Hamilton-Jacobi Equation Research Articles

From Second-Order Differential Geometry to Stochastic Geometric Mechanics

Classical geometric mechanics, including the study of symmetries, Lagrangian and Hamiltonian mechanics, and the Hamilton–Jacobi theory, are founded on geometric structures such as jets, symplectic and contact ones. In this paper, we shall use a partly forgotten framework of second-order (or stochastic) differential geometry, developed originally by L. Schwartz and P.-A. Meyer, to construct second-order counterparts of those classical structures. These will allow us to study symmetries of stochastic differential equations (SDEs), to establish stochastic Lagrangian and Hamiltonian mechanics and their key relations with second-order Hamilton–Jacobi–Bellman (HJB) equations. Indeed, stochastic prolongation formulae will be derived to study symmetries of SDEs and mixed-order Cartan symmetries. Stochastic Hamilton’s equations will follow from a second-order symplectic structure and canonical transformations will lead to the HJB equation. A stochastic variational problem on Riemannian manifolds will provide a stochastic Euler–Lagrange equation compatible with HJB one and equivalent to the Riemannian version of stochastic Hamilton’s equations. A stochastic Noether’s theorem will also follow. The inspirational example, along the paper, will be the rich dynamical structure of Schrödinger’s problem in optimal transport, where the latter is also regarded as a Euclidean version of hydrodynamical interpretation of quantum mechanics.

Homogenization of some periodic Hamilton-Jacobi equations with defects

We study homogenization for a class of stationary Hamilton-Jacobi equations in which the Hamiltonian is obtained by perturbing near the origin an otherwise periodic Hamiltonian. We prove that the limiting problem consists of a Hamilton-Jacobi equation outside the origin, with the same effective Hamiltonian as in periodic homogenization, supplemented at the origin with an effective Dirichlet condition that keeps track of the perturbation. Various comments and extensions are discussed.

Singularity formation and regularization at multiple times in the viscous Hamilton–Jacobi equation

The Cauchy–Dirichlet problem for the superquadratic viscous Hamilton–Jacobi equation (VHJ) from stochastic control theory, admits a unique, global viscosity solution. Solutions thus exist in the weak sense after appearance of singularity, which occurs through gradient blow-up (GBU) on the boundary. Whereas viscosity solution theory has been extensively applied to many PDEs, there seem to be less results on refined singular behavior of solutions. Although occurrence of two types of GBU, with or without loss of boundary condition (LBC), are known, detailed behavior after GBU has remained open except for a strongly restricted special class of one dimensional solutions. In this paper, in general dimensions, we construct solutions which undergo GBU with LBC arbitrarily many times and then recover regularity, as well as solutions without LBC at first GBU time. In one space dimension, we obtain the complete classification of viscosity solutions at each time, which extends to radial case in higher dimensions. Furthermore we show the existence of solutions exhibiting an arbitrarily given combination of GBU types with/without LBC at multiple times, in which a new type of behavior called “bouncing” is discovered. Global weak solutions of VHJ with multiple times singularity turn out to display larger variety of behaviors than in Fujita equation. We introduce a method based on an arbitrary number of critical parameters, whose continuity requires delicate arguments. Since we do not rely on any known special solution unlike in Fujita equation, our method is expected to apply to other equations. Singular behaviors at multiple times are completely new in the context of VHJ but also of stochastic control theory. Our results imply that for certain spatial distributions of rewards, if a controlled Brownian particle starts near the boundary, then the net gain attains profitable values on different time horizons but not on some intermediate times.

Symmetric Reduction and Hamilton-Jacobi Equations of the Controlled Underwater Vehicle-Rotor System

Symmetric Reduction and Hamilton-Jacobi Equations of the Controlled Underwater Vehicle-Rotor System

Large‐scale asymptotics of velocity‐jump processes and nonlocal Hamilton–Jacobi equations

AbstractWe investigate a simple velocity jump process in the regime of large deviation asymptotics. New velocities are taken randomly at a constant, large, rate from a Gaussian distribution with vanishing variance. The Kolmogorov forward equation associated with this process is the linear BGK kinetic transport equation. We derive a new type of Hamilton–Jacobi equation which is nonlocal with respect to the velocity variable. We introduce a suitable notion of viscosity solution, and we prove well‐posedness in the viscosity sense. We also prove convergence of the logarithmic transformation toward this limit problem. Furthermore, we identify the variational formulation of the solution by means of an action functional supported on piecewise linear curves. As an application of this theory, we compute the exact rate of acceleration in a kinetic version of the celebrated F–KPP equation in the one‐dimensional case.

