp. 1507 - 1524

We present a proof of qualitative stochastic homogenization for a nonconvex Hamilton–Jacobi equations. The new idea is to introduce a family of “sub-equations”...

35B27 | Mathematics | Systems Theory, Control | Calculus of Variations and Optimal Control; Optimization | Analysis | Theoretical, Mathematical and Computational Physics | MATHEMATICS | MATHEMATICS, APPLIED | BELLMAN EQUATIONS | MEDIA | Analysis of PDEs

35B27 | Mathematics | Systems Theory, Control | Calculus of Variations and Optimal Control; Optimization | Analysis | Theoretical, Mathematical and Computational Physics | MATHEMATICS | MATHEMATICS, APPLIED | BELLMAN EQUATIONS | MEDIA | Analysis of PDEs

Journal Article

Journal of Process Control, ISSN 0959-1524, 03/2014, Volume 24, Issue 3, pp. 172 - 187

•A dissipative-based decentralized nonlinear control for process networks developed.•Candidate dissipativity functions distributed for the global...

Plantwide processes | Hamilton–Jacobi inequality | Dissipative systems | Decentralized control | Nonlinear process systems | Process networks | Hamilton-Jacobi inequality | FEEDBACK-CONTROL | PASSIVITY-BASED CONTROL | OPERABILITY ANALYSIS | BELLMAN EQUATION | CHEMICAL-PROCESSES | ISAACS EQUATION | ENGINEERING, CHEMICAL | H-INFINITY CONTROL | DISTURBANCE ATTENUATION | DYNAMICAL-SYSTEMS | STATE-FEEDBACK | AUTOMATION & CONTROL SYSTEMS

Plantwide processes | Hamilton–Jacobi inequality | Dissipative systems | Decentralized control | Nonlinear process systems | Process networks | Hamilton-Jacobi inequality | FEEDBACK-CONTROL | PASSIVITY-BASED CONTROL | OPERABILITY ANALYSIS | BELLMAN EQUATION | CHEMICAL-PROCESSES | ISAACS EQUATION | ENGINEERING, CHEMICAL | H-INFINITY CONTROL | DISTURBANCE ATTENUATION | DYNAMICAL-SYSTEMS | STATE-FEEDBACK | AUTOMATION & CONTROL SYSTEMS

Journal Article

SIAM Journal on Scientific Computing, ISSN 1064-8275, 2018, Volume 40, Issue 2, pp. A629 - A652

A procedure for the numerical approximation of high-dimensional Hamilton-Jacobi-Bellman (HJB) equations associated to optimal feedback control problems for...

High-dimensional approximation | Nonlinear dynamics | Hamilton–Jacobi–Bellman equations | Polynomial approximation | Optimal feedback control | MATHEMATICS, APPLIED | polynomial approximation | high-dimensional approximation | STABILIZATION | optimal feedback control | Hamilton-Jacobi-Bellman equations | nonlinear dynamics

High-dimensional approximation | Nonlinear dynamics | Hamilton–Jacobi–Bellman equations | Polynomial approximation | Optimal feedback control | MATHEMATICS, APPLIED | polynomial approximation | high-dimensional approximation | STABILIZATION | optimal feedback control | Hamilton-Jacobi-Bellman equations | nonlinear dynamics

Journal Article

Journal of Nonlinear Science, ISSN 0938-8974, 8/2019, Volume 29, Issue 4, pp. 1563 - 1619

High-dimensional partial differential equations (PDEs) appear in a number of models from the financial industry, such as in derivative pricing models, credit...

