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

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

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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

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

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

Journal of Scientific Computing, ISSN 0885-7474, 1/2018, Volume 74, Issue 1, pp. 145 - 174

We analyse a class of nonoverlapping domain decomposition preconditioners for nonsymmetric linear systems arising from discontinuous Galerkin finite element...

Discontinuous Galerkin | Computational Mathematics and Numerical Analysis | Theoretical, Mathematical and Computational Physics | Mathematics | Finite element methods | Hamilton–Jacobi–Bellman equations | 35J66 | Approximation in discontinuous spaces | Algorithms | 65F10 | GMRES | Mathematical and Computational Engineering | Preconditioners | Domain decomposition | 65N22 | 65N55 | 65N30 | MATHEMATICS, APPLIED | ALGORITHM | ADDITIVE SCHWARZ METHODS | Hamilton-Jacobi-Bellman equations | FINITE-ELEMENT APPROXIMATION | Analysis | Differential equations | Mathematics - Numerical Analysis | Numerical Analysis

Discontinuous Galerkin | Computational Mathematics and Numerical Analysis | Theoretical, Mathematical and Computational Physics | Mathematics | Finite element methods | Hamilton–Jacobi–Bellman equations | 35J66 | Approximation in discontinuous spaces | Algorithms | 65F10 | GMRES | Mathematical and Computational Engineering | Preconditioners | Domain decomposition | 65N22 | 65N55 | 65N30 | MATHEMATICS, APPLIED | ALGORITHM | ADDITIVE SCHWARZ METHODS | Hamilton-Jacobi-Bellman equations | FINITE-ELEMENT APPROXIMATION | Analysis | Differential equations | Mathematics - Numerical Analysis | Numerical Analysis

Journal Article

Journal of Differential Equations, ISSN 0022-0396, 12/2017, Volume 263, Issue 12, pp. 8418 - 8466

This paper studies optimal control problems on networks without controllability assumptions at the junctions. The Value Function associated with the control...

Stratified structure | Hamilton–Jacobi equations | State constraint problems | Control problem on networks | MATHEMATICS | Hamilton-Jacobi equations | R-N | BELLMAN APPROACH | EIKONAL EQUATIONS | Analysis of PDEs | Mathematics | Optimization and Control

Stratified structure | Hamilton–Jacobi equations | State constraint problems | Control problem on networks | MATHEMATICS | Hamilton-Jacobi equations | R-N | BELLMAN APPROACH | EIKONAL EQUATIONS | Analysis of PDEs | Mathematics | Optimization and Control

Journal Article

SIAM Journal on Numerical Analysis, ISSN 0036-1429, 1/2013, Volume 51, Issue 1, pp. 137 - 162

We study the convergence of monotone P1 finite element methods on unstructured meshes for fully nonlinear Hamilton-Jacobi-Bellman equations arising from...

Viscosity | Finite element method | Mathematical problems | Approximation | Mathematical monotonicity | Optimal control | Numerical methods | Mathematical functions | Numerical schemes | Partial differential equations | Finite element methods | Hamilton-Jacobi- Bellman equations | Viscosity solutions | MATHEMATICS, APPLIED | GRIDS | PARABOLIC EQUATIONS | APPROXIMATION | TIME | finite element methods | PARTIAL-DIFFERENTIAL EQUATIONS | COEFFICIENTS | Hamilton-Jacobi-Bellman equations | DIFFUSION | partial differential equations | viscosity solutions | DISCRETE MAXIMUM PRINCIPLE | SCHEMES | Operators | Discretization | Mathematical analysis | Consistency | Projection | Diffusion | Convergence

Viscosity | Finite element method | Mathematical problems | Approximation | Mathematical monotonicity | Optimal control | Numerical methods | Mathematical functions | Numerical schemes | Partial differential equations | Finite element methods | Hamilton-Jacobi- Bellman equations | Viscosity solutions | MATHEMATICS, APPLIED | GRIDS | PARABOLIC EQUATIONS | APPROXIMATION | TIME | finite element methods | PARTIAL-DIFFERENTIAL EQUATIONS | COEFFICIENTS | Hamilton-Jacobi-Bellman equations | DIFFUSION | partial differential equations | viscosity solutions | DISCRETE MAXIMUM PRINCIPLE | SCHEMES | Operators | Discretization | Mathematical analysis | Consistency | Projection | Diffusion | Convergence

