Reduction theory has played a major role in the study of Hamiltonian systems. On the other hand, the Hamilton-Jacobi theory is one of the main tools to integrate the dynamics of certain Hamiltonian problems and a topic of research on its own. Moreover, the construction of several symplectic integrators rely on approxim…
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48 results for “Hamilton-Jacobi”
Hamilton-Jacobi Diffietiesmath.DG
Diffieties formalize geometrically the concept of differential equations. We introduce and study Hamilton-Jacobi diffieties. They are finite dimensional subdiffieties of a given diffiety and appear to play a special role in the field theoretic version of the geometric Hamilton-Jacobi theory.
Survey reviews Hamilton-Jacobi theory in various geometric settings, focusing on Jacobi and Leibniz identities.
problem Analyzing Hamilton-Jacobi theory across different geometric backgrounds.
method Geometric review of Hamilton-Jacobi theory, focusing on Jacobi and Leibniz identities.
result Novel Hamilton-Jacobi equation for conformal Hamiltonian vector fields.
In this paper we develop a Hamilton-Jacobi theory in the setting of almost Poisson manifolds. The theory extends the classical Hamilton-Jacobi theory and can be also applied to very general situations including nonholonomic mechanical systems and time dependent systems with external forces.
The first widely used financial model is linked to dynamical Hamilton jacobi model
This paper studies Hamilton-Jacobi equations for magnetic systems with constraints.
problem Understanding dynamics of magnetic systems with geometric constraints.
method Developed Hamilton-Jacobi equations for magnetic systems with nonholonomic constraints.
result Revealed relationships between magnetic structures, constraints, and dynamics.
The Hamilton-Jacobi equation for a Hamiltonian section on a Lie affgebroid is introduced and some examples are discussed.
I briefly review my proposal about how to extend the geometric Hamilton-Jacobi theory to higher derivative field theories on fiber bundles.
The interplay between the Hamilton-Jacobi theory of orthogonal separation of variables and the theory of group actions is investigated based on concrete examples.
In this paper, we give precisely the geometric constraint conditions of canonical symplectic form and regular reduced symplectic forms for the dynamical vector fields of a regular controlled Hamiltonian (RCH) system and its regular reduced systems, which are called the Type I and Type II of Hamilton-Jacobi equations. A…
Paper introduces stochastic HJB on Jacobi structures.
problem Stochastic analysis on Jacobi manifolds.
method Global stochastic analysis techniques, extending Bismut and Lázaro-Camí work.
result Proposes a stochastic HJB framework.
Hamilton-Jacobi Equations of Controlled Magnetic Hamiltonian System with Nonholonomic Constraintmath.SG
Study magnetic Hamiltonian systems with constraints, deriving Hamilton-Jacobi equations.
problem Understanding dynamics of controlled magnetic Hamiltonian systems with constraints.
method Defined CMH system, derived Hamilton-Jacobi equations for different constraints.
result Invariant solutions of Hamilton-Jacobi equations under CMH-equivalence.
Model quantifies uncertainty's impact on European option prices.
problem Uncertainty in market volatility risk affects option pricing.
method Hamilton-Jacobi-Bellman framework and finite element method.
result Dependence of Delta on uncertainty is nonlinear and varied.
Study solves optimal portfolio selection using HJB equation.
problem Optimal portfolio selection problem.
method Maximal monotone operator method, Banach fixed-point theorem, Fourier transform, monotone operators technique.
result Existence and uniqueness of solution to HJB equation.
Researchers tackle the globalization problem of locally cosymplectic Hamiltonian dynamics.
problem Globalization problem of locally cosymplectic Hamiltonian dynamics.
method Investigate the geometry of locally conformally cosymplectic manifolds and provide a geometric Hamilton-Jacobi theory.
result Provide a geometric Hamilton-Jacobi theory on locally conformally cosymplectic manifolds.
We establish some perturbed minimization principles, and we develop a theory of subdifferential calculus, for functions defined on Riemannian manifolds. Then we apply these results to show existence and uniqueness of viscosity solutions to Hamilton-Jacobi equations defined on Riemannian manifolds.
We extend the geometric Hamilton-Jacobi formalism for hamiltonian mechanics to higher order field theories with regular lagrangian density. We also investigate the dependence of the formalism on the lagrangian density in the class of those yelding the same Euler-Lagrange equations.
Researchers find metric lines in SE(2) using Hamilton-Jacobi theory.
problem Identifying metric lines in the Special Euclidean group on the plane.
method Alternative proof using Hamilton-Jacobi theory.
result Metric lines in SE(2) are identified.
