Hamilton-Jacobi Equation Research Articles

Time-Optimal Rendezvous for Cooperative UUV Swarms Using Hamilton-Jacobi Partial Differential Equations

A new approach to time-optimal path planning for cooperative swarms of underwater vehicles in the presence of ocean currents is presented. Specifically, the determination of a time-optimal rendezvous among a swarm of vehicles is carried out by solving a family of Hamilton-Jacobi partial differential equations. The resulting algorithm is fast, globally time-optimal, and scales linearly with the number of vehicles. The method is validated through several examples for which optimal trajectories are known from the calculus of variations. Additional examples are also included.

Large-deviation principles of switching Markov processes via Hamilton-Jacobi equations

Large-deviation principles of switching Markov processes via Hamilton-Jacobi equations

Well-posedness for a transmission problem connecting first and second-order operators

We establish the existence and uniqueness of viscosity solutions within a domain Ω ⊆ ℝn for a class of equations governed by elliptic and eikonal type equations in disjoint regions. Our primary motivation stems from the Hamilton-Jacobi equation that arises in the context of a stochastic optimal control problem.

Conservation laws and Hamilton-Jacobi equations on a junction: The convex case

Conservation laws and Hamilton-Jacobi equations on a junction: The convex case

Combination Therapy-Based Adaptive Control for Organism Using Medicine Dosage Regulation Mechanism.

In this article, the optimal control strategy for organism is investigated by using the adaptive dynamic programming (ADP) method under the architecture of nonzero-sum games (NZSGs). First, a tumor model is established to formulate the interaction relationships among normal cells, tumor cells, endothelial cells, and the concentrations of drugs. Then, the ADP-based method of single-critic network architecture is proposed to approximate the coupled Hamilton-Jacobi equations (HJEs) under the medicine dosage regulation mechanism (MDRM). According to the game theory, the approximate MDRM-based optimal strategy can be derived, which is of great practical significance. Owing to the proposed mechanism, the dosages of the chemotherapy and anti-angiogenic drugs can be regulated timely and necessarily. Furthermore, the stability of the closed-loop system with the obtained strategy is analyzed via the Lyapunov theory. Finally, a simulation experiment is conducted to verify the effectiveness of the proposed method.

Large deviations for trajectory observables of diffusion processes in dimension d > 1 in the double limit of large time and small diffusion coefficient

For diffusion processes in dimension d > 1, the statistics of trajectory observables over the time-window [0,T] can be studied via the Feynman–Kac deformations of the Fokker–Planck generator, which can be interpreted as Euclidean non-Hermitian electromagnetic quantum Hamiltonians. It is interesting to compare the four regimes corresponding to the time T, either finite or large, and to the diffusion coefficient D, either finite or small. (1) For finite T and finite D, one needs to consider the full time-dependent quantum problem that involves the full spectrum of the Hamiltonian. (2) For large time T→+∞ and finite D, one only needs to consider the ground-state properties of the quantum Hamiltonian to obtain the generating function of rescaled cumulants and to construct the corresponding canonical conditioned processes. (3) For finite T and D → 0, one only needs to consider the dominant classical trajectory and its action satisfying the Hamilton–Jacobi equation, as in the semi-classical Wentzel–Kramers–Brillouin (WKB) approximation of quantum mechanics. (4) In the double limit T→+∞ and D → 0, the simplifications in the large deviations in TD of trajectory observables can be analyzed via the two orders of limits, i.e. either from the limit D → 0 of the ground-state properties of the quantum Hamiltonians of (2), or from the limit of long classical trajectories T→+∞ in the semi-classical WKB approximation of (3). This general framework is illustrated in dimension d = 2 with rotational invariance.

Event-Triggered Robust Adaptive Dynamic Programming for Multiplayer Stackelberg-Nash Games of Uncertain Nonlinear Systems.

In this article, an event-triggered robust adaptive dynamic programming (ETRADP) algorithm is developed to solve a class of multiplayer Stackelberg-Nash games (MSNGs) for uncertain nonlinear continuous-time systems. Considering the different roles of players in the MSNG, the hierarchical decision-making process is described as the designed value functions for the leader and all followers, which assist to transform the robust control problem of the uncertain nonlinear system into an optimal regulation problem of the nominal system. Then, an online policy iteration algorithm is formulated to solve the derived coupled Hamilton-Jacobi equation. Meanwhile, an event-triggered mechanism is designed to alleviate computational and communication burdens. Moreover, critic neural networks (NNs) are constructed to obtain the event-triggered approximate optimal control polices for all players, which constitute the Stackelberg-Nash equilibrium of the MSNG. By using Lyapunov's direct method, the stability of the closed-loop uncertain nonlinear system is guaranteed under the ETRADP-based control scheme in the sense of uniform ultimate boundedness. Finally, a numerical simulation is provided to demonstrate the effectiveness of the present ETRADP-based control scheme.

