Jacobi Equation Research Articles

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.

Barrier-critic-disturbance approximate optimal control of nonzero-sum differential games for modular robot manipulators

In this paper, for addressing the safe control problem of modular robot manipulators (MRMs) system with uncertain disturbances, an approximate optimal control scheme of nonzero-sum (NZS) differential games is proposed based on the control barrier function (CBF). The dynamic model of the manipulator system integrates joint subsystems through the utilization of joint torque feedback (JTF) technique, incorporating interconnected dynamic coupling (IDC) effects. By integrating the cost functions relevant to each player with the CBF, the evolution of system states is ensured to remain within the safe region. Subsequently, the optimal tracking control problem for the MRM system is reformulated as an NZS game involving multiple joint subsystems. Based on the adaptive dynamic programming (ADP) algorithm, a cost function approximator for solving Hamilton–Jacobi (HJ) equation using only critic neural networks (NN) is established, which promotes the feasible derivation of the approximate optimal control strategy. The Lyapunov theory is utilized to demonstrate that the tracking error is uniformly ultimately bounded (UUB). Utilizing the CBF’s state constraint mechanism prevents the robot from deviating from the safe region, and the application of the NZS game approach ensures that the subsystems of the MRM reach a Nash equilibrium. The proposed control method effectively addresses the problem of safe and approximate optimal control of MRM system under uncertain disturbances. Finally, the effectiveness and superiority of the proposed method are verified through simulations and experiments.

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.

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

Emergent fault friction and supershear in a continuum model of geophysical rupture

Important physical observations in rupture dynamics such as static fault friction, short-slip, self-healing, and the supershear phenomenon in cracks are studied. A continuum model of rupture dynamics is developed using the field dislocation mechanics (FDM) theory. The energy density function in our model encodes accepted and simple physical facts related to rocks and granular materials under compression. We work within a 2-dimensional ansatz of FDM where the rupture front is allowed to move only in a horizontal fault layer sandwiched between elastic blocks. Damage via the degradation of elastic modulus is allowed to occur only in the fault layer, characterized by the amount of plastic slip. The theory dictates the evolution equation of the plastic shear strain to be a Hamilton–Jacobi (H-J) equation, resulting in the representation of a propagating rupture front. A Central-Upwind scheme is used to solve the H-J equation. The rupture propagation is fully coupled to elastodynamics in the whole domain, and our simulations recover static friction laws as emergent features of our continuum model, without putting in by hand any such discontinuous criteria in the formulation. Estimates of material parameters of cohesion and friction angle are deduced. Short-slip and slip-weakening (crack-like) behaviors are also reproduced as a function of the degree of damage behind the rupture front. The long-time behavior of a moving rupture front is probed, and it is deduced that equilibrium profiles under no shear stress are not traveling wave profiles under non-zero shear load in our model. However, it is shown that a traveling wave structure is likely attained in the limit of long times. Finally, a crack-like damage front is driven by an initial impact loading, and it is observed in our numerical simulations that an upper bound to the crack speed is the dilatational wave speed of the material unless the material is put under pre-stressed conditions, in which case supersonic motion can be obtained. Without pre-stress, intersonic (supershear) motion is recovered under appropriate conditions.

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.

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.

Monge solutions for discontinuous Hamilton-Jacobi equations in Carnot groups

In this paper we study Monge solutions to stationary Hamilton–Jacobi equations associated to discontinuous Hamiltonians in the framework of Carnot groups. After showing the equivalence between Monge and viscosity solutions in the continuous setting, we prove existence and uniqueness for the Dirichlet problem, together with a comparison principle and a stability result.

Madelung mechanics and superoscillations

In single-particle Madelung mechanics, the single-particle quantum state Ψ(x→,t)=R(x→,t)eiS(x→,t)/ℏ is interpreted as comprising an entire conserved fluid of classical point particles, with local density R(x→,t)2 and local momentum ∇→S(x→,t) (where R and S are real). The Schrödinger equation gives rise to the continuity equation for the fluid, and the Hamilton–Jacobi equation for particles of the fluid, which includes an additional density-dependent quantum potential energy term Q(x→,t)=−ℏ22m∇→R(x→,t)R(x→,t) , which is all that makes the fluid behavior nonclassical. In particular, the quantum potential can become negative and create a nonclassical boost in the kinetic energy. This boost is related to superoscillations in the wavefunction, where the local frequency of Ψ exceeds its global band limit. Berry showed that for states of definite energy E, the regions of superoscillation are exactly the regions where Q(x→,t)<0 . For energy superposition states with band-limit E+ , the situation is slightly more complicated, and the bound is no longer Q(x→,t)<0 . However, the fluid model provides a definite local energy for each fluid particle which allows us to define a local band limit for superoscillation, and with this definition, all regions of superoscillation are again regions where Q(x→,t)<0 for general superpositions. An alternative interpretation of these quantities involving a reduced quantum potential is reviewed and advanced, and a parallel discussion of superoscillation in this picture is given. Detailed examples are given which illustrate the role of the quantum potential and superoscillations in a range of scenarios.