Minimax solutions are weak solutions to Cauchy problems involving Hamilton--Jacobi equations, constructed from generating families quadratic at infinity of their geometric solutions. We give a complete description of minimax solutions and we classify their generic singularities of codimension not greater than 2.
Develops Hamilton-Jacobi theory for non-conservative field theories in k-contact geometry.
problem Analyzes non-conservative field theories, especially dissipative systems.
method Introduces evolution k-contact k-vector fields and develops two Hamilton-Jacobi theories.
result Recover ordinary contact Hamilton-Jacobi theory as k=1, and enlarges application range.
New bracket unifies nonholonomic dynamics and Hamilton-Jacobi theory.
problem Unified description of nonholonomic dynamics and Hamilton-Jacobi theory.
method Defined and proved coincidence of three nonholonomic brackets.
result Three nonholonomic brackets coincide.
Paper designs Poisson integrators using machine learning.
problem Designing integrators that preserve Poisson geometry.
method Reformulated as an optimization problem in Hamilton-Jacobi PDE, solved using machine learning.
result Machine learning approximates solutions to Hamilton-Jacobi PDE.
Deep neural network approximation for high-dimensional parabolic Hamilton-Jacobi-Bellman equationsmath.NA
Deep neural nets approximate high-dimensional HJB equations efficiently.
problem Approximating solutions to high-dimensional HJB equations.
method Deep neural networks for approximating solutions.
result Deep neural networks can approximate solutions without the curse of dimensionality.
A Hamilton-Jacobi theory for general dynamical systems, defined on fibered phase spaces, has been recently developed. In this paper we shall apply such a theory to contact Hamiltonian systems, as those appearing in thermodynamics and on geodesic flows in fluid mechanics. We first study the partial and complete solution…
Nonholonomic mechanical systems have been attracting more interest in recent years because of their rich geometric properties and their applications in Engineering. In all generality, we discuss the reduction of a Hamilton-Jacobi theory for systems subject to nonholonomic constraints and that are invariant under the ac…
In this paper we develope, in a geometric framework, a Hamilton-Jacobi Theory for general dynamical systems. Such a theory contains the classical theory for Hamiltonian systems on a cotangent bundle and recent developments in the framework of general symplectic, Poisson and almost-Poisson manifolds (including some appr…
The concept of subdifferentiability is studied in the context of Finsler manifolds (modeled on a Banach space with a Lipschitz bump function). A class of Hamilton-Jacobi equations defined on Finsler manifolds is studied and several results related to the existence and uniqueness of viscosity solutions…
Market makers optimize trading with a new implicit scheme for complex inequalities.
problem Optimizing trading in a limit order book with stochastic and impulse control.
method Implicit numerical scheme coupled with policy iteration algorithm.
result Convergence to the unique viscosity solution of the HJBQVI.
Nonholonomic Controlled Hamiltonian System: Symmetric Reduction and Hamilton-Jacobi Equationsmath.SG
This paper studies nonholonomic constraints in Hamiltonian systems, deriving equations and theorems.
problem Analyzing nonholonomic constraints in Hamiltonian systems.
method Deriving distributional RCH systems, geometric constraint conditions, and Hamilton-Jacobi theorems.
result Derives precise geometric constraint conditions and Hamilton-Jacobi theorems for nonholonomic systems.
Study on Tukey depth in machine learning using Hamilton-Jacobi equations.
problem Understanding Tukey depth in machine learning applications.
method Derive necessary conditions for Tukey depth in continuum limit, formulating them as a Hamilton-Jacobi equation.
result Prove existence and uniqueness of viscosity solutions for the derived equation, which bounds Tukey depth.
Deep learning for HJB PDEs using synthetic data and residual minimization.
problem Solving Hamilton-Jacobi-Bellman PDEs for optimal control problems.
method Gradient-augmented synthetic dataset for supervised learning, residual minimization.
result Improves accuracy and efficiency of deep learning for HJB PDEs.
A close relationship between the classical Hamilton-Jacobi theory and the kinematic reduction of control systems by decoupling vector fields is shown in this paper. The geometric interpretation of this relationship relies on new mathematical techniques for mechanics defined on a skew-symmetric algebroid. This geometric…
Paper solves investment strategy optimization with deep learning.
problem Maximizing investor utility with optimal asset allocation.
method Solves PDEs with Deep Galerkin method.
result Deep learning algorithm outperforms finite difference method.
Deep learning method proves convergence for solving HJI equations.
problem Solving Hamilton-Jacobi-Isaacs equations for reachability analysis.
method Uniform convergence guarantee for DeepReach algorithm.
result DeepReach algorithm converges to classical solution of HJI equation.
