Article
Mathematics
Kaizhi Wang, Lin Wang, Jun Yan
Summary: The paper provides necessary and sufficient conditions for the existence of viscosity solutions of nonlinear first order PDEs, proving compactness of the set of solutions. Furthermore, it explores the long-term behavior of viscosity solutions for Cauchy problems using weak KAM theory and dynamic methods.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2021)
Article
Mathematics, Applied
Yang Xu, Jun Yan, Kai Zhao
Summary: This paper studies the relationship between the stability of viscosity solutions and the set structure of weak KAM solutions to the contact Hamilton-Jacobi equation.
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY
(2023)
Article
Mathematics, Applied
Kaizhi Wang, Jun Yan, Kai Zhao
Summary: This paper deals with the long-time behavior of viscosity solutions of evolutionary contact Hamilton-Jacobi equations. It shows the connection between viscosity solutions of the ergodic contact Hamilton-Jacobi equation and solutions of the evolutionary equation.
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY
(2022)
Article
Mathematics, Applied
Maxime Zavidovique
Summary: This article studies the properties of solutions to Hamilton-Jacobi equations and proves that the function u lambda converges to a function u0 satisfying a specific equation under certain conditions.
Article
Mathematics
Andrea Davini, Elena Kosygina
Summary: We prove homogenization for a class of nonconvex (possibly degenerate) viscous Hamilton-Jacobi equations in stationary ergodic random environments in one space dimension. The results concern Hamiltonians of the form G(p) + V (x, omega), where the nonlinearity G is the minimum of two or more convex functions with the same absolute minimum, and the potential V is a bounded stationary process satisfying an additional scaled hill and valley condition. This condition is trivially satisfied in the inviscid case, while it is equivalent to the original hill and valley condition of A. Yilmaz and O. Zeitouni [32] in the uniformly elliptic case. Our approach is based on PDE methods and does not rely on representation formulas for solutions. Using only comparison with suitably constructed super- and sub- solutions, we obtain tight upper and lower bounds for solutions with linear initial data x ? theta x. Another important ingredient is a general result of P. Cardaliaguet and P. E. Souganidis [13] which guarantees the existence of sublinear correctors for all theta outside flat parts of effective Hamiltonians associated with the convex functions from which G is built. We derive crucial derivative estimates for these correctors which allow us to use them as correctors for G. (c) 2022 Elsevier Inc. All rights reserved.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2022)
Article
Mathematics, Applied
Yves Achdou, Claude Le Bris
Summary: We study the homogenization problem for a certain class of stationary Hamilton-Jacobi equations. By perturbing a periodic Hamiltonian near the origin, we prove that the limiting problem consists of a Hamilton-Jacobi equation outside the origin, with the same effective Hamiltonian as in periodic homogenization, and a Dirichlet condition at the origin to account for the perturbation. Various comments and extensions are provided.
COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS
(2023)
Article
Mathematics, Applied
Kaizhi Wang, Jun Yan, Kai Zhao
Summary: This article discusses the existence and multiplicity of nontrivial time periodic viscosity solutions to a contact Hamilton-Jacobi equation, and investigates the long time behavior of these solutions. It is found that for a certain class of initial data, the corresponding viscosity solutions converge to asymptotic time periodic viscosity solutions. The article also analyzes a bifurcation phenomenon for a parameter-dependent Hamilton-Jacobi equation.
JOURNAL DE MATHEMATIQUES PURES ET APPLIQUEES
(2023)
Article
Mathematics
William M. Feldman, Jean-Baptiste Fermanian, Bruno Ziliotto
Summary: The article provides an example of the failure of homogenization for a viscous Hamilton-Jacobi equation with a non-convex Hamiltonian.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2021)
Article
Mathematics, Applied
Panrui Ni
Summary: This paper discusses the convergence of viscosity solutions for the generalized discounted Hamilton-Jacobi equation and provides two examples where the solutions do not converge as lambda tends to zero.
