Asymptotic Solutions of Hamilton–Jacobi Equations with Semi-Periodic Hamiltonians

We study the long time behavior of viscosity solutions of the Cauchy problem for Hamilton–Jacobi equations in ? ⁿ . We prove that if the Hamiltonian H(x, p) is coercive and strictly convex in a mild sense in p and upper semi-periodic in x, then any solution of the Cauchy problem “converges” to an asymptotic solution for any lower semi-almost periodic initial function.     

Asymptotic Solutions of Viscous Hamilton–Jacobi Equations with Ornstein–Uhlenbeck Operator

We study the long time behavior of solutions of the Cauchy problem for semilinear parabolic equations with the Ornstein–Uhlenbeck operator in ? ^N . The long time behavior in the main results is stated with help of the corresponding to ergodic problem, which complements, in the case of unbounded domains, the recent developments on long time behaviors of solutions of (viscous) Hamilton–Jacobi equations due to Namah (1996 Namah , G. ( 1996 ). Asymptotic solution of a Hamilton–Jacobi equation . Asymptotic Anal. 12 ( 4 ): 355 – 370 . [CSA] [Web of Science ®] , [Google Scholar]), Namah and Roquejoffre (1999 Namah , G. , Roquejoffre , J.-M . ( 1999 ). Remarks on the long-time behavior of the solutions of Hamilton–Jacobi equations . Comm. PDE 24 ( 5–6 ): 883 – 893 . [CSA] [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]), Roquejoffre (1998 Roquejoffre , J.-M . ( 1998 ). Comportement asymptotique des solutions d’équations de Hamilton–Jacobi monodimensionnelles . C. R. Acad. Sci. Paris Sér. I Math. 326 ( 2 ): 185 – 189 . [CSA] [Crossref] , [Google Scholar]), Fathi (1998 Fathi , A. ( 1998 ). Sur la convergence du semi-groupe de Lax–Oleinik semigroup . C. R. Acad. Sci. Paris Sér. I Math. 327 ( 3 ): 267 – 270 . [CSA] [Crossref] , [Google Scholar]), Barles and Souganidis (2000 Barles , G. , Souganidis , P. E. ( 2000 ). On the large time behavior of solutions of Hamilton–Jacobi equaitons . SIAM J. Math. Anal. 31 ( 4 ): 925 – 939 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar] 2001 Barles , G. , Souganidis , P. E. ( 2001 ). Space-time periodic solutions and long-time behavior of solutions to quasi-periodic parabolic equations . SIAM J. Math. Anal. 32 ( 6 ): 1311 – 1323 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]). We also establish existence and uniqueness results for solutions of the Cauchy problem and ergodic problem for semilinear parabolic equations with the Ornstein–Uhlenbeck operator.     

Optimal L ∞-Error Estimate of a Finite Element Method for Hamilton–Jacobi–Bellman Equations

This paper is concerned with the piecewise linear finite element approximation of Hamilton–Jacobi–Bellman equations. We establish the optimal L ^∞-error estimate, combining the concepts of subsolution and discrete regularity.     

Optimal Control of Non-well-posed Heat Equations

This work is concerned with Pontryagin＇s maximum principle of optimal control problems governed by some non-well-posed semilinear heat equations. A type of approach to the non-well-posed optimal control problem is given.     

A Characterization of the Existence of Solutions for Hamilton—Jacobi Equations in Ergodic Control Problems with Applications

We give a characterization of the existence of bounded solutions for Hamilton—Jacobi equations in ergodic control problems with state-constraint. This result is applied to the reexamination of the counterexample given in [5] concerning the existence of solutions for ergodic control problems in infinite-dimensional Hilbert spaces and also establishing results on effective Hamiltonians in periodic homogenization of Hamilton—Jacobi equations. Accepted 1 December 1999     

Variational Solutions of Coupled Hamilton—Jacobi Equations

In this note we study variational solutions of weakly coupled Hamilton—Jacobi equations in the case where the Hamiltonians are convex. More precisely, we build the variational solution by an approximation scheme. Accepted 24 April 1998     

Two Fast Marching Methods for Hamilton–Jacobi Equations

We introduce and analyze two new Fast Marching (FM) methods based on a semi-Lagrangian (SL) approximation (see [2] for a more complete presentation). The Characteristics driven Fast Marching method accepts more than one node at every iteration using a dynamic condition which leads to a faster convergence. The Buffered Fast Marching method allows to deal with convex and non–convex Hamilton–Jacobi equations including anisotropic front propagation and differential games. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Nonlinear Potentials for Hamilton–Jacobi–Bellman Equations. II

A formal method of constructing the viscosity solutions for abstract nonlinear equations of Hamilton–Jacobi–Bellman (HJB) type was developed in the previous work of the author. A new advantage of this method (which was called an nonlinear potentials method) is that it gives a possibility to choose at the first step an expected regularity of the solution and then – to construct this solution. This makes the whole procedure more simple because an analysis of regularity of viscosity solutions is usually the most complicated step.Nonlinear potentials method is a generalization of Krylov's approach to study HJB equations.In this article nonlinear potentials method is applied to elliptic degenerate HJB equations in R^d with variable coefficients.     

