Hamilton-Jacobi Equation Research Articles

Finite dimensional Hamiltonian systems and the algebro-geometry solution of the nonlocal reverse space–time sine-Gordon equation

In this paper, we investigate the algebro-geometric solution of the novel integrable nonlocal reverse space–time sine-Gordon equation through the framework of nonlocal finite-dimensional Lie-Poisson Hamiltonian systems, which are generated by the nonlinearization method. The nonlocal reverse space–time sine-Gordon equation is introduced via the Lenard recursion equations. Under suitable constraints, this equation is nonlinearized into two nonlocal finite-dimensional Lie-Poisson Hamiltonian systems. By applying the separated variables, the nonlocal finite-dimensional Lie-Poisson Hamiltonian systems are reformulated as the canonical Hamiltonian systems with the standard symplectic structures. Furthermore, the relationship between the action–angle coordinates and the Jacobi inversion problem is established based on the Hamilton–Jacobi theory. In the end, the algebraic-geometric solution of the nonlocal reverse space–time sine-Gordon equation is obtained using the Riemann-Jacobi inversion.

Optimal Rate of Convergence in Periodic Homogenization of Viscous Hamilton-Jacobi Equations

Optimal Rate of Convergence in Periodic Homogenization of Viscous Hamilton-Jacobi Equations

Quantitative Convergence of a Discretization of Dynamic Optimal Transport Using the Dual Formulation

AbstractWe present a discretization of the dynamic optimal transport problem for which we can obtain the convergence rate for the value of the transport cost to its continuous value when the temporal and spatial stepsize vanish. This convergence result does not require any regularity assumption on the measures, though experiments suggest that the rate is not sharp. Via an analysis of the duality gap we also obtain the convergence rates for the gradient of the optimal potentials and the velocity field under mild regularity assumptions. To obtain such rates, we discretize the dual formulation of the dynamic optimal transport problem and use the mature literature related to the error due to discretizing the Hamilton–Jacobi equation.

Some regularity results of Hamilton-Jacobi equations with variable-order Caputo time derivative

Some regularity results of Hamilton-Jacobi equations with variable-order Caputo time derivative

Monotonicity, asymptotics and level sets for principal eigenvalues of some elliptic operators with shear flow

We investigate the joint effects of diffusion and advection on principal eigenvalues of some elliptic operators with shear flow. Some monotonicity and asymptotic behaviors of principal eigenvalues, with respect to diffusion rate and flow amplitude, are established. These analyses lead to a classification of topological structures of level sets for principal eigenvalues, as a function of diffusion rate and flow amplitude. Our analytical results provide a unifying viewpoint to understand mixing enhancement and dispersal-induced growth, which are apparently two unrelated phenomena, one in fluid mechanics and the other in population dynamics. In our analysis, some limiting Hamilton-Jacobi equations, blowup argument and limiting generalized principal eigenvalue problems play critical roles.

General Definitions of Information, Intelligence, and Consciousness from the Perspective of Generalized Natural Computing

Atoms themselves have no thoughts and cannot be thinking. Why does the human body, which is composed of atoms, have consciousness? The widely used concepts of information and intelligence in today’s science, which are related to this, do not yet have appropriate general definitions. Answering these interesting questions is a crucial issue for technological development in the historical context of human society entering the era of intelligence. The key lies in how to fully utilize the existing fundamental theories subtly related to information science. Here we attempt to give the definition of general information and general intelligence from the perspective of generalized natural computing, based on the least action principle, Hamilton-Jacobi equation, dynamic programming, reinforcement learning, and point out the relationship between the two. The least action principle for describing conservative systems can be seen as an intelligent manifestation of natural matter, and its equivalent form, the Hamilton-Jacobi equation, can be extended to describe quantum phenomena and is a special case of continuous dynamic programming equations. Dynamic programming is an efficient optimization method under deterministic models, while reinforcement learning, as a manifestation of biological intelligence, is its model-free version. The statement that reinforcement learning is the most promising machine learning method has a profound physical foundation. General information is defined as the degree to which a certain environmental element determines the behavior of the subject. General intelligence is defined as the automatic optimization ability of the action or value function of a system with a certain degree of conservatism. Intelligence is a basic property of material systems, rather than an emergent property that only complex systems possess. Consciousness is an advanced intelligent phenomenon, a reconstruction of quasi conservative systems based on complex systems.