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY
(2023)
Article
Automation & Control Systems
Jianjun Zhou
Summary: This article introduces the concept of viscosity solutions for first-order path-dependent Hamilton-Jacobi-Bellman (PHJB) equations associated with optimal control problems for path-dependent evolution equations in Hilbert space. We identify the value functional of optimal control problems as the unique viscosity solution to the associated PHJB equations without a specific assumption. We also demonstrate that our notion of viscosity solutions is consistent with classical solutions and exhibits a stability property.
Article
Mathematics, Applied
Italo Capuzzo Dolcetta, Andrea Davini
Summary: We study the asymptotic behavior of viscosity solutions u(G)(lambda) of the Hamilton-Jacobi (HJ) equation as the positive discount factor lambda tends to 0. We prove the local uniform convergence of the functions u(G)(lambda) to a specific solution u(G)(0) of the critical equation. Our work also includes a qualitative analysis of the critical equation.
COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS
(2023)
Article
Mathematics, Applied
Kaizhi Wang, L. I. N. Wang, J. U. N. Yan
Summary: In this paper, Aubry-Mather and weak KAM theories for contact Hamiltonian systems with certain dependence on the contact variable are further developed. Various properties of the Mane set and the Aubry set are discussed, along with the identification of a new flow-invariant set. Examples are provided to demonstrate the differences between solutions in the contact case, revealing new phenomena and differences in the vanishing discount problem.
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS
(2022)
Article
Mathematics
M. I. Gomoyunov, A. R. Plaksin
Summary: The paper discusses a path-dependent Hamilton-Jacobi equation with coinvariant derivatives over the space of continuous functions. It studies generalized solutions of the equation in both the minimax and viscosity senses, and proves the equivalence between these two notions. The paper also obtains comparison and uniqueness results for viscosity solutions of a Cauchy problem with a right-end boundary condition.
JOURNAL OF FUNCTIONAL ANALYSIS
(2023)
Article
Mathematics, Applied
William Cooperman
Summary: We studied a problem concerning the Hamilton-Jacobian equation, where the Hamiltonian is periodic, coercive, and convex. By combining the representation formula from optimal control theory and a theorem by Alexander, we obtained a homogenized rate that is close to optimal and holds in all dimensions.
ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS
(2022)
Article
Automation & Control Systems
Anton Plaksin
Summary: This paper addresses a zero-sum differential game for a dynamical system described by a nonlinear delay differential equation under a initial condition defined by a piecewise continuous function. It derives the corresponding Cauchy problem for Hamilton-Jacobi-Bellman-Isaacs equation with coinvariant derivatives, and considers the definition of a viscosity solution for this problem. It proves that the differential game has a unique viscosity solution value, and obtains an infinitesimal description of the viscosity solution based on notions of sub- and superdifferentials corresponding to coinvariant derivatives.
SIAM JOURNAL ON CONTROL AND OPTIMIZATION
(2021)
Article
Mathematics, Applied
Miguel Brozos-Vazquez, Diego Mojon-Alvarez
Summary: We study the geometric structure of weighted Einstein smooth metric measure spaces with weighted harmonic Weyl tensor. A complete local classification is provided, showing that either the underlying manifold is Einstein, or decomposes as a warped product in a specific way. Moreover, if the manifold is complete, then it either is a weighted analogue of a space form, or it belongs to a particular family of Einstein warped products.
JOURNAL DE MATHEMATIQUES PURES ET APPLIQUEES
(2024)
Article
Mathematics, Applied
Domenec Ruiz-Balet, Enrique Zuazua
Summary: Inspired by normalising flows, we analyze the bilinear control of neural transport equations using time-dependent velocity fields constrained by a simple neural network assumption. We prove the L1 approximate controllability property, showing that any probability density can be driven arbitrarily close to any other one within any given time horizon. The control vector fields are explicitly and recursively constructed, providing quantitative estimates of their complexity and amplitude. This also leads to statistical error bounds when only random samples of the target probability density are available.
JOURNAL DE MATHEMATIQUES PURES ET APPLIQUEES
(2024)
