Some results on the large-time behavior of weakly coupled systems of first-order Hamilton–Jacobi equations

Systems of Hamilton–Jacobi equations arise naturally when we study optimal control problems with pathwise deterministic trajectories with random switching. In this work, we are interested in the large-time behavior of weakly coupled systems of first-order Hamilton–Jacobi equations in the periodic setting. First results have been obtained by Camilli et al. (NoDEA Nonlinear Diff Eq Appl, 2012) and Mitake and Tran (Asymptot Anal, 2012) under quite strict conditions. Here, we use a PDE approach to extend the convergence result proved by Barles and Souganidis (SIAM J Math Anal 31(4):925–939 (electronic), 2000) in the scalar case. This result permits us to treat general cases, for instance, systems of nonconvex Hamiltonians and systems of strictly convex Hamiltonians. We also obtain some other convergence results under different assumptions. These results give a clearer view on the large-time behavior for systems of Hamilton–Jacobi equations.     

Large time behavior of weakly coupled systems of first-order Hamilton–Jacobi equations

We show a large time behavior result for class of weakly coupled systems of first-order Hamilton–Jacobi equations in the periodic setting. We use a PDE approach to extend the convergence result proved by Namah and Roquejoffre (Commun. Partial. Differ. Equ. 24(5–6):883–893, 1999) in the scalar case. Our proof is based on new comparison, existence and regularity results for systems. An interpretation of the solution of the system in terms of an optimal control problem with switching is given.     

Long-time behavior of solutions to Hamilton–Jacobi equations with quadratic gradient term

We study a rate of convergence appearing in the long-time behavior of viscosity solutions of the Cauchy problem for the Hamilton–Jacobi equation

Minimax solutions of Hamilton–Jacobi functional equations for neutral-type systems

A functional Hamilton–Jacobi equation with covariant derivatives which corresponds to neutral-type dynamical systems is obtained. The definition of a minimax solution of this equation is given. Conditions under which such a solution exists and is unique and well defined are found.     

Propagation of singularities for solutions of Hamilton–Jacobi equations

We consider the viscosity solution of the Cauchy problem for a class of Hamilton–Jacobi equations and we show that the points of the C¹$C^1$ singular support of such a function propagate along the generalized characteristics for all the times.     

From pointwise to local regularity for solutions of Hamilton–Jacobi equations

It is well-known that solutions to the Hamilton–Jacobi equation $$\begin{aligned} u_{t}(t,x)+H(x,u_{x}(t,x))=0 \end{aligned}$$ fail to be everywhere differentiable. Nevertheless, suppose a solution $u$ turns out to be differentiable at a given point $(t,x)$ in the interior of its domain. May then one deduce that $u$ must be continuously differentiable in a neighborhood of $(t,x)$ ? Although this question has a negative answer in general, our main result shows that it is indeed the case when the proximal subdifferential of $u(t,\cdot )$ at $x$ is nonempty. Our approach uses the representation of $u$ as the value function of a Bolza problem in the calculus of variations, as well as necessary conditions for such a problem.     

Lax–Hopf formula and Max-Plus properties of solutions to Hamilton–Jacobi equations

We state and prove a “Lax–Hopf formula” characterizing viable capture basins of targets investigated in viability theory and derive a “Max-Plus” morphism of capture basins with respect to the target. Capture basins are used to define “viability solutions” to Hamilton–Jacobi equations satisfying “trajectory conditions” (initial, boundary or Lagrangian conditions). The Max-Plus morphism property of Lax–Hopf formula implies the fact that the solution associated with inf-convolution of trajectory conditions is the inf-convolution of the solutions for each trajectory condition. For instance, Lipschitz regularization or decreasing envelopes of trajectory condition imply the Lipschitz regulation or decreasing envelopes of the solutions.     

Probabilistic interpretation of a coupled system of Hamilton–Jacobi–Bellman equations

In this paper, we give a probabilistic interpretation for a coupled system of Hamilton–Jacobi–Bellman equations using the value function of a stochastic control problem. First we introduce this stochastic control problem. Then we prove that the value function of this problem is deterministic and satisfies a (strong) dynamic programming principle. And finally, the value function is shown to be the unique viscosity solution of the coupled system of Hamilton–Jacobi–Bellman equations.     

Local and global properties of solutions of quasilinear Hamilton–Jacobi equations

We study some properties of the solutions of (E) −_Δpu+|∇u|^q=0$-{\mathrm{\Delta }}_{p}u+{|\mathrm{\nabla }u|}^q=0$ in a domain Ω⊂R^N$\Omega \subset {\mathbb{R}}^N$, mostly when p≥q>p−1$p\ge q>p-1$. We give a universal a priori estimate of the gradient of the solutions with respect to the distance to the boundary. We give a full classification of the isolated singularities of the nonnegative solutions of (E), a partial classification of isolated singularities of the negative solutions. We prove a general removability result expressed in terms of some Bessel capacity of the removable set. We extend our estimates to equations on complete noncompact manifolds satisfying a lower bound estimate on the Ricci curvature, and derive some Liouville type theorems.     

Layered viscosity solutions of nonautonomous Hamilton–Jacobi equations: Semiconvexity and relations to characteristics

We construct an explicit representation of viscosity solutions of the Cauchy problem for the Hamilton–Jacobi equation (H,σ)$(H,\sigma )$ on a given domain Ω=(0,T)×Rⁿ$\Omega =(0,T)\times {\mathbb{R}}^n$. It is known that, if the Hamiltonian H=H(t,p)$H=H(t,p)$ is not a convex (or concave) function in p  , or H(⋅,p)$H(\cdot ,p)$ may change its sign on (0,T)$(0,T)$, then the Hopf-type formula does not define a viscosity solution on Ω  . Under some assumptions for H(t,p)$H(t,p)$ on the subdomains (_ti,_ti+1)×Rⁿ⊂Ω$(t_i,t_{i+1})\times {\mathbb{R}}^n\subset \Omega$, we are able to arrange “partial solutions” given by the Hopf-type formula to get a viscosity solution on Ω. Then we study the semiconvexity of the solution as well as its relations to characteristics.