On the optimal Lq-regularity for viscous Hamilton–Jacobi equations with subquadratic growth in the gradient

This paper studies a maximal [Formula: see text]-regularity property for nonlinear elliptic equations of second order with a zeroth order term and gradient nonlinearities having superlinear and subquadratic growth, complemented with Dirichlet boundary conditions. The approach is based on the combination of linear elliptic regularity theory and interpolation inequalities, so that the analysis of the maximal regularity estimates boils down to determine lower order integral bounds. The latter are achieved via a [Formula: see text] duality method, which exploits the regularity properties of solutions to stationary Fokker–Planck equations. For the latter problems, we discuss both global and local estimates. Our main novelties for the regularity properties of this class of nonlinear elliptic boundary-value problems are the analysis of the end-point summability threshold [Formula: see text], [Formula: see text] being the dimension of the ambient space and [Formula: see text] the growth of the first-order term in the gradient variable, along with the treatment of the full integrability range [Formula: see text].

The propagation of a flame front with a p-Laplacian operator

The goal of the presented study is to provide a mathematical analysis for modeling a flame propagation dynamics with a p-Laplacian operator. We extend some results already contemplated in the literature. Given the introduction of a new nonlinear operator, we first discuss the regularity and boundedness of solutions. Afterward, we obtain results concerning the existence and uniqueness of solutions. The second part of the paper is focused on the exploration of stationary and nonstationary solutions. The stationary solutions are constructed based on the definition of a Hamiltonian and a perturbation procedure. The non-stationary solutions are obtained with a single-point exponential scaling that ends in a Hamilton-Jacobi equation. This last equation is solved based on an asymptotic separation of variables technique.

Homogenization for sub-riemannian Lagrangians

We consider Lagrangians that are defined only on the horizontal distribution of a sub-riemannian manifold. The associated Hamiltonian is neither strictly convex nor coercive. We prove a result on homogenization of the Hamilton–Jacobi equation following the approach in Contreras et al (2015 Calc. Var. Partial Differ. Equ. 52 237–52). To establish the result, we need to extend weak KAM and Aubry–Mather theories to the sub-riemannian setting. We obtain a Tonelli theorem to dispend of the Lagrangian dynamics tools used in standard weak KAM theory. In the way towards our main result we observe that, the long time convergence of the Lax-Oleinik semigroup, and the vanishing discount convergence of the discounted value function hold.

Event‐triggered neural experience replay learning for nonzero‐sum tracking games of unknown continuous‐time nonlinear systems

AbstractIn this paper, an online event‐triggered integral reinforcement learning is proposed to solve the nonzero‐sum tracking games of the two‐player nonlinear system with unknown dynamics and constrained input. Firstly, an augmented system of the nonzero‐sum game is constructed to describe the tracking problem. Thus, the optimal tracking problem is treated as solving the coupled Hamilton‐Jacobi (HJ) equations of the augmented system. To restrict the number of sampling states, a novel triggering threshold containing the augmented states is designed, and it avoids the existence of Zeno behavior. Secondly, a single‐critic network is utilized to approximate the solution of the coupled HJ equations. The critic network turning law with experience replay technology relaxes the dependence on the persistence of excitation (PE) conditions in traditional integral reinforcement learning (IRL) by using historical data in the stack. Moreover, the augmented system states and the critic NN weight errors are uniformly ultimate boundedness (UUB) by Lyapunov theory. Simulation examples are provided to demonstrate the availability of the proposed algorithm.

Hamilton–Jacobi theory and integrability for autonomous and non-autonomous contact systems

In this paper, we study the integrability of contact Hamiltonian systems, both time-dependent and independent. In order to do so, we construct a Hamilton--Jacobi theory for these systems following two approaches, obtaining two different Hamilton--Jacobi equations. Compared to conservative Hamiltonian systems, contact Hamiltonian systems depend of one additional parameter. The fact of obtaining two equations reflects whether we are looking for solutions depending on this additional parameter or not. In order to illustrate the theory developed in this paper, we study three examples: the free particle with a linear external force, the freely falling particle with linear dissipation and the damped and forced harmonic oscillator.