Journal Article

Applied Mathematics and Optimization, ISSN 0095-4616, 06/2018, Volume 77, Issue 3, pp. 599 - 611

An example of a nonunique solution of the Cauchy problem of Hamilton-Jacobi-Bellman (HJB) equation with surprisingly regular Hamiltonian is presented. The...

Hamilton–Jacobi–Bellman equation | Nonsmooth analysis | Optimal control theory | Viscosity solution | VISCOSITY SOLUTIONS | CONVEX HAMILTONIANS | MATHEMATICS, APPLIED | Hamilton-Jacobi-Bellman equation | BOLZA PROBLEMS | UNIQUENESS | Mathematical analysis | Cauchy problem | MATHEMATICAL SOLUTIONS | THAILAND | HAMILTONIANS | CAUCHY PROBLEM | CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS

Hamilton–Jacobi–Bellman equation | Nonsmooth analysis | Optimal control theory | Viscosity solution | VISCOSITY SOLUTIONS | CONVEX HAMILTONIANS | MATHEMATICS, APPLIED | Hamilton-Jacobi-Bellman equation | BOLZA PROBLEMS | UNIQUENESS | Mathematical analysis | Cauchy problem | MATHEMATICAL SOLUTIONS | THAILAND | HAMILTONIANS | CAUCHY PROBLEM | CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS

Journal Article

SIAM JOURNAL ON CONTROL AND OPTIMIZATION, ISSN 0363-0129, 2019, Volume 57, Issue 1, pp. 23 - 52

In this paper we study the ergodic problem for viscous Hamilton-Jacobi equations with superlinear Hamiltonian and inward drift. We investigate (i) existence...

MATHEMATICS, APPLIED | stochastic ergodic control | LARGE TIME BEHAVIOR | ergodic problem | generalized principal eigenvalue | BELLMAN EQUATIONS | viscous Hamilton-Jacobi equation | AUTOMATION & CONTROL SYSTEMS | Analysis of PDEs | Mathematics

MATHEMATICS, APPLIED | stochastic ergodic control | LARGE TIME BEHAVIOR | ergodic problem | generalized principal eigenvalue | BELLMAN EQUATIONS | viscous Hamilton-Jacobi equation | AUTOMATION & CONTROL SYSTEMS | Analysis of PDEs | Mathematics

Journal Article

International Journal of Control, ISSN 0020-7179, 10/2019, Volume 92, Issue 10, pp. 2263 - 2273

In this article, optimal control problems of differential equations with delays are investigated for which the associated Hamilton-Jacobi-Bellman (HJB)...

optimal control | Hamilton-Jacobi-Bellman equations | differential equations with delays | viscosity solutions | EXISTENCE | INFINITE DIMENSIONS | TIME-TO-BUILD | AUTOMATION & CONTROL SYSTEMS | UNIQUENESS | Viscosity | Nonlinear equations | Equations of state | Partial differential equations | Production planning | Mathematical analysis | Optimal control | Nonlinear differential equations | Control theory

optimal control | Hamilton-Jacobi-Bellman equations | differential equations with delays | viscosity solutions | EXISTENCE | INFINITE DIMENSIONS | TIME-TO-BUILD | AUTOMATION & CONTROL SYSTEMS | UNIQUENESS | Viscosity | Nonlinear equations | Equations of state | Partial differential equations | Production planning | Mathematical analysis | Optimal control | Nonlinear differential equations | Control theory

Journal Article

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Weak solution for a class of fully nonlinear stochastic Hamilton–Jacobi–Bellman equations

Stochastic Processes and their Applications, ISSN 0304-4149, 06/2017, Volume 127, Issue 6, pp. 1926 - 1959

This paper is concerned with a class of stochastic Hamilton–Jacobi–Bellman equations with controlled leading coefficients, which are fully nonlinear backward...