Viscosity Solutions of Hamilton–Jacobi Equation in $$\textsf{RCD}(K,\infty )$$ Spaces and Applications to Large Deviations

Viscosity Solutions of Hamilton–Jacobi Equation in $$\textsf{RCD}(K,\infty )$$ Spaces and Applications to Large Deviations

An advection–diffusion equation with a generalized advection term: Well-posedness analysis and examples

We consider a Cauchy–Dirichlet problem for a semilinear advection–diffusion equation with a generalized advection term. Specific examples include an incision–diffusion landscape evolution model and a viscous Hamilton–Jacobi equation with an absorbing gradient term. We establish existence and uniqueness of a weak solution and its continuous dependence on initial data and a source term.

Coercive Hamilton–Jacobi equations in domains: the twin blow-ups method

Coercive Hamilton–Jacobi equations in domains: the twin blow-ups method

Dynamic event-triggered finite-horizon robust suboptimal control of multi-player systems with input disturbances

This paper presents an adaptive dynamic event-triggered robust finite-horizon suboptimal control scheme for multi-player nonzero-sum game of nonlinear systems with input disturbances based on dynamic programming (ADP) and integral sliding mode (ISM) control techniques. Firstly, dynamic event-triggered adaptive ISM controllers are designed to handle input disturbances that can force the system state to remain on the sliding manifold and relax the known upper bounds condition of input disturbances. Corresponding event-triggered condition is employed for ISM controller of each player to reduce the update frequency. Then, utilizing the obtained ISM dynamics, we employ ADP-based single critic neural networks to approximate the time-varying solution of the Hamilton–Jacobi equations. Thus, the event-triggered finite-horizon suboptimal controllers can be obtained, and corresponding state-based dynamic event-triggered condition is designed to improve the resource utilization. The stability property of the closed-loop system is demonstrated through the Lyapunov technique. Finally, the effectiveness of the proposed control scheme is confirmed through two simulation examples.

Optimal strategies design of active target defense differential game

In this paper, the optimal strategies are investigated for the differential game of active target defense. Specifically, we describe a pursuitevasion game with three agents (namely, an attacker, a target and a defender). For the three agents engagement scenario, two different differential game guidance laws–pure pursuit (PP) and proportional navigation (PN)–are presented for defender. Firstly, the relative motion kinematic models of the three agents are established. We subsequently derive the coupled Hamilton-Jacobi (HJ) equations to design the optimal control strategies for the three agents. Through the construction of actor-critic neural networks, optimal solutions are obtained. Then, the stability of the three-agent system is ensured, and the convergence to the Nash equilibrium equations is demonstrated. Finally, we simulate the optimal strategies employed by the attacker and the defender-target team, thereby highlighting the contrasting outcomes and superiority of different defensive approaches.

Mass minimization using the reaction-diffusion level set method by considering local stress constraints in an integral form

Research in topology optimization with stress constraints has shown that defining the stress constraint locally yields better performance compared to global methods. However, an examination of the formulas for local stress constraints reveals a limitation - the constraint is only satisfied at points where it is explicitly defined, failing to guarantee satisfaction across the entire design domain. To address this shortcoming, this paper proposes utilizing an integral form of the stress constraint. The integral formulation theoretically ensures that the stress constraint is satisfied over whole design domain. The objective is to minimize the mass of plane stress structures using reaction-diffusion level set method while incorporating local stress constraints in this integral form. The methodology utilizes finite element approximations for geometry and displacements, defining local stress constraints through an integral formulation. A Lagrangian function combines objective and constraint functions, with sensitivity analysis performed during optimization based on level set function changes. Structural boundaries are updated using the Hamilton-Jacobi equation. The paper presents numerical examples with varying loads and support conditions to demonstrate the effectiveness of this integral stress constraint approach. Results illustrate the capability of the proposed method to generate optimal topologies while satisfying stress constraints across the entire design space.