A hyperkähler 4-metric with a triholomorphic SU(2) action gives rise to a family of confocal quadrics in Euclidean 3-space when cast in the canonical form of a hyperkähler 4-metric metric with a triholomorphic circle action. Moreover, at least in the case of geodesics orthogonal to the U(1) fibres, both the covariant S…
The aim of this paper is to construct and analyze solutions to a class of Hamilton-Jacobi-Bellman equations with range bounds on the optimal response variable. Using the Riccati transformation we derive and analyze a fully nonlinear parabolic partial differential equation for the optimal response function. We construct…
We solve continuous-time reinforcement learning using distributional Hamilton-Jacobi-Bellman equations.
problem Predicting the distribution of returns in continuous-time, stochastic environments.
method We derive a distributional Hamilton-Jacobi-Bellman equation for Itô diffusions and Feller-Dynkin processes, and propose an algorithm based on a JKO scheme.
result We propose an online control algorithm that can be used to approximately solve the distributional HJB equation.
Introduces intrinsic Hopf-Lax semigroup linking to intrinsic slope.
problem Understanding intrinsic Hopf-Lax semigroup and its relation to intrinsic slope.
method Introduces and proves the link between intrinsic Hopf-Lax semigroup and intrinsic slope.
result Intrinsic Hopf-Lax semigroup is a subsolution of Hamilton-Jacobi type equality.
Hamilton-Jacobi Equations for Nonholonomic Reducible Hamiltonian Systems on a Cotangent Bundlemath.SG
In this paper, for a variety of nonholonomic (reducible) Hamiltonian systems, we first give to various distributional Hamiltonian systems, by analyzing carefully the dynamics and structures of the nonholonomic Hamiltonian systems. Secondly, we derive precisely the geometric constraint conditions of the induced distribu…
Geometric structures help in understanding thermodynamics.
problem Understanding thermodynamic systems using geometric methods.
method Using almost cosymplectic structures and variational arguments.
result Evolution equations are derived and discussed.
Symmetric Reduction and Hamilton-Jacobi Equations of the Controlled Underwater Vehicle-Rotor Systemmath.SG
In this paper, we first give the regular point reduction and the two types of Hamilton-Jacobi equation for a regular controlled Hamiltonian (RCH) system with symmetry and momentum map on the generalization of a semidirect product Lie group. Next, as an application of the theoretical results, we consider the underwater …
Symplectic groupoids create Poisson integrators for complex systems.
problem Creating efficient integrators for non-linear Poisson structures.
method Recursive solutions of Hamilton-Jacobi equation, interpreted as Lagrangian bisections.
result Constructs Poisson integrators using symplectic groupoids.
We study the singular locus of solutions to Hamilton-Jacobi equations with a Hamiltonian independent of . In a previous paper, we proved that the singular locus is what we call a balanced split locus. In this paper, we find and classify all balanced split sets, identifying the cases where the only balanced split loc…
The Hamilton-Jacobi problem is revisited bearing in mind the consequences arising from a possible bi-Hamiltonian structure. The problem is formulated on the tangent bundle for Lagrangian systems in order to avoid the bias of the existence of a natural symplectic structure on the cotangent bundle. First it is developed …
We define a Hamilton-Jacobi semigroup acting on continuous functions on a compact length space. Following a strategy of Bobkov, Gentil and Ledoux, we use some basic properties of the semigroup to study geometric inequalities related to concentration of measure. Our main results are that (1) a Talagrand inequality on a …
Study of Hamilton-Jacobi Theory with symmetries and integrability by quadratures.
problem Hamilton-Jacobi equation in systems with symmetries.
method Constructing complete solutions and solving reconstruction equations.
result Explicit expressions for exponential curves in Lie groups, valid for all elements in the Lie algebra.
In this paper we propose and analyze a method based on the Riccati transformation for solving the evolutionary Hamilton-Jacobi-Bellman equation arising from the stochastic dynamic optimal allocation problem. We show how the fully nonlinear Hamilton-Jacobi-Bellman equation can be transformed into a quasi-linear paraboli…
Machine Learning and Hamilton-Jacobi-Bellman Equation for Optimal Decumulation: a Comparison Studymath.OC
A neural network approach solves optimal decumulation problems for pension plans.
problem Optimal asset allocation and withdrawal strategies for DC pension holders.
method Data-driven neural network optimization with customized activation functions.
result The neural network approach learns near-optimal solutions comparable to HJB PDE methods.