Discontinuous solutions to the hamilton jacobi equation under second order interiority hypotheses

Discontinuous solutions to the hamilton jacobi equation under second order interiority hypotheses

Optimal Control of Infinite-Dimensional Differential Systems with Randomness and Path-Dependence and Stochastic Path-Dependent Hamilton–Jacobi Equations

This paper is devoted to the stochastic optimal control problem of infinite-dimensional differential systems allowing for both path-dependence and measurable randomness. As opposed to the deterministic path-dependent cases studied by Bayraktar and Keller [J. Funct. Anal. 275 (2018) 2096-2161], the value function turns out to be a random field on the path space and it is characterized by a stochastic path-dependent Hamilton-Jacobi (SPHJ) equation. A notion of viscosity solution is proposed and the value function is proved to be the unique viscosity solution to the associated SPHJ equation.

Analysis, Symmetry and Asymptotic Behaviour of Solutions for the Belousov-Zhabotinsky Model

This research provides analytical insights in connection with the solutions of the Oregonator model, a refined iteration of the iconic Belousov-Zhabotinsky (BZ) reaction problem. This chemical process, initially observed by B. P. Belousov while replicating the Krebs cycle in vitro and later modified by Zhabotinsky using Fe-phenanthroline (ferroin), has become a hallmark example of non-linear dynamics, chaos theory, and has parallels in various biological systems. Our study systematically delves into the boundedness, regularity, and possible symmetries of weak solutions. We explore traveling waves using the Tanh-method, alongside examining asymptotic solutions entrenched in self-similar forms and exponential scaling leading to a Hamilton-Jacobi equation. This research emphasizes on mathematical arguments along with the dynamics of the involved chemical concentrations. We provide new forms of analytical solutions showing them in a comprehensive manner that connects with the interpretation of the Oregonator model and its broader implications in chemical systems.

MHD equilibria with nonconstant pressure in nondegenerate toroidal domains

We prove the existence of piecewise smooth MHD equilibria in three-dimensional toroidal domains of \mathbb R^3 where the pressure is constant on the boundary but not in the interior. The pressure is piecewise constant and the plasma current exhibits an arbitrary number of current sheets. We also establish the existence of free boundary steady states surrounded by vacuum with an external surface current. The toroidal domains where these equilibria are shown to exist need not be small perturbations of an axisymmetric domain, and in fact they can have any knotted topology. The building blocks we use in our construction are analytic toroidal domains satisfying a certain nondegeneracy condition, which roughly states that there exists a force-free field that is ergodic on the surface of the domain. The proof involves three main ingredients: a gluing construction of piecewise smooth MHD equilibria, a Hamilton–Jacobi equation on the two-dimensional torus that can be understood as a nonlinear deformation of the cohomological equation (so the nondegeneracy assumption plays a major role in the corresponding analysis), and a new KAM theorem tailored for the study of divergence-free fields in three dimensions whose Poincaré map cannot be computed explicitly.

Optimization-based level-set re-initialization: A robust interface preserving approach in multiphase problems

In spite of its overall efficiency and robustness for capturing the interface in multiphase fluid dynamics simulations, the well-known shortcoming of the level-set method is associated with the lack of a systematic approach for preserving the regularity of the distance function. This is mainly due to the stretching (or compressing) effect of the strain rate especially in the vicinity of the liquid-gas interface. Level-set re-initialization is an effective treatment for this issue. However, the traditional approach based on the hyperbolic Hamilton–Jacobi equation is a computationally expensive procedure. Crucially, due to the hyperbolic nature of the formulation, the accuracy of the results hinges significantly on the specialized handling of blind spots near the liquid-gas interface intersecting the substrate. The present work proposes a two-step elliptic level-set re-initialization approach that strictly preserves the location of zero level-set via incorporation of an element splitting process. The primary initialization step helps remove any non-smoothness in the to-be regularized level-set function dramatically improving the efficiency of the secondary optimization step. Geometric representation of the boundary conditions is utilized in the initialization step, while the optimization step minimizes the reliance of the results on the treatment of the blind spots. The performance of the proposed method is examined for free and sessile three-dimensional droplets.

A new third-order EXP-WENO scheme for Hamilton–Jacobi equations

The authors of this research paper have introduced a new method for solving Hamilton–Jacobi (HJ) equations called third-order weighted essentially nonoscillatory (WENO) method. This method uses exponential polynomials to construct numerical fluxes and smoothness indicators, which helps distinguish between singular and smooth regions more efficiently. The smoothness indicators are created using a finite difference operator that eliminates exponential polynomials. The numerical flux is constructed using an interpolation method based on exponential polynomials, which results in better outcomes around steep gradients. The new method maintains a high level of accuracy (i.e. three) in smooth regions, even near critical points. The authors have presented some numerical results to demonstrate the effectiveness of the new method and compared it with other WENO methods.