On the Solution of a System of Hamilton–Jacobi Equations of Special Form

The paper is concerned with the investigation of a system of first-order Hamilton–Jacobi equations. We consider a strongly coupled hierarchical system: the first equation is independent of the second, and the Hamiltonian of the second equation depends on the gradient of the solution of the first equation. The system can be solved sequentially. The solution of the first equation is understood in the sense of the theory of minimax (viscosity) solutions and can be obtained with the help of the Lax–Hopf formula. The substitution of the solution of the first equation in the second Hamilton–Jacobi equation results in a Hamilton–Jacobi equation with discontinuous Hamiltonian. This equation is solved with the use of the idea of M-solutions proposed by A. I. Subbotin, and the solution is chosen from the class of multivalued mappings. Thus, the solution of the original system of Hamilton–Jacobi equations is the direct product of a single-valued and multivalued mappings, which satisfy the first and second equations in the minimax and M-solution sense, respectively. In the case when the solution of the first equation is nondifferentiable only along one Rankine–Hugoniot line, existence and uniqueness theorems are proved. A representative formula for the solution of the system is obtained in terms of Cauchy characteristics. The properties of the solution and their dependence on the parameters of the problem are investigated.     

Functional Inequalities and Hamilton–Jacobi Equations in Geodesic Spaces

We study the connection between the p-Talagrand inequality and the q-logarithmic Sololev inequality for conjugate exponents p ≥ 2, q ≤ 2 in proper geodesic metric spaces. By means of a general Hamilton–Jacobi semigroup we prove that these are equivalent, and moreover equivalent to the hypercontractivity of the Hamilton–Jacobi semigroup. Our results generalize those of Lott and Villani. They can be applied to deduce the p-Talagrand inequality in the sub-Riemannian setting of the Heisenberg group.     

Homogenization of “Viscous” Hamilton–Jacobi Equations in Stationary Ergodic Media

ABSTRACT

We study the homogenization of “viscous” Hamilton–Jacobi equations in stationary ergodic media. The “viscosity” and the spatial oscillations are assumed to be of the same order. We identify the asymptotic (effective) equation, which is a first-order deterministic Hamilton–Jacobi equation. We also provide examples that show that the associated macroscopic problem does not admit suitable solutions (correctors). Finally, we present as applications results about large deviations of diffusion processes and front propagation (asymptotics of reaction-diffusion equations) in random environments.     

Equations for the Missing Boundary Values in the Hamiltonian Formulation of Optimal Control Problems

Partial differential equations for the unknown final state and initial costate arising in the Hamiltonian formulation of regular optimal control problems with a quadratic final penalty are found. It is shown that the missing boundary conditions for Hamilton’s canonical ordinary differential equations satisfy a system of first-order quasilinear vector partial differential equations (PDEs), when the functional dependence of the H-optimal control in phase-space variables is explicitly known. Their solutions are computed in the context of nonlinear systems with ℝ ⁿ -valued states. No special restrictions are imposed on the form of the Lagrangian cost term. Having calculated the initial values of the costates, the optimal control can then be constructed from on-line integration of the corresponding 2n-dimensional Hamilton ordinary differential equations (ODEs). The off-line procedure requires finding two auxiliary n×n matrices that generalize those appearing in the solution of the differential Riccati equation (DRE) associated with the linear-quadratic regulator (LQR) problem. In all equations, the independent variables are the finite time-horizon duration T and the final-penalty matrix coefficient S, so their solutions give information on a whole two-parameter family of control problems, which can be used for design purposes. The mathematical treatment takes advantage from the symplectic structure of the Hamiltonian formalism, which allows one to reformulate Bellman’s conjectures concerning the “invariant-embedding” methodology for two-point boundary-value problems. Results for LQR problems are tested against solutions of the associated differential Riccati equation, and the attributes of the two approaches are illustrated and discussed. Also, nonlinear problems are numerically solved and compared against those obtained by using shooting techniques.     

A stochastic Hamilton–Jacobi equation with infinite speed of propagation

We give an example of a stochastic Hamilton–Jacobi equation $\mathrm{d}u=H(\mathrm{D}u)\mathrm{d}\xi$ which has an infinite speed of propagation as soon as the driving signal ξ is not of bounded variation.     

Ergodic Type Problems and Large Time Behaviour of Unbounded Solutions of Hamilton–Jacobi Equations

We study the large time behavior of Lipschitz continuous, possibly unbounded, viscosity solutions of Hamilton–Jacobi Equations in the whole space ? ^N . The associated ergodic problem has Lipschitz continuous solutions if the analogue of the ergodic constant is larger than a minimal value λ_min. We obtain various large-time convergence and Liouville type theorems, some of them being of completely new type. We also provide examples showing that, in this unbounded framework, the ergodic behavior may fail, and that the asymptotic behavior may also be unstable with respect to the initial data.