Lax Formula Research Articles

On representation formulas for optimal control: A Lagrangian perspective

In this paper, we study representation formulas for finite-horizon optimal control problems with or without state constraints, unifying two different viewpoints: the Lagrangian and dynamic programming (DP) frameworks. In a recent work [1], the generalized Lax formula is obtained via DP for optimal control problems with state constraints and nonlinear systems. We revisit the formula from the Lagrangian perspective to provide a unified framework for understanding and implementing the nontrivial representation of the value function. Our simple derivation makes direct use of the Lagrangian formula from the theory of Hamilton-Jacobi (HJ) equations. We also discuss a rigorous way to construct an optimal control using a $\delta$-net, as well as a numerical scheme for controller synthesis via convex optimization.

A Survey of Hopf-Lax Formulas and Quasiconvexity in PDEs

MSC: 35C05,35D40,49L12,49N70 DOI: 10.21538/0134-4889-2021-27-3-237-245 This is a short survey of recent results obtained by the author and collaborators primarily on Hopf–Lax formulas for Hamilton–Jacobi equations and obstacle problems. The initiation of the use of quasiconvex (i.e., level convex) functions in $L^\infty$ control and differential games led to such formulas and

From Hopf--Lax Formula to Optimal Weak Transfer Plan

We study the properties of the Hopf--Lax formula restricted to convex functions and provide a characterization of the optimal transfer plan for weak transport problems on the real line. On $n$ dimensional real space, we also provide a sufficient condition on the potential function such that the optimal plan of the classical Monge--Kantorovich problem does not depend on the cost function. As a byproduct, we establish a link between the combinatorial object (the permutation polytope) and the Hamilton--Jacobi equation.

On periodic subsolutions to steady second-order systems and applications

Bostan and Namah (Remarks on bounded solutions of steady Hamilton–Jacobi equations, C. R. Acad. Sci. Paris, Ser. I 347(15–16) (2009) 873–878) proved that constant functions are the only bounded solutions to H(Du)=H(0) when H is superlinear and strictly convex. In this short note, we present a proof other than that of Bostan and Namah for equations that can be easily applied to some types of possibly degenerate parabolic systems. Our proof applies for periodic subsolutions instead of bounded solutions like that of Bostan and Namah; however, we need periodic subsolutions, which is quite restrictive. We do not consider Hopf–Lax's formula in our proof, so we can relax some restrictions on H. We also present an application to the large-time behavior of solutions to degenerate parabolic systems.

Computing the Homology of Semialgebraic Sets. I: Lax Formulas

We describe and analyze an algorithm for computing the homology (Betti numbers and torsion coefficients) of closed semialgebraic sets given by Boolean formulas without negations over lax polynomial inequalities. The algorithm works in weak exponential time. This means that outside a subset of data having exponentially small measure, the cost of the algorithm is single exponential in the size of the data. All previous algorithms solving this problem have doubly exponential complexity (and this is so for almost all input data). Our algorithm thus represents an exponential acceleration over state-of-the-art algorithms for all input data outside a set that vanishes exponentially fast.

A splitting method for overcoming the curse of dimensionality in Hamilton–Jacobi equations arising from nonlinear optimal control and differential games with applications to trajectory generation

Recent observations have been made that bridge splitting methods arising from optimization, to the Hopf and Lax formulas for Hamilton-Jacobi Equations with Hamiltonians $H(p)$. This has produced extremely fast algorithms in computing solutions of these PDEs. More recent observations were made in generalizing the Hopf and Lax formulas to state-and-time-dependent cases $H(x,p,t)$. In this article, we apply a new splitting method based on the Primal Dual Hybrid Gradient algorithm (a.k.a. Chambolle-Pock) to nonlinear optimal control and differential games problems, based on techniques from the derivation of the new Hopf and Lax formulas, which allow us to compute solutions at points $(x,t)$ directly, i.e. without the use of grids in space. This algorithm also allows us to create trajectories directly. Thus we are able to lift the curse of dimensionality a bit, and therefore compute solutions in much higher dimensions than before. And in our numerical experiments, we actually observe that our computations scale polynomially in time. Furthermore, this new algorithm is embarrassingly parallelizable.

Revisiting the redistancing problem using the Hopf–Lax formula

This article presents a fast new numerical method for redistancing objective functions based on the Hopf–Lax formula [1]. The algorithm suggested here is a special case of the previous work in [2] and an extension that applies the Hopf–Lax formula for computing the signed distance to the front. We propose the split Bregman approach to solve the minimization problem as a solution of the eikonal equation obtained from Hopf–Lax formula. Our redistancing procedure is expected to be generalized and widely applied to many fields such as computational fluid dynamics, the minimal surface problem, and elsewhere.