Quasinormal modes and universality of the Penrose limit of black hole photon rings

We study the physics of photon rings in a wide range of axisymmetric black holes admitting a separable Hamilton-Jacobi equation for the geodesics. Utilizing the Killing-Yano tensor, we derive the Penrose limit of the black holes, which describes the physics near the photon ring. The obtained plane wave geometry is directly linked to the frequency matrix of the massless wave equation, as well as the instabilities and Lyapunov exponents of the null geodesics. Consequently, the Lyapunov exponents and frequencies of the photon geodesics, along with the quasinormal modes, can be all extracted from a Hamiltonian in the Penrose limit plane wave metric. Additionally, we explore potential bounds on the Lyapunov exponent, the orbital and precession frequencies, in connection with the corresponding inverted harmonic oscillators and we discuss the possibility of photon rings serving as effective holographic horizons in a holographic duality framework for astrophysical black holes. Our formalism is applicable to spacetimes encompassing various types of black holes, including stationary ones like Kerr, Kerr-Newman, as well as static black holes such as Schwarzschild, Reissner-Nordström, among others.

On the improvement of Hölder seminorms in superquadratic Hamilton-Jacobi equations

We show in this paper that maximal Lq-regularity for time-dependent viscous Hamilton-Jacobi equations with unbounded right-hand side and superquadratic γ-growth in the gradient holds in the full range q>(N+2)γ−1γ. Our approach is based on new γ−2γ−1-Hölder estimates, which are consequence of the decay at small scales of suitable nonlinear space and time Hölder quotients. This is obtained by proving suitable oscillation estimates, that also give in turn some Liouville type results for entire solutions.

Space-time structures, light trajectories and shadows for Kerr-Newman-AdS black hole with surrounding perfect fluid matter in Rastall gravity

We present the horizon structures and the shadows for Kerr-Newman-AdS black hole located in perfect fluid matter field in Rastall gravity. By considering the Hamilton-Jacobi equation and Carter constant separable method, we obtain the null geodesic equations of the black hole space-time as well as the unstable circular photon orbits in terms of the impact parameters. The horizon structures of the space-time are calculated numerically. The results show that there are four roots for a general case of the horizon equation (the Cauchy horizon r−h , the event horizon r+h , the cosmological horizon r q of Rastall gravity and the cosmological horizon r c of cosmological constant), which lead to a specific characteristic with the subregions of the space-time. The influences of the cosmological horizons r q and r c on the light trajectories around the black hole are also investigated. Furthermore, we study the effects of the parameters, including the perfect fluid matter intensity N s , the Rastall parameter k λ, the cosmological constant Λ, the black hole spin a and the black hole charge Q on black hole shadows in detail.

A Representation Formula for Viscosity Solutions of Nonlocal Hamilton–Jacobi Equations and Applications

A Representation Formula for Viscosity Solutions of Nonlocal Hamilton–Jacobi Equations and Applications

A physics-informed neural SDE network for learning cellular dynamics from time-series scRNA-seq data.

Learning cellular dynamics through reconstruction of the underlying cellular potential energy landscape (aka Waddington landscape) from time-series single-cell RNA sequencing (scRNA-seq) data is a current challenge. Prevailing data-driven computational methods can be hampered by the lack of physical principles to guide learning from complex data, resulting in reduced prediction accuracy and interpretability when applied to infer cell population dynamics. Here, we propose PI-SDE, a physics-informed neural stochastic differential equation (SDE) framework that combines the Hamilton-Jacobi (HJ) equation and neural SDE to learn cellular dynamics. Grounded in potential energy theory of biological systems, PI-SDE integrates the principle of least action by enforcing the HJ equation when reconstructing cellular potential energy function. This approach not only facilitates accurate predictions, but also improves interpretability, especially in the reconstructed potential energy landscape. Through benchmarking on two real scRNA-seq datasets, we demonstrate the importance of incorporating the HJ regularization term in dynamic inference, especially in predicting gene expression at held-out time points. Meanwhile, the learned potential energy landscape provides biologically interpretable insights into the process of cell differentiation. Our framework enhances model performance, while maintaining robustness and stability. PI-SDE software is available at https://github.com/QiJiang-QJ/PI-SDE.