Non-Markovian control | Weak solution | Potential | Stochastic Hamilton–Jacobi–Bellman equation | Backward stochastic partial differential equation | Stochastic Hamilton-Jacobi-Bellman equation | VISCOSITY SOLUTIONS | PARTIAL-DIFFERENTIAL-EQUATIONS | COEFFICIENTS | BACKWARD SPDES | SDES | STATISTICS & PROBABILITY | REGULARITY THEORY | Differential equations

Non-Markovian control | Weak solution | Potential | Stochastic Hamilton–Jacobi–Bellman equation | Backward stochastic partial differential equation | Stochastic Hamilton-Jacobi-Bellman equation | VISCOSITY SOLUTIONS | PARTIAL-DIFFERENTIAL-EQUATIONS | COEFFICIENTS | BACKWARD SPDES | SDES | STATISTICS & PROBABILITY | REGULARITY THEORY | Differential equations

Journal Article

SIAM Journal on Control and Optimization, ISSN 0363-0129, 2017, Volume 55, Issue 5, pp. 2981 - 3012

Stochastic optimal control problems governed by delay equations with delay in the control are usually more difficult to study than those in which the delay...

Optimal control of stochastic delay equations | Second order Hamilton-Jacobi-Bellman equations in infinite dimension | Delay in the control | Lack of structure condition | Smoothing properties of transition semigroups | optimal control of stochastic delay equations | HILBERT-SPACES | MATHEMATICS, APPLIED | PARABOLIC EQUATIONS | STATE CONSTRAINTS | HAMILTON-JACOBI EQUATIONS | DIFFERENTIAL-EQUATIONS | HORIZON | lack of structure condition | second order Hamilton Jacobi Bellman equations in infinite dimension | delay in the control | REGULARITY | smoothing properties of transition semigroups | BELLMAN EQUATIONS | SYSTEMS | KOLMOGOROV EQUATIONS | AUTOMATION & CONTROL SYSTEMS

Optimal control of stochastic delay equations | Second order Hamilton-Jacobi-Bellman equations in infinite dimension | Delay in the control | Lack of structure condition | Smoothing properties of transition semigroups | optimal control of stochastic delay equations | HILBERT-SPACES | MATHEMATICS, APPLIED | PARABOLIC EQUATIONS | STATE CONSTRAINTS | HAMILTON-JACOBI EQUATIONS | DIFFERENTIAL-EQUATIONS | HORIZON | lack of structure condition | second order Hamilton Jacobi Bellman equations in infinite dimension | delay in the control | REGULARITY | smoothing properties of transition semigroups | BELLMAN EQUATIONS | SYSTEMS | KOLMOGOROV EQUATIONS | AUTOMATION & CONTROL SYSTEMS

Journal Article

SIAM Journal on Scientific Computing, ISSN 1064-8275, 2016, Volume 38, Issue 3, pp. A1587 - A1615

The numerical realization of the dynamic programming principle for continuous-time optimal control leads to nonlinear Hamilton-Jacobi-Bellman equations which...

Semismooth Newton methods | Hamilton-Jacobi-Bellman equations | Dynamic programming | Semi-lagrangian schemes | First-order primal-dual methods | MATHEMATICS, APPLIED | dynamic programming | semismooth Newton methods | semi-Lagrangian schemes | first-order primal-dual methods | SCHEMES | Numerical Analysis | Mathematics | Optimization and Control

Semismooth Newton methods | Hamilton-Jacobi-Bellman equations | Dynamic programming | Semi-lagrangian schemes | First-order primal-dual methods | MATHEMATICS, APPLIED | dynamic programming | semismooth Newton methods | semi-Lagrangian schemes | first-order primal-dual methods | SCHEMES | Numerical Analysis | Mathematics | Optimization and Control

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Convergent semi-lagrangian methods for the Monge-Ampere equation on unstructured grids

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

Journal of Machine Learning Research, ISSN 1532-4435, 09/2018, Volume 19, pp. 1 - 49

We consider off-policy temporal-difference (TD) learning in discounted Markov decision processes, where the goal is to evaluate a policy in a model-free way by...