A Hopf–Lax formula for the time evolution of the level-set equation and a new approach to shape sensitivity analysis

The level-set method is used in many different applications to describe the propagation of shapes and domains. When scalar speed fields are used to encode the desired shape evolution, this leads to the classical level-set equation. We present a concise Hopf–Lax representation formula that can be used to characterise the evolved domains at arbitrary times. This result is also applicable for the case of speed fields without a fixed sign, even though the level-set equation has a non-convex Hamiltonian in these situations. The representation formula is based on the same idea that underpins the Fast-Marching Method, and it provides a strong theoretical justification for a generalised Composite Fast-Marching method. Based on our Hopf–Lax formula, we are also able to present new theoretical results. In particular, we show non-fattening of the zero level set in a measure-theoretic sense, derive a very general shapesensitivity calculus that does not require the usual regularity assumptions on the domains, prove optimal Lipschitz constants for the evolved level-set function and discuss the effect of perturbations in both the speed field and the initial geometry.

Differential Games in $$L^{\infty }$$

The Isaacs equations for differential games in \({L^{\infty }}\) first derived in Barron (Nonlinear Anal 14:971–989, 1990) are reformulated so that the Hamiltonians are continuous and result in a simpler problem to analyze numerically. Relaxed differential games in \({L^{\infty }}\) are considered. \(L^{\infty }\) differential games with time and state independent dynamics and convex or quasiconvex terminal data are solved explicitly using a type of Hopf–Lax formula. The stochastic differential game in \({L^{\infty }}\) connected to stochastic target problems is also discussed.

Continuous formulations and analytical properties of the link transmission model

The link transmission model (LTM) has great potential for simulating traffic flow in large-scale networks since it is much more efficient and accurate than the Cell Transmission Model (CTM). However, there lack general continuous formulations of LTM, and there has been no systematic study on its analytical properties such as stationary states and stability of network traffic flow. In this study we attempt to fill the gaps. First we apply the Hopf–Lax formula to derive Newell’s simplified kinematic wave model with given boundary cumulative flows and the triangular fundamental diagram. We then apply the Hopf–Lax formula to define link demand and supply functions, as well as link queue and vacancy functions, and present two continuous formulations of LTM, by incorporating boundary demands and supplies as well as invariant macroscopic junction models. With continuous LTM, we define and solve the stationary states in a road network. We also apply LTM to directly derive a Poincaré map to analyze the stability of stationary states in a diverge-merge network. Finally we present an example to show that LTM is not well-defined with non-invariant junction models. We can see that Newell’s model and continuous LTM complement each other and provide an alternative formulation of the network kinematic wave theory. This study paves the way for further extensions, analyses, and applications of LTM in the future.

The inviscid, compressible and rotational, 2D isotropic Burgers and pressureless Euler–Coriolis fluids: Solvable models with illustrations

The coupling between dilatation and vorticity, two coexisting and fundamental processes in fluid dynamics (Wu et al., 2006, pp. 3, 6) is investigated here, in the simplest cases of inviscid 2D isotropic Burgers and pressureless Euler–Coriolis fluids respectively modeled by single vortices confined in compressible, local, inertial and global, rotating, environments. The field equations are established, inductively, starting from the equations of the characteristics solved with an initial Helmholtz decomposition of the velocity fields namely a vorticity free and a divergence free part (Wu et al., 2006, Sects. 2.3.2, 2.3.3) and, deductively, by means of a canonical Hamiltonian Clebsch like formalism (Clebsch, 1857, 1859), implying two pairs of conjugate variables. Two vector valued fields are constants of the motion: the velocity field in the Burgers case and the momentum field per unit mass in the Euler–Coriolis one. Taking advantage of this property, a class of solutions for the mass densities of the fluids is given by the Jacobian of their sum with respect to the actual coordinates. Implementation of the isotropy hypothesis entails a radial dependence of the velocity potentials and of the stream functions associated to the compressible and to the rotational part of the fluids and results in the cancellation of the dilatation-rotational cross terms in the Jacobian. A simple expression is obtained for all the radially symmetric Jacobians occurring in the theory. Representative examples of regular and singular solutions are shown and the competition between dilatation and vorticity is illustrated. Inspired by thermodynamical, mean field theoretical analogies, a genuine variational formula is proposed which yields unique measure solutions for the radially symmetric fluid densities investigated. We stress that this variational formula, unlike the Hopf–Lax formula, enables us to treat systems which are both compressible and rotational. Moreover in the one-dimensional case, we show for an interesting application that both variational formulas are equivalent.