Semiclassical theory predicts stochastic ghosts scaling

Slowing down occurs in dynamical systems close to bifurcations or phase transitions. The time length ( τ ) of ghost transients close to a saddle-node bifurcation in deterministic systems follows τ ∼ | ϵ − ϵ c | − 1 / 2 , ϵ being the bifurcation parameter and ϵ c its critical value. Recently, we numerically investigated how intrinsic noise affected the deterministic picture, finding a more complicated scaling law. We here provide a theoretical basis for this new law with two models of cooperation using a Wentzel–Kramers–Brillouin asymptotic approximation of the Master Equation. A study of the phase space of the Hamiltonian derived from the Hamilton–Jacobi equation shows that the statistically significant orbits (paths) reproduce the scaling function observed in the stochastic simulations. The flight times tied to these orbits underpin the scaling law of the stochastic system, and the same properties should extend in a universal way to all stochasticsystems whose associated Hamiltonian exhibits the same behaviour. Our approach allows to make useful theoretical predictions of transient times in stochastic systems close to a bifurcation.

Limit of solutions for semilinear Hamilton–Jacobi equations with degenerate viscosity

Abstract In the paper we prove the convergence of viscosity solutions u λ {u_{\lambda}} as λ → 0 + {\lambda\rightarrow 0_{+}} for the parametrized degenerate viscous Hamilton–Jacobi equation H ⁢ ( x , d x ⁢ u , λ ⁢ u ) = α ⁢ ( x ) ⁢ Δ ⁢ u , α ⁢ ( x ) ≥ 0 , x ∈ 𝕋 n H(x,d_{x}u,\lambda u)=\alpha(x)\Delta u,\quad\alpha(x)\geq 0,\quad x\in\mathbb% {T}^{n} under suitable convex and monotonic conditions on H : T * ⁢ M × ℝ → ℝ {H:T^{*}M\times\mathbb{R}\rightarrow\mathbb{R}} . Such a limit can be characterized in terms of stochastic Mather measures associated with the critical equation H ⁢ ( x , d x ⁢ u , 0 ) = α ⁢ ( x ) ⁢ Δ ⁢ u . H(x,d_{x}u,0)=\alpha(x)\Delta u.

Value-Gradient Based Formulation of Optimal Control Problem and Machine Learning Algorithm

Optimal control problem is typically solved by first finding the value function through the Hamilton–Jacobi equation (HJE) and then taking the minimizer of the Hamiltonian to obtain the control. In this work, instead of focusing on the value function, we propose a new formulation for the gradient of the value function (value-gradient) as a decoupled system of partial differential equations in the context of a continuous-time deterministic discounted optimal control problem. We develop an efficient iterative scheme for this system of equations in parallel by utilizing the fact that they share the same characteristic curves as the HJE for the value function. For the theoretical part, we prove that this iterative scheme converges linearly in sense for some suitable exponent in a weight function. For the numerical method, we combine a characteristic line method with machine learning techniques. Specifically, we generate multiple characteristic curves at each policy iteration from an ensemble of initial states and compute both the value function and its gradient simultaneously on each curve as the labeled data. Then supervised machine learning is applied to minimize the weighted squared loss for both the value function and its gradients. Experimental results demonstrate that this new method not only significantly increases the accuracy but also improves the efficiency and robustness of the numerical estimates, particularly with less characteristics data or fewer training steps.

Evaluation of the Feynman Propagator by Means of the Quantum Hamilton-Jacobi Equation

It is shown that the complex phase of the Feynman propagator is a solution of the quantum Hamilton–Jacobi equation, namely, it is the quantum Hamilton's principal function (or quantum action). Therefore, the Feynman propagator can be computed either by means of the path integration, or by the way of the Hamilton–Jacobi equation. This is analogous to what happens in classical mechanics, where the Hamilton's principal function can be computed either by integrating the Lagrangian along the extremal paths, or as a solution of partial differential equation, namely the classical Hamilton–Jacobi equation. If the path is decomposed in the classical one and quantum fluctuations, the contribution of these quantum fluctuations satisfies a non-linear partial differential equation, whose coefficients depend on the classical action. When the contribution of the quantum fluctuations depend only on the time, it can be computed by means of a simple integration. The final results for the propagators in this case are equal to the Van Vleck–Pauli–Morette expressions, even though the two derivations are quite different.Quanta 2023; 12: 22–26.

