Second order Hamilton–Jacobi–Bellman equations with an unbounded operator

This work is devoted to the study of a class of Hamilton–Jacobi–Bellman equations associated to an optimal control problem where the state equation is a stochastic differential inclusion with a maximal monotone operator. We show that the value function minimizing a Bolza-type cost functional is a viscosity solution of the HJB equation. The proof is based on the perturbation of the initial problem by approximating the unbounded operator. Finally, by providing a comparison principle we are able to show that the solution of the equation is unique.     

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.     

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.     

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.     

Hamilton–Jacobi equations in space of measures associated with a system of conservation laws

We introduce a class of action integrals defined over probability measure-valued path space. We show that extremal point of such action exits and satisfies a type of compressible Euler equation in a weak sense. Moreover, we prove that both Cauchy and resolvent formulations of the associated Hamilton–Jacobi equations, in the space of probability measures, are well-posed.     

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.     

Envelopes and nonconvex Hamilton–Jacobi equations

This paper introduces a new representation formula for viscosity solutions of nonconvex Hamilton–Jacobi PDE using “generalized envelopes” of affine solutions. We study as well envelope and singular characteristic constructions of equivocal surfaces and discuss also differential game theoretic interpretations. In memory of Arik A. Melikyan.      

Hamilton–Jacobi equations constrained on networks

We consider continuous-state and continuous-time control problems where the admissible trajectories of the system are constrained to remain on a network. In our setting, the value function is continuous. We define a notion of constrained viscosity solution of Hamilton–Jacobi equations on the network and we study related comparison principles. Under suitable assumptions, we prove in particular that the value function is the unique constrained viscosity solution of the Hamilton–Jacobi equation on the network.     

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.