Foldy–Lax approximation on multiple scattering by many small scatterers

The multiple scattering of time harmonic wave emitted by a localized source through a medium with many scatterers can be approximated by an Foldy–Lax self-consistent system when the relative radius of each scatterer is small and the distribution of scatterers is sparse. The scattering amplitude in the Foldy–Lax self-consistent system will be specified in terms of scatterer volume and scattering strength. By neglecting the self-interaction effect, the difference from the exciting field in the Foldy–Lax formula to the analytic wave field given implicitly by the Lippmann–Schwinger integral equation is compared. An upper bound of the difference is obtained in terms of scaled radius and sparsity of the distribution of the scatterers.

The Hopf–Lax formula in Carnot groups: a control theoretic approach

The purpose of this paper is to bring a new light on the state-dependent Hamilton–Jacobi equation and its connection with the Hopf–Lax formula in the framework of a Carnot group \((\mathbf G ,\circ ).\) The equation we shall consider is of the form $$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} u_{t}+ \Psi (X_{1}u, \ldots , X_{m}u)=0\qquad &{}(x,t)\in \mathbf G \times (0,\infty ) {u}(x,0)=g(x)&{}x\in \mathbf G , \end{array} \right. \end{aligned}$$ where \(X_{1},\ldots , X_{m}\) are a basis of the first layer of the Lie algebra of the group \(\mathbf G ,\) and \(\Psi : \mathbb{R }^{m} \rightarrow \mathbb{R }\) is a superlinear, convex function. The main result shows that the unique viscosity solution of the Hamilton–Jacobi equation can be given by the Hopf–Lax formula $$\begin{aligned} u(x,t) = \inf _{y\in \mathbf G }\left\{ t \Psi ^\mathbf{G }\left( \delta _{\frac{1}{t}}(y^{-1}\circ x)\right) + g(y) \right\} , \end{aligned}$$ where \(\Psi ^\mathbf{G }:\mathbf G \rightarrow \mathbb{R }\) is the \(\mathbf G \)-Legendre–Fenchel transform of \(\Psi ,\) defined by a control theoretical approach. We recover, as special cases, some known results like the classical Hopf–Lax formula in the Euclidean spaces \(\mathbb{R }^n,\) showing that \(\Psi ^{\mathbb{R }^n}\) is the Legendre–Fenchel transform \(\Psi ^*\) of \(\Psi ;\) moreover, we recover the result by Manfredi and Stroffolini in the Heisenberg group for particular Hamiltonian function \(\Psi .\) In this paper we follow an optimal control problem approach and we obtain several properties for the value functions \(u\) and \(\Psi ^\mathbf G .\)

Sharp asymptotic growth laws of turbulent flame speeds in cellular flows by inviscid Hamilton–Jacobi models

We study the large time asymptotic speeds (turbulent flame speeds sT) of the simplified Hamilton–Jacobi (HJ) models arising in turbulent combustion. One HJ model is G-equation describing the front motion law in the form of local normal velocity equal to a constant (laminar speed) plus the normal projection of fluid velocity. In level set formulation, G-equations are HJ equations with convex (L1 type) but non-coercive Hamiltonians. The other is the quadratically nonlinear (L2 type) inviscid HJ model of Majda–Souganidis derived from the Kolmogorov–Petrovsky–Piskunov reactive fronts. Motivated by a question posed by Embid, Majda and Souganidis (1995) [10], we compare the turbulent flame speeds sTʼs from these inviscid HJ models in two-dimensional cellular flows or a periodic array of steady vortices via sharp asymptotic estimates in the regime of large amplitude. The estimates are obtained by analyzing the action minimizing trajectories in the Lagrangian representation of solutions (Lax formula and its extension) in combination with delicate gradient bound of viscosity solutions to the associated cell problem of homogenization. Though the inviscid turbulent flame speeds share the same leading order asymptotics, their difference due to nonlinearities is identified as a subtle double logarithm in the large flow amplitude from the sharp growth laws. The turbulent flame speeds differ much more significantly in the corresponding viscous HJ models.