Knightian uncertainty | Theoretical, Mathematical and Computational Physics | Classical Mechanics | Economic Theory/Quantitative Economics/Mathematical Methods | Mathematics | Numerical method | Black–Scholes–Barenblatt equation | Nonlinear expectation | Deep learning | Hamiltonian–Jacobi–Bellman equation | Analysis | HJB equation | Mathematical and Computational Engineering | Second-order backward stochastic differential equation | 2BSDE | G -Brownian motion | G-Brownian motion | BROWNIAN-MOTION | MATHEMATICS, APPLIED | Black-Scholes-Barenblatt equation | ORDER NUMERICAL SCHEMES | Hamiltonian-Jacobi-Bellman equation | BSDES | SIMULATION | PHYSICS, MATHEMATICAL | DISCRETIZATION | MECHANICS | DEEP NEURAL-NETWORKS | DISCRETE-TIME APPROXIMATION | Investment analysis | Big data | Algorithms | Differential equations | Machine learning

Knightian uncertainty | Theoretical, Mathematical and Computational Physics | Classical Mechanics | Economic Theory/Quantitative Economics/Mathematical Methods | Mathematics | Numerical method | Black–Scholes–Barenblatt equation | Nonlinear expectation | Deep learning | Hamiltonian–Jacobi–Bellman equation | Analysis | HJB equation | Mathematical and Computational Engineering | Second-order backward stochastic differential equation | 2BSDE | G -Brownian motion | G-Brownian motion | BROWNIAN-MOTION | MATHEMATICS, APPLIED | Black-Scholes-Barenblatt equation | ORDER NUMERICAL SCHEMES | Hamiltonian-Jacobi-Bellman equation | BSDES | SIMULATION | PHYSICS, MATHEMATICAL | DISCRETIZATION | MECHANICS | DEEP NEURAL-NETWORKS | DISCRETE-TIME APPROXIMATION | Investment analysis | Big data | Algorithms | Differential equations | Machine learning

Journal Article

SIAM Journal on Numerical Analysis, ISSN 0036-1429, 1/2014, Volume 52, Issue 2, pp. 993 - 1016

We propose an hp-version discontinuous Galerkin finite element method for fully nonlinear second-order elliptic Hamilton–Jacobi–Bellman equations with Cordes...

Ellipticity | Approximation | Mathematical discontinuity | Mathematical monotonicity | Fens | Polynomials | Coefficients | Newtons method | Stencils | Degrees of polynomials | Semismooth Newton methods | Fully nonlinear equations | Hamilton-Jacobi-Bellman equations | Cordes condition | Hp-version discontinuous Galerkin finite element methods | VISCOSITY SOLUTIONS | MATHEMATICS, APPLIED | PARABOLIC EQUATIONS | fully nonlinear equations | semismooth Newton methods | CONVERGENT DIFFERENCE-SCHEMES | ELLIPTIC-EQUATIONS | hp-version discontinuous Galerkin finite element methods | Finite element method | Rope | Mathematical analysis | Nonlinearity | Mathematical models | Computational efficiency | Galerkin methods

Ellipticity | Approximation | Mathematical discontinuity | Mathematical monotonicity | Fens | Polynomials | Coefficients | Newtons method | Stencils | Degrees of polynomials | Semismooth Newton methods | Fully nonlinear equations | Hamilton-Jacobi-Bellman equations | Cordes condition | Hp-version discontinuous Galerkin finite element methods | VISCOSITY SOLUTIONS | MATHEMATICS, APPLIED | PARABOLIC EQUATIONS | fully nonlinear equations | semismooth Newton methods | CONVERGENT DIFFERENCE-SCHEMES | ELLIPTIC-EQUATIONS | hp-version discontinuous Galerkin finite element methods | Finite element method | Rope | Mathematical analysis | Nonlinearity | Mathematical models | Computational efficiency | Galerkin methods

Journal Article

Automatica, ISSN 0005-1098, 01/2019, Volume 99, pp. 181 - 187

In this paper, for a given (time-dependent) Schrödinger equation in quantum mechanics, an interpretation of it is investigated from the perspective of...