Markov chain | Generalized bellman equation | Approximate policy evaluation | Randomized stopping time | Reinforcement learning | Markov decision process | Temporal-difference method | approximate policy evaluation | randomized stopping time | temporal-difference method | STOCHASTIC-APPROXIMATION | generalized Bellman equation | AUTOMATION & CONTROL SYSTEMS | COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE | reinforcement learning

Markov chain | Generalized bellman equation | Approximate policy evaluation | Randomized stopping time | Reinforcement learning | Markov decision process | Temporal-difference method | approximate policy evaluation | randomized stopping time | temporal-difference method | STOCHASTIC-APPROXIMATION | generalized Bellman equation | AUTOMATION & CONTROL SYSTEMS | COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE | reinforcement learning

Journal Article

Journal de mathématiques pures et appliquées, ISSN 0021-7824, 02/2019, Volume 122, pp. 164 - 197

We consider a family of optimal control problems in the plane with dynamics and running costs possibly discontinuous across a two-scale oscillatory interface....

Homogenization | Discontinuous Hamiltonian | Hamilton–Jacobi equations | Effective transmission condition | Oscillatory interface | MATHEMATICS | Hamilton-Jacobi equations | MATHEMATICS, APPLIED | BELLMAN APPROACH | JUNCTION PROBLEMS | Numerical Analysis | Computer Science

Homogenization | Discontinuous Hamiltonian | Hamilton–Jacobi equations | Effective transmission condition | Oscillatory interface | MATHEMATICS | Hamilton-Jacobi equations | MATHEMATICS, APPLIED | BELLMAN APPROACH | JUNCTION PROBLEMS | Numerical Analysis | Computer Science

Journal Article

SIAM Journal on Scientific Computing, ISSN 1064-8275, 2015, Volume 37, Issue 1, pp. A181 - A200

We present an accelerated algorithm for the solution of static Hamilton-Jacobi-Bellman equations related to optimal control problems. Our scheme is based on a...

Semi-Lagrangian schemes | Policy iteration | Dynamic programming | Hamilton–Jacobi equations | Optimal control | Hamilton-Jacobi equations | JACOBI-BELLMAN EQUATIONS | SCHEME | MATHEMATICS, APPLIED | dynamic programming | optimal control | CONVERGENCE | semi-Lagrangian schemes | policy iteration | Policies | Algorithms | Computation | Mathematical analysis | Joining | Mathematical models | Iterative methods | Convergence | Numerical Analysis | Mathematics | Optimization and Control

Semi-Lagrangian schemes | Policy iteration | Dynamic programming | Hamilton–Jacobi equations | Optimal control | Hamilton-Jacobi equations | JACOBI-BELLMAN EQUATIONS | SCHEME | MATHEMATICS, APPLIED | dynamic programming | optimal control | CONVERGENCE | semi-Lagrangian schemes | policy iteration | Policies | Algorithms | Computation | Mathematical analysis | Joining | Mathematical models | Iterative methods | Convergence | Numerical Analysis | Mathematics | Optimization and Control

Journal Article

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Multidimensional viscosity solution theory of semi-linear partial differential equations

Journal of Mathematical Analysis and Applications, ISSN 0022-247X, 12/2017, Volume 456, Issue 2, pp. 1347 - 1364

In this paper, we concern the multidimensional viscosity solution theory of a kind of semi-linear partial differential equations (PDEs). A new definition of...

BSDEs | Multidimensional PDEs | Comparison theorem | Viscosity solution | MATHEMATICS | JACOBI-BELLMAN EQUATIONS | MATHEMATICS, APPLIED | MONOTONE SYSTEMS | GAMES | PDES | UNIQUENESS | Differential equations

BSDEs | Multidimensional PDEs | Comparison theorem | Viscosity solution | MATHEMATICS | JACOBI-BELLMAN EQUATIONS | MATHEMATICS, APPLIED | MONOTONE SYSTEMS | GAMES | PDES | UNIQUENESS | Differential equations

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

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