Complete Integrability of Shock Clustering and Burgers Turbulence

We consider scalar conservation laws with convex flux and random initial data. The Hopf–Lax formula induces a deterministic evolution of the law of the initial data. In a recent article, we derived a kinetic theory and Lax equations to describe the evolution of the law under the assumption that the initial datum is a spectrally negative Markov process. Here we show that: (i) the Lax equations are Hamiltonian and describe a principle of least action on the Markov group; (ii) the Lax equations are completely integrable and linearized via a loop-group factorization of operators; (iii) the associated zero-curvature equations can be solved via inverse scattering. Our results are rigorous for N-dimensional approximations of the Lax equations, and yield formulas for the limit N → ∞. The main observation is that the Lax equations and zero-curvature equations are a Markovian analog of known integrable systems (geodesic flow on Lie groups and the N-wave model respectively). This allows us to introduce a variety of methods from the theory of integrable systems.

ON LeFLOCH'S SOLUTIONS TO THE INITIAL-BOUNDARY VALUE PROBLEM FOR SCALAR CONSERVATION LAWS

We consider the initial-boundary value problem for scalar conservation laws on the strip (0, ∞) × [0, 1] with strictly convex smooth flux of a superlinear growth. We show that an associated Hamilton–Jacobi equation with initial and (appropriately defined) boundary conditions has a unique generalized solution V that can be obtained as minimum of three value functions of the calculus of variation. Each of these functions, in turn, can be expressed using Lax's formula. The traces of the gradients Vxsatisfy generalized boundary conditions (as in LeFloch (1988)) in a pointwise manner when the initial and boundary data are continuous and in a weak sense when they are discontinuous. It is also shown that Vxis continuous almost everywhere, and a result concerning the traces of the sign of f′(Vx(t, ⋅)) is proven.

Metric techniques for convex stationary ergodic Hamiltonians

We adapt the metric approach to the study of stationary ergodic Hamilton–Jacobi equations, for which a notion of admissible random (sub)solution is defined. For any level of the Hamiltonian greater than or equal to a distinguished critical value, we define an intrinsic random semidistance and prove that an asymptotic norm does exist. Taking as source region a suitable class of closed random sets, we show that the Lax formula provides admissible subsolutions. This enables us to relate the degeneracies of the critical stable norm to the existence/nonexistence of exact or approximate critical admissible solutions.

Numerical solution of a Cauchy problem for nonlinear reaction diffusion processes

In this paper, the authors propose a numerical method to compute the solution of the Cauchy problem: w t - ( w m w x ) x = w p , the initial condition is a nonnegative function with compact support, m > 0 , p ⩾ m + 1 . The problem is split into two parts: a hyperbolic term solved by using the Hopf and Lax formula and a parabolic term solved by a backward linearized Euler method in time and a finite element method in space. The convergence of the scheme is obtained. Further, it is proved that if m + 1 ⩽ p < m + 3 , any numerical solution blows up in a finite time as the exact solution, while for p > m + 3 , if the initial condition is sufficiently small, a global numerical solution exists, and if p ⩾ m + 3 , for large initial condition, the solution is unbounded.

Mesoscale Averaging of Nucleation and Growth Models

The aim of this paper is to derive a general theory for the averaging of heterogeneous processes with stochastic nucleation and deterministic growth. We start by generalizing the classical Johnson--Mehl--Avrami--Kolmogorov theory based on the causal cone to heterogeneous growth situations. Moreover, we relate the computation of the causal cone to a Hopf--Lax formula for Hamilton--Jacobi equations describing the growth of grains. As an outcome of the approach we obtain formulae for the expected values of geometric densities describing the growth processes; in particular we generalize the standard Avrami--Kolmogorov relations for the degree of crystallinity. By relating the computation of expected values to mesoscale averaging, we obtain a suitable description of the process at the mesoscale. We show how the variance of these mesoscale averages can be estimated in terms of quotients of the typical length on the microscale and on the mesoscale. Moreover, we discuss the efficient computation of the mesoscale averages in the typical case when the nucleation and growth rates are obtained from mesoscopic fields (such as, e.g., temperature). Finally, we give a brief outlook to possible extensions such as polycrystalline growth, which turns out to be rather straightforward when starting from our general framework.

Numerical solution of a nonlinear reaction diffusion equation

In this paper, the authors propose a numerical method to compute the solution of a Cauchy problem with blow-up of the solution. The problem is split in two parts: a hyperbolic problem which is solved by using Hopf and Lax formula and a parabolic problem solved by a backward linearized Euler method in time and a finite element method in space. It is proved that the numerical solution blows up in a finite time as the exact solution and the support of the approximation of a self-similar solution remains bounded. The convergence of the scheme is obtained.