Nelson stochastic mechanics | Hamilton–Jacobi–Bellman equation | Stochastic control | Dynamic programming | Schrödinger equation | DERIVATION | Schrodinger equation | Hamilton-Jacobi-Bellman equation | QUANTIZATION | AUTOMATION & CONTROL SYSTEMS | ENGINEERING, ELECTRICAL & ELECTRONIC | Quantum theory

Nelson stochastic mechanics | Hamilton–Jacobi–Bellman equation | Stochastic control | Dynamic programming | Schrödinger equation | DERIVATION | Schrodinger equation | Hamilton-Jacobi-Bellman equation | QUANTIZATION | AUTOMATION & CONTROL SYSTEMS | ENGINEERING, ELECTRICAL & ELECTRONIC | Quantum theory

Journal Article

Neural Networks, ISSN 0893-6080, 11/2015, Volume 71, pp. 150 - 158

The constrained optimal control problem depends on the solution of the complicated Hamilton–Jacobi–Bellman equation (HJBE). In this paper, a data-based...

Constrained optimal control | Data-based | Hamilton–Jacobi–Bellman equation | The method of weighted residuals | Off-policy reinforcement learning | Hamilton-Jacobi-Bellman equation | DESIGN | POLICY ITERATION | ADAPTIVE OPTIMAL-CONTROL | STABILIZATION | TIME NONLINEAR-SYSTEMS | DYNAMIC-PROGRAMMING ALGORITHM | NEUROSCIENCES | COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE | Models, Theoretical | Algorithms | Computer Simulation | Problem Solving | Reinforcement (Psychology) | Nonlinear Dynamics | Machine Learning | Neural Networks (Computer) | Electrical engineering | Neural networks | Convergence (Social sciences)

Constrained optimal control | Data-based | Hamilton–Jacobi–Bellman equation | The method of weighted residuals | Off-policy reinforcement learning | Hamilton-Jacobi-Bellman equation | DESIGN | POLICY ITERATION | ADAPTIVE OPTIMAL-CONTROL | STABILIZATION | TIME NONLINEAR-SYSTEMS | DYNAMIC-PROGRAMMING ALGORITHM | NEUROSCIENCES | COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE | Models, Theoretical | Algorithms | Computer Simulation | Problem Solving | Reinforcement (Psychology) | Nonlinear Dynamics | Machine Learning | Neural Networks (Computer) | Electrical engineering | Neural networks | Convergence (Social sciences)

Journal Article

Journal of Scientific Computing, ISSN 0885-7474, 9/2018, Volume 76, Issue 3, pp. 1839 - 1867

In this paper, we propose a monotone mixed finite difference scheme for solving the two-dimensional Monge–Ampère equation. In order to accomplish this, we...

Mixed schemes | Computational Mathematics and Numerical Analysis | Monotone schemes | Algorithms | Monge–Ampère equations | Theoretical, Mathematical and Computational Physics | Mathematical and Computational Engineering | Mathematics | Nonlinear elliptic partial differential equations | Hamilton–Jacobi–Bellman equations | Viscosity solutions | Finite difference methods | Analysis | Differential equations

Mixed schemes | Computational Mathematics and Numerical Analysis | Monotone schemes | Algorithms | Monge–Ampère equations | Theoretical, Mathematical and Computational Physics | Mathematical and Computational Engineering | Mathematics | Nonlinear elliptic partial differential equations | Hamilton–Jacobi–Bellman equations | Viscosity solutions | Finite difference methods | Analysis | Differential equations

Journal Article

ESAIM - Control, Optimisation and Calculus of Variations, ISSN 1292-8119, 04/2015, Volume 21, Issue 2, pp. 442 - 464

An optimal finite-time horizon feedback control problem for (semi-linear) wave equations is presented. The feedback law can be derived from the dynamic...

