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# Hamilton–Jacobi–Bellman Equations

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arXiv Nov 12, 2026
Newton's method linearizes viscous Hamilton–Jacobi equations directly: geometric convergence for nonconvex Hamiltonians, quadratic near the solution
For the Cauchy problem for viscous Hamilton–Jacobi equations with Hamiltonians that need not be convex, each Newton step solves a linear parabolic equation obtained by linearizing at the current iterate, without a control representation or min–max structure. Any admissible initial guess gives geometric convergence, and initial guesses close enough to the solution give quadratic convergence in the parabolic Hölder norm; when the Hamiltonian admits a control representation, the Newton iterates coincide with classical policy iteration, which therefore inherits both rates.
For the Cauchy problem for viscous Hamilton–Jacobi equations with Hamiltonians that need not be convex, each Newton step solves a linear parabolic equation obtained by linearizing at the current iterate, without a control representation or min–max structure. Any admissible initial guess gives geometric convergence, and initial guesses close enough to the solution give quadratic convergence in the parabolic Hölder norm; when the Hamiltonian admits a control representation, the Newton iterates coincide with classical policy iteration, which therefore inherits both rates.
Diogo A. Gomes（First author）
Newton's Method Hamilton–Jacobi–Bellman Equations Policy Iteration Physics-Informed Neural Networks
arXiv Oct 8, 2026
Kernel and neural-network solvers for HJB equations on unbounded domains: convergence and error bounds from empirical residuals
For two non-monotone, mesh-free approximations of Hamilton–Jacobi–Bellman equations on unbounded spatial domains—Wendland kernel collocation under a native space norm constraint and physics-informed neural networks with smooth activations under a Sobolev-norm constraint—the work derives error bounds from computable empirical squared residuals: norm constraints, an interpolation inequality and a controlled-diffusion truncation estimate convert the empirical loss into the pointwise residual control needed for viscosity-solution error estimates, yielding uniform convergence on compact sets plus a posteriori and high-probability quantitative bounds governed by the fill distance for kernels and by network and sample sizes for neural networks.
For two non-monotone, mesh-free approximations of Hamilton–Jacobi–Bellman equations on unbounded spatial domains—Wendland kernel collocation under a native space norm constraint and physics-informed neural networks with smooth activations under a Sobolev-norm constraint—the work derives error bounds from computable empirical squared residuals: norm constraints, an interpolation inequality and a controlled-diffusion truncation estimate convert the empirical loss into the pointwise residual control needed for viscosity-solution error estimates, yielding uniform convergence on compact sets plus a posteriori and high-probability quantitative bounds governed by the fill distance for kernels and by network and sample sizes for neural networks.
Information and Mathematical Science and Bioinformatics (Japan)
Yumiharu Nakano（First author, Corresponding author）
Kernel Collocation Physics-Informed Neural Networks Hamilton–Jacobi–Bellman Equations Convergence Analysis Viscosity Solutions

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