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.     

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.     

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.     

Semiconcavity estimates for viscous Hamilton–Jacobi equations

We present sharp Hessian estimates of the form D² S^e(t,x) ￡ g(t)I{D^2 S^\varepsilon(t,x)\leq g(t)I} for the solution of the viscous Hamilton–Jacobi equation

Finite volume schemes for Hamilton–Jacobi equations

Summary. We introduce two classes of monotone finite volume schemes for Hamilton-Jacobi equations. The corresponding approximating functions are piecewise linear defined on a mesh consisting of triangles. The schemes are shown to converge to the viscosity solution of the Hamilton–Jacobi equation. Received February 25, 1998 / Published online: June 29, 1999     

On unbounded solutions of ergodic problems in ℝm for viscous Hamilton–Jacobi equations

In this article, we study ergodic problems in the whole space ?^m for viscous Hamilton–Jacobi equations in the case of locally Lipschitz continuous and coercive right-hand sides. We prove in particular the existence of a critical value λ* for which (i) the ergodic problem has solutions for all λ≤λ*, (ii) bounded from below solutions exist and are associated to λ*, (iii) such solutions are unique (up to an additive constant). We obtain these properties without additional assumptions in the superquadratic case, while, in the subquadratic one, we assume the right-hand side to behave like a power. These results are slight generalizations of analogous results by Ichihara but they are proved in the present paper by partial differential equation (pde) methods, contrarily to Ichihara who is using a combination of pde technics with probabilistic arguments.     

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.     

Application of variational iteration method for Hamilton–Jacobi–Bellman equations

In this paper, we use the variational iteration method (VIM) for optimal control problems. First, optimal control problems are transferred to Hamilton–Jacobi–Bellman (HJB) equation as a nonlinear first order hyperbolic partial differential equation. Then, the basic VIM is applied to construct a nonlinear optimal feedback control law. By this method, the control and state variables can be approximated as a function of time. Also, the numerical value of the performance index is obtained readily. In view of the convergence of the method, some illustrative examples are presented to show the efficiency and reliability of the presented method.     

Regularity properties of viscosity solutions of integro-partial differential equations of Hamilton–Jacobi–Bellman type

We study the regularity properties of integro-partial differential equations of Hamilton–Jacobi–Bellman type with the terminal condition, which can be interpreted through a stochastic control system, composed of a forward and a backward stochastic differential equation, both driven by a Brownian motion and a compensated Poisson random measure. More precisely, we prove that, under appropriate assumptions, the viscosity solution of such equations is jointly Lipschitz and jointly semiconcave in (t,x)∈Δ×R^d$(t,x)\in \Delta \times {\mathbb{R}}^d$, for all compact time intervals Δ$\Delta$ excluding the terminal time. Our approach is based on the time change for the Brownian motion and on Kulik’s transformation for the Poisson random measure.     

Null controllability of viscous Hamilton–Jacobi equations

We study the problem of null controllability for viscous Hamilton–Jacobi equations in bounded domains of the Euclidean space in any space dimension and with controls localized in an arbitrary open nonempty subset of the domain where the equation holds. We prove the null controllability of the system in the sense that, every bounded (and in some cases uniformly continuous) initial datum can be driven to the null state in a sufficiently large time. The proof combines decay properties of the solutions of the uncontrolled system and local null controllability results for small data obtained by means of Carleman inequalities. We also show that there exists a waiting time so that the time of control needs to be large enough, as a function of the norm of the initial data, for the controllability property to hold. We give sharp asymptotic lower and upper bounds on this waiting time both as the size of the data tends to zero and infinity. These results also establish a limit on the growth of nonlinearities that can be controlled uniformly on a time independent of the initial data.     

Asymptotic solutions for large time of Hamilton–Jacobi equations in Euclidean n space

We study the large time behavior of solutions of the Cauchy problem for the Hamilton–Jacobi equation ut+H(x,Du)=0$u_t+H(x,Du)=0$ in Rn×(0,∞)${\mathbf{R}}^n\times (0,\infty )$, where H(x,p)$H(x,p)$ is continuous on Rn×Rn${\mathbf{R}}^n\times {\mathbf{R}}^n$ and convex in p  . We establish a general convergence result for viscosity solutions u(x,t)$u(x,t)$ of the Cauchy problem as t→∞$t\to \infty$.     

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