In this paper, we study a nonlinear Langevin equation involving n-parameter singular fractional orders ?i(i=1,2), and ? with initial conditions. By means of an interesting fixed point theorem, establish sufficient conditions for the existence uniqueness solutions equations.

We consider a two-player zero-sum game with integral payoff and with incomplete information on one side, where the payoff is chosen among a continuous set of possible payoffs. We prove that the value function of this game is solution of an auxiliary optimization problem over a set of measure-valued processes. Then we use this equivalent formulation to characterize the value function as the visc...

We present a formula for the viscosity subdiierential of the sum of two uniformly continuous functions on smooth Banach spaces. This formula is deduced from a new variational principle with constraints. We obtain as a consequence a weak form of Preiss' theorem for uniformly continuous functions. We use these results to give simple proofs of some uniqueness results of viscosity solutions of Hami...

We perform a systematic study of optimization problems in the Wasserstein spaces that are analogs of infinite horizon, deterministic control problems. We derive necessary conditions on action minimizing paths and present a sufficient condition for their existence. We also verify that the corresponding generalized value functions are a type of viscosity solution of a time independent, Hamilton–J...

Adomian decomposition method has been used intensively to solve nonlinear boundary and initial value problems. It has been proved to be very efficient in generating series solutions of the problem under consideration under the assumption that such series solution exits. However, very little has been done to address the mathematical foundation of the method and its error analysis. In this articl...

We present new theorems characterizing robust Lyapunov functions and infinite horizon value functions in optimal control as unique viscosity solutions of partial differential equations. We use these results to further extend Zubov’s method for representing domains of attraction in terms of partial differential equation solutions.

We study the long-time behavior of the unique viscosity solution u of the viscous Hamilton-Jacobi Equation ut−∆u+|Du| m = f in Ω×(0,+∞) with inhomogeneous Dirichlet boundary conditions, where Ω is a bounded domain of RN . We mainly focus on the superquadratic case (m > 2) and consider the Dirichlet conditions in the generalized viscosity sense. Under rather natural assumptions on f, the initial...

this paper presents a new numerical method for solution of eikonal equation in two dimensions.in contrast to the previously developed methods which try to define the solution surface by its level sets(contour curves), the developed methodology identifies the solution surface by resorting to its characteristics. the suggested procedure is based on the geometric properties of the solution surface...

We consider a situation where dislocations are parallel lines moving in a single plane. For this simple geometry, dislocations dynamics is modeled by a one-dimensional non-local transport equation. We prove a result of existence and uniqueness for all time of the continuous viscosity solution for this equation. A finite difference scheme is proposed to approximate the continuous viscosity solut...