Continuous Generalized Solution of the Hamilton–Jacobi Equation with a Noncoercive Hamiltonian

Continuous Generalized Solution of the Hamilton–Jacobi Equation with a Noncoercive Hamiltonian

Existence and Uniqueness Results to a System of Hamilton–Jacobi Equations with Application to Dislocation Dynamics

Existence and Uniqueness Results to a System of Hamilton–Jacobi Equations with Application to Dislocation Dynamics

A Discrete Hamilton–Jacobi Theory for Contact Hamiltonian Dynamics

In this paper, we propose a discrete Hamilton–Jacobi theory for (discrete) Hamiltonian dynamics defined on a (discrete) contact manifold. To this end, we first provide a novel geometric Hamilton–Jacobi theory for continuous contact Hamiltonian dynamics. Then, rooting on the discrete contact Lagrangian formulation, we obtain the discrete equations for Hamiltonian dynamics by the discrete Legendre transformation. Based on the discrete contact Hamilton equation, we construct a discrete Hamilton–Jacobi equation for contact Hamiltonian dynamics. We show how the discrete Hamilton–Jacobi equation is related to the continuous Hamilton–Jacobi theory presented in this work. Then, we propose geometric foundations of the discrete Hamilton–Jacobi equations on contact manifolds in terms of discrete contact flows. At the end of the paper, we provide a numerical example to test the theory.

Eulerian and Lagrangian Stability in Zeitlin’s Model of Hydrodynamics

The two-dimensional (2-D) Euler equations of a perfect fluid possess a beautiful geometric description: they are reduced geodesic equations on the infinite-dimensional Lie group of symplectomorphims with respect to a right-invariant Riemannian metric. This structure enables insights to Eulerian and Lagrangian stability via sectional curvature and Jacobi equations. The Zeitlin model is a finite-dimensional analogue of the 2-D Euler equations; the only known discretization that preserves the rich geometric structure. Theoretical and numerical studies indicate that Zeitlin’s model provides consistent long-time behaviour on large scales, but to which extent it truly reflects the Euler equations is mainly open. Towards progress, we give here two results. First, convergence of the sectional curvature in the Euler–Zeitlin equations on the Lie algebra su(N)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathfrak {su}(N)$$\\end{document} to that of the Euler equations on the sphere. Second, L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2$$\\end{document}-convergence of the corresponding Jacobi equations for Lagrangian and Eulerian stability. The results allow geometric conclusions about Zeitlin’s model to be transferred to Euler’s equations and vice versa, which could expedite the ultimate aim: to characterize the generic long-time behaviour of perfect 2-D fluids.

Tipler naked singularities in N dimensions

Abstract A spacetime singularity, identified by the existence of incomplete causal geodesics in the spacetime, is called a (Tipler) strong curvature singularity if the volume form acting on independent Jacobi fields along causal geodesics vanishes in the approach of the singularity. It is called naked if at least one of these causal geodesics is past incomplete. Here, we study the formation of strong curvature naked singularities arising from spherically symmetric gravitational collapse of general type-I matter fields in an arbitrarily finite number of dimensions. In the spirit of Joshi and Dwivedi (1993 Phys. Rev. D 47 5357), and Goswami and Joshi (2007 Phys. Rev. D 76 084026), beginning with regular initial data, we derive two distinct (but not mutually exclusive) conditions, which we call the positive root condition (PRC) and the simple positive root condition (SPRC), that serve as necessary and sufficient conditions, respectively, for the existence of naked singularities. In doing so, we generalize the results of both the aforementioned works. We further restrict the PRC and the SPRC by imposing the curvature growth condition (CGC) of Clarke and Krolak (1985 J. Geom. Phys. 2(2) 127) on all causal curves that satisfy the causal convergence condition. The CGC then gives a sufficient condition ensuring that the naked singularities implying the PRC and implied by the SPRC, are of strong curvature type and hence correspond to the inextendibility of the spacetime. Using the CGC, we extend the results of Mosani et. al. (2020 Phys. Rev. D 101 044052) (that hold for dimension N=4) to the case N=5, showing that strong curvature naked singularities can occur in this case. However, for the case N≥6, we show that past-incomplete causal curves that identify naked singularities do not satisfy the CGC. These results shed light on the validity of the cosmic censorship conjectures in arbitrary dimensions.

EPDIFF-JF-NET: Adjoint Jacobi Fields for Diffeomorphic Registration Networks

This paper presents a deep learning unsupervisedapproach for diffeomorphic image registrationcalled EPDiff-JF-Net. We propose a novel paralleltransport layer to compute the gradients necessaryfor training with adjoint Jacobi fields. We test ourmethod on two independent brain MRI datasets andobtain state-of-the-art results.

Distributed optimal control of nonlinear multi‐agent systems based on integral reinforcement learning

AbstractIn this article, a distributed optimal control approach is proposed for a class of affine nonlinear multi‐agent systems (MASs) with unknown nonlinear dynamics. The game theory is used to formulate the distributed optimal control problem into a differential graphical game problem with synchronized updates of all nodes. A data‐based integral reinforcement learning (IRL) algorithm is used to learn the solution of the coupled Hamilton–Jacobi (HJ) equation without prior knowledge of the drift dynamics, and the actor‐critic neural networks (A‐C NNs) are used to approximate the control law and the cost function, respectively. To update the parameters synchronously, the gradient descent algorithm is used to design the weight update laws of the A‐C NNs. Combining the IRL and the A‐C NNs, a distributed consensus optimal control method is designed. By using the Lyapunov stability theory, the developed optimal control method can show that all signals in the considered system are uniformly ultimately bounded (UUB), and the systems can achieve Nash equilibrium when all agents update their controllers simultaneously. Finally, simulation results are given to illustrate the effectiveness of the developed optimal control approach.

Well-posedness for Hamilton–Jacobi equations on the Wasserstein space on graphs

Well-posedness for Hamilton–Jacobi equations on the Wasserstein space on graphs