Wave equation | Hamilton-Jacobi Bellman equation | Spectral elements | Optimal control | spectral elements | MATHEMATICS, APPLIED | wave equation | CONVERGENCE | RICCATI EQUATION | CONTROLLABILITY | AUTOMATION & CONTROL SYSTEMS | Approximation | Computer simulation | Mathematical analysis | Wave equations | Control systems | Dynamic programming | Feedback control | Analysis of PDEs | Mathematics | Optimization and Control

Wave equation | Hamilton-Jacobi Bellman equation | Spectral elements | Optimal control | spectral elements | MATHEMATICS, APPLIED | wave equation | CONVERGENCE | RICCATI EQUATION | CONTROLLABILITY | AUTOMATION & CONTROL SYSTEMS | Approximation | Computer simulation | Mathematical analysis | Wave equations | Control systems | Dynamic programming | Feedback control | Analysis of PDEs | Mathematics | Optimization and Control

Journal Article

Journal of Functional Analysis, ISSN 0022-1236, 10/2018, Volume 275, Issue 8, pp. 2096 - 2161

We propose notions of minimax and viscosity solutions for a class of fully nonlinear path-dependent PDEs with nonlinear, monotone, and coercive operators on...

Nonlinear evolution equations | Path-dependent PDEs | Minimax solutions | Optimal control | BOUNDARY CONTROL-PROBLEMS | HJB EQUATIONS | VISCOSITY SOLUTIONS | STATE CONSTRAINTS | NONLINEAR 2ND-ORDER EQUATIONS | MATHEMATICS | RISK-SENSITIVE CONTROL | PARTIAL-DIFFERENTIAL-EQUATIONS | BELLMAN EQUATIONS | OPTIMAL STOCHASTIC-CONTROL | SADDLE-POINT

Nonlinear evolution equations | Path-dependent PDEs | Minimax solutions | Optimal control | BOUNDARY CONTROL-PROBLEMS | HJB EQUATIONS | VISCOSITY SOLUTIONS | STATE CONSTRAINTS | NONLINEAR 2ND-ORDER EQUATIONS | MATHEMATICS | RISK-SENSITIVE CONTROL | PARTIAL-DIFFERENTIAL-EQUATIONS | BELLMAN EQUATIONS | OPTIMAL STOCHASTIC-CONTROL | SADDLE-POINT

Journal Article

15.
Full Text
Stochastic homogenization of nonconvex Hamilton–Jacobi equations in one space dimension

Journal of Differential Equations, ISSN 0022-0396, 09/2016, Volume 261, Issue 5, pp. 2702 - 2737

We prove stochastic homogenization for a general class of coercive, nonconvex Hamilton–Jacobi equations in one space dimension. Some properties of the...

Nonconvex Hamilton–Jacobi equation | Dynamical properties of effective Hamiltonians | Metric problem | Stochastic homogenization | Nonconvex Hamilton-Jacobi equation | MATHEMATICS | BELLMAN EQUATIONS | MEDIA | Analysis of PDEs | Mathematics

Nonconvex Hamilton–Jacobi equation | Dynamical properties of effective Hamiltonians | Metric problem | Stochastic homogenization | Nonconvex Hamilton-Jacobi equation | MATHEMATICS | BELLMAN EQUATIONS | MEDIA | Analysis of PDEs | Mathematics

Journal Article

Automatica, ISSN 0005-1098, 04/2016, Volume 66, pp. 205 - 217

We study simple quadratic approximations for general Hamilton–Jacobi–Bellman equations. The theoretical error bounds are shown to be composed of the time...

Stochastic control | Quadratic approximations | Error analysis | Partial differential equations | Hamilton–Jacobi–Bellman equations | Hamilton-Jacobi-Bellman equations | AUTOMATION & CONTROL SYSTEMS | SCHEMES | ENGINEERING, ELECTRICAL & ELECTRONIC | Quadratic equations | Errors | Approximation | Discretization | Mathematical analysis | Boundaries | Hamiltonian functions

Stochastic control | Quadratic approximations | Error analysis | Partial differential equations | Hamilton–Jacobi–Bellman equations | Hamilton-Jacobi-Bellman equations | AUTOMATION & CONTROL SYSTEMS | SCHEMES | ENGINEERING, ELECTRICAL & ELECTRONIC | Quadratic equations | Errors | Approximation | Discretization | Mathematical analysis | Boundaries | Hamiltonian functions

Journal Article

SIAM JOURNAL ON CONTROL AND OPTIMIZATION, ISSN 0363-0129, 2019, Volume 57, Issue 1, pp. 693 - 717

In this paper we study stochastic control problems with delayed information, that is, the control at time t can depend only on the information observed before...