Minimal entropy conditions for scalar conservation laws with general convex fluxes

We are concerned with the minimal entropy conditions for one-dimensional scalar conservation laws with general convex flux functions. For such scalar conservation laws, we prove that a single entropy-entropy flux pair ( η ( u ) , q ( u ) ) (\eta (u),q(u)) with η ( u ) \eta (u) of strict convexity is sufficient to single out an entropy solution from a broad class of weak solutions in L l o c ∞ L^\infty _{\mathrm { loc}} that satisfy the inequality: η ( u ) t + q ( u ) x ≤ μ \eta (u)_t+q(u)_x\leq \mu in the distributional sense for some non-negative Radon measure μ \mu . Furthermore, we extend this result to the class of weak solutions in L l o c p L^p_{\mathrm {loc}} , based on the asymptotic behavior of the flux function f ( u ) f(u) and the entropy function η ( u ) \eta (u) at infinity. The proofs are based on the equivalence between the entropy solutions of one-dimensional scalar conservation laws and the viscosity solutions of the corresponding Hamilton-Jacobi equations, as well as the bilinear form and commutator estimates as employed similarly in the theory of compensated compactness.

Time complexity analysis of quantum algorithms via linear representations for nonlinear ordinary and partial differential equations

We construct quantum algorithms to compute the solution and/or physical observables of nonlinear ordinary differential equations (ODEs) and nonlinear Hamilton-Jacobi equations (HJE) via linear representations or exact mappings between nonlinear ODEs/HJE and linear partial differential equations (the Liouville equation and the Koopman-von Neumann equation). The connection between the linear representations and the original nonlinear system is established through the Dirac delta function or the level set mechanism. We compare the quantum linear systems algorithms based methods and the quantum simulation methods arising from different numerical approximations, including the finite difference discretisations and the Fourier spectral discretisations for the two different linear representations, with the result showing that the quantum simulation methods usually give the best performance in time complexity. We also propose the Schrödinger framework to solve the Liouville equation for the HJE with the Hamiltonian formulation of classical mechanics, since it can be recast as the semiclassical limit of the Wigner transform of the Schrödinger equation. Comparison between the Schrödinger and the Liouville framework will also be made.

Viscosity Solutions of Hamilton–Jacobi Equations for Neutral-Type Systems

Viscosity Solutions of Hamilton–Jacobi Equations for Neutral-Type Systems

Time-Dependent Hamiltonian Mechanics on a Locally Conformal Symplectic Manifold

In this paper we aim at presenting a concise but also comprehensive study of time-dependent (t-dependent) Hamiltonian dynamics on a locally conformal symplectic (lcs) manifold. We present a generalized geometric theory of canonical transformations in order to formulate an explicitly time-dependent geometric Hamilton-Jacobi theory on lcs manifolds, extending our previous work with no explicit time-dependence. In contrast to previous papers concerning locally conformal symplectic manifolds, the introduction of the time dependency that this paper presents, brings out interesting geometric properties, as it is the case of contact geometry in locally symplectic patches. To conclude, we show examples of the applications of our formalism, in particular, we present systems of differential equations with time-dependent parameters, which admit different physical interpretations as we shall point out.

A Hamilton–Jacobi-based proximal operator

First-order optimization algorithms are widely used today. Two standard building blocks in these algorithms are proximal operators (proximals) and gradients. Although gradients can be computed for a wide array of functions, explicit proximal formulas are known for only limited classes of functions. We provide an algorithm, HJ-Prox, for accurately approximating such proximals. This is derived from a collection of relations between proximals, Moreau envelopes, Hamilton-Jacobi (HJ) equations, heat equations, and Monte Carlo sampling. In particular, HJ-Prox smoothly approximates the Moreau envelope and its gradient. The smoothness can be adjusted to act as a denoiser. Our approach applies even when functions are accessible only by (possibly noisy) black box samples. We show that HJ-Prox is effective numerically via several examples.

Applications of the invariant subspace method on searching explicit solutions to certain special-type non-linear evolution equations

We extend the invariant subspace method (ISM) to a class of Hamilton–Jacobi equations (HJEs) and a family of third-order time-fractional dispersive PDEs with the Caputo fractional derivative in this letter. More precisely, the complete classification is presented for such HJEs that admit invariant subspaces governed by solutions of the second-order and third-order linear ordinary differential equations (ODEs). Meanwhile, some concrete equations are derived for the construction of new exact solutions u(x,t)=∑i=1nCi(t)fi(x). Then a set of invariant subspaces of the considered third-order time-fractional non-linear dispersive equations are obtained. Based on the Laplace transform method (LTM) and applying several properties of the well known Mitta-Leffer (ML) function, the different types of explicit solutions of a family of third-order time-fractional dispersive PDEs are finally derived.