A continuous dependence estimate for viscous Hamilton–Jacobi equations on networks with applications

A continuous dependence estimate for viscous Hamilton–Jacobi equations on networks with applications

Initial boundary value problem for 1D scalar balance laws with strictly convex flux

A Lax-Oleĭnik type explicit formula for 1D scalar balance laws has been recently obtained for the pure initial value problem by Adimurthi et al. in [2]. In this article, by introducing a suitable boundary functional, we establish a Lax-Oleĭnik type formula for the initial-boundary value problem. For the pure initial value problem, the solution for the corresponding Hamilton-Jacobi equation turns out to be the minimizer of a functional on the set of curves known as h-curves. In the present situation, part of the h-curve joining any two points in the quarter plane may cross the boundary x=0. This phenomenon breaks the simplicity of the minimization process through the boundary functional compared to the case of conservation laws. Moreover, this complicates the verification of the boundary condition in the sense of Bardos, le Roux, and Nédélec [4]. To verify the boundary condition, the boundary points are classified into three types depending on the structure of the minimizers at those points. Finally, by introducing characteristic triangles, we construct generalized characteristics and show that the explicit solution is entropy admissible.

Melting interfaces in induction heated bodies

A numerical study of the solution of the mathematical problem corresponding to the melting of a solid metallic body, heated by the action of an exterior electromagnetic field, is presented. The extended finite element method is used for the numerical analysis. The melting interface is geometrically modeled by a level set function whose evolution is obtained by applying a discontinuous Galerkin method to the Hamilton-Jacobi equations.

Revisiting Schrödinger's fourth-order, real-valued wave equation and the implication from the resulting energy levels.

In his seminal part IV, Annalen der Physik vol. 81, 1926 paper, Schrödinger has developed a clear understanding about the wave equation that produces the correct quadratic dispersion relation for matter-waves and he first presents a real-valued wave equation that is fourth-order in space and second-order in time. In the view of the mathematical difficulties associated with the eigenvalue analysis of a fourth-order, differential equation in association with the structure of the Hamilton-Jacobi equation, Schrödinger splits the fourth-order real operator into the product of two, second-order, conjugate complex operators and retains only one of the two complex operators to construct his iconic second-order, complex-valued wave equation. In this paper, we show that Schrödinger's original fourth-order, real-valued wave equation is a stiffer equation that produces higher energy levels than his second-order, complex-valued wave equation that predicts with remarkable accuracy the energy levels observed in the atomic line spectra of the chemical elements. Accordingly, the fourth-order, real-valued wave equation is too stiff to predict the emitted energy levels from the electrons of the chemical elements; therefore, the paper concludes that quantum mechanics can only be described with the less stiff, second-order, complex-valued wave equation.

Fine Properties of Geodesics and Geodesic λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda $$\\end{document}-Convexity for the Hellinger–Kantorovich Distance

We study the fine regularity properties of optimal potentials for the dual formulation of the Hellinger–Kantorovich problem (HK\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf{H}\\!\\!\ extsf{K}$$\\end{document}), providing sufficient conditions for the solvability of the primal Monge formulation. We also establish new regularity properties for the solution of the Hamilton–Jacobi equation arising in the dual dynamic formulation of HK\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf{H}\\!\\!\ extsf{K}$$\\end{document}, which are sufficiently strong to construct a characteristic transport-growth flow driving the geodesic interpolation between two arbitrary positive measures. These results are applied to study relevant geometric properties of HK\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf{H}\\!\\!\ extsf{K}$$\\end{document} geodesics and to derive the convex behaviour of their Lebesgue density along the transport flow. Finally, exact conditions for functionals defined on the space of measures are derived that guarantee the geodesic λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda $$\\end{document}-convexity with respect to the Hellinger–Kantorovich distance. Examples of geodesically convex functionals are provided.

Long-time behavior of stochastic Hamilton-Jacobi equations

The long-time behavior of stochastic Hamilton-Jacobi equations is analyzed, including the stochastic mean curvature flow as a special case. In a variety of settings, new and sharpened results are obtained. Among them are (i) a regularization by noise phenomenon for the mean curvature flow with homogeneous noise which establishes that the inclusion of noise speeds up the decay of solutions, and (ii) the long-time convergence of solutions to spatially inhomogeneous stochastic Hamilton-Jacobi equations. A number of motivating examples about nonlinear stochastic partial differential equations are presented in the appendix.

Stability and weak KAM solutions of contact Hamilton-Jacobi equation

We are concerned with the stability of viscosity solutions to contact Hamilton-Jacobi equation H ( x , ∂ x u ( x ) , u ( x ) ) = 0 , x ∈ M , \begin{align*} H(x,\partial _x u(x),u(x))=0, \quad x\in M, \end{align*} where H = H ( x , p , u ) H=H(x,p,u) satisfies Tonelli conditions. We study the relationship between Lyapunov stability of viscosity solutions and the structure of the set of weak KAM solutions to the contact Hamilton-Jacobi equation.