VISCOSITY SOLUTIONS | MATHEMATICS, APPLIED | MAXIMUM PRINCIPLE | information delay | BELLMAN EQUATION | McKean-Vlasov SDE | FIELD STACKELBERG GAMES | partial observation | master equation | functional Ito formula | QUADRATIC PROBLEM | SYSTEMS | AUTOMATION & CONTROL SYSTEMS

VISCOSITY SOLUTIONS | MATHEMATICS, APPLIED | MAXIMUM PRINCIPLE | information delay | BELLMAN EQUATION | McKean-Vlasov SDE | FIELD STACKELBERG GAMES | partial observation | master equation | functional Ito formula | QUADRATIC PROBLEM | SYSTEMS | AUTOMATION & CONTROL SYSTEMS

Journal Article

Journal of Vibration and Control, ISSN 1077-5463, 5/2018, Volume 24, Issue 9, pp. 1741 - 1756

The performance index of both the state and control variables with a constrained dynamic optimization problem of a fractional order system with fixed final...

Hamilton–Jacobi–Bellman equation | Legendre–Gauss collocation | linear quadratic regulator system | Fractional optimal control problem | Riemann–Liouville fractional derivative | DIFFERENTIAL-EQUATIONS | EVOLUTION-EQUATIONS | FORMULATION | HOMOTOPY ANALYSIS METHOD | ENGINEERING, MECHANICAL | NUMERICAL SCHEME | CAPUTO | ACOUSTICS | ORDER | MECHANICS | Hamilton-Jacobi-Bellman equation | COLLOCATION METHOD | Riemann-Liouville fractional derivative | SYSTEMS | Legendre-Gauss collocation | Performance indices | Collocation methods | Optimal control | Optimization

Hamilton–Jacobi–Bellman equation | Legendre–Gauss collocation | linear quadratic regulator system | Fractional optimal control problem | Riemann–Liouville fractional derivative | DIFFERENTIAL-EQUATIONS | EVOLUTION-EQUATIONS | FORMULATION | HOMOTOPY ANALYSIS METHOD | ENGINEERING, MECHANICAL | NUMERICAL SCHEME | CAPUTO | ACOUSTICS | ORDER | MECHANICS | Hamilton-Jacobi-Bellman equation | COLLOCATION METHOD | Riemann-Liouville fractional derivative | SYSTEMS | Legendre-Gauss collocation | Performance indices | Collocation methods | Optimal control | Optimization

Journal Article

Nonlinear Differential Equations and Applications NoDEA, ISSN 1021-9722, 6/2013, Volume 20, Issue 3, pp. 413 - 445

We consider continuous-state and continuous-time control problems where the admissible trajectories of the system are constrained to remain on a network. In...

Networks | Secondary 34H05 | 35F21 | Analysis | Optimal control | Graphs | Mathematics | Hamilton–Jacobi equations | Primary 35R02 | 35Q93 | 49J15 | Viscosity solutions | Hamilton-Jacobi equations | MATHEMATICS, APPLIED | STATE CONSTRAINTS | BELLMAN EQUATIONS | Analysis of PDEs

Networks | Secondary 34H05 | 35F21 | Analysis | Optimal control | Graphs | Mathematics | Hamilton–Jacobi equations | Primary 35R02 | 35Q93 | 49J15 | Viscosity solutions | Hamilton-Jacobi equations | MATHEMATICS, APPLIED | STATE CONSTRAINTS | BELLMAN EQUATIONS | Analysis of PDEs

Journal Article