On the optimal rate for the convergence problem in mean field control

The goal of this work is to obtain (nearly) optimal rates for the convergence problem in mean field control. Our analysis covers cases where the solutions to the limiting problem may not be unique nor stable. Equivalently the value function of the limiting problem might not be differentiable on the entire space. Our main result is then to derive sharp rates of convergence in two distinct regimes. First, when the data is sufficiently regular, we obtain rates proportional to N−1/2, with N being the number of particles, and we verify that N−1/2 is indeed optimal in this setting. Second, when the data is merely Lipschitz and semi-concave with respect to the first Wasserstein distance, we obtain rates proportional to N−2/(3d+6). We do not expect this second estimate to be optimal, but it improves substantially on the existing literature. Moreover, we construct an example showing that the optimal rate is no faster than N−1/d, and we conjecture that the optimal rate should indeed be exactly N−1/d (at least when d≥3). The key argument in our approach consists in mollifying the value function of the limiting problem in order to produce functions that are almost classical sub-solutions to the limiting Hamilton-Jacobi equation (which is a PDE set on the space of probability measures). These sub-solutions can be projected onto finite dimensional spaces and then compared with the value functions associated with the particle systems. In the end, this comparison is used to prove the most demanding bound in the estimates. The key challenge therein is thus to exhibit an appropriate form of mollification. We do so by employing sup-convolution within a convenient functional Hilbert space. To make the whole easier, we limit ourselves to the periodic setting. We also provide some examples to show that our results are sharp up to some extent.

Adaptive dynamic programming for containment control with robustness analysis to iterative error: A global Nash equilibrium solution

Global Nash equilibrium is an optimal solution for each player in a graphical game. This paper proposes an iterative adaptive dynamic programming-based algorithm to solve the global Nash equilibrium solution for optimal containment control problem with robustness analysis to the iterative error. The containment control problem is transferred into the graphical game formulation. Sufficient conditions are given to decouple the Hamilton–Jacobi equations, which guarantee the solvability of the global Nash equilibrium solution. The iterative algorithm is designed to obtain the solution without any knowledge of system dynamics. Conditions of iterative error for global stability are given with rigorous proof. Compared with existing works, the design procedures of control gain and coupling strength are separated, which avoids trivial cases in the design procedure. The robustness analysis exactly quantifies the effect of the iterative error caused by various sources in engineering practice. The theoretical results are validated by two numerical examples with marginally stable and unstable dynamics of the leader.

Optimal Asymptotic Tracking Control for Nonzero-Sum Differential Game Systems with Unknown Drift Dynamics via Integral Reinforcement Learning

This paper employs an integral reinforcement learning (IRL) method to investigate the optimal tracking control problem (OTCP) for nonlinear nonzero-sum (NZS) differential game systems with unknown drift dynamics. Unlike existing methods, which can only bound the tracking error, the proposed approach ensures that the tracking error asymptotically converges to zero. This study begins by constructing an augmented system using the tracking error and reference signal, transforming the original OTCP into solving the coupled Hamilton–Jacobi (HJ) equation of the augmented system. Because the HJ equation contains unknown drift dynamics and cannot be directly solved, the IRL method is utilized to convert the HJ equation into an equivalent equation without unknown drift dynamics. To solve this equation, a critic neural network (NN) is employed to approximate the complex value function based on the tracking error and reference information data. For the unknown NN weights, the least squares (LS) method is used to design an estimation law, and the convergence of the weight estimation error is subsequently proven. The approximate solution of optimal control converges to the Nash equilibrium, and the tracking error asymptotically converges to zero in the closed system. Finally, we validate the effectiveness of the proposed method in this paper based on MATLAB using the ode45 method and least squares method to execute Algorithm 2.