On Viscosity Solution of HJB Equations with State Constraints and Reflection Control

Abstract

Abstract. Motivated by a control problem of a certain queueing network we consider a control problem where the dynamics is constrained in the nonnegative orthant R

Nonlinear Neumann problems for fully nonlinear elliptic PDEs on a quadrant

“…The condition that h ′ i (t) ≥ 0 a.e. on R can be stated that h i is nondecreasing on R. In [2], the authors have studied the nonlinear elliptic partial differential equation (1.2) F (D 2 u, Du, u, x) = 0 in Ω := (0, ∞) n , with the nonlinear Neumann type boundary condition, similar to the above, stated as…”

“…As shown in [2], the boundary value problem (1.2)-(1.3) arises in a scaling limit of stochastic control of queuing systems. A main contribution in [2] is that, under suitable hypotheses, the value function of the stochastic control problem, obtained in the scaling limit, is identified as the unique viscosity solution of (1.2) and (1.3), with an appropriate choice of F . This identification result heavily depends on the uniqueness or comparison theorem for viscosity solutions, subsolutions, and supersolutions of (1.2)- (1.3).…”

“…This identification result heavily depends on the uniqueness or comparison theorem for viscosity solutions, subsolutions, and supersolutions of (1.2)- (1.3). The comparison theorem in [2] is basically valid only for the H i having the positive homogeneity of degree one.…”

“…We remark that there are a great amount of contributions to the Neumann boundary value problem on relatively regular domains even in the framework of viscosity solutions. We refer for some general results to [1,3,5], and for some results on domains with corners to [2,4] and the references therein.…”

An Ergodic Control Problem for Many-Server Multiclass Queueing Systems with Cross-Trained Servers

“…A legitimate question for these processor sharing type model is that whether the GPS type policy is optimal or not for the pay-off function considered above. Motivated by this question a similar control problem is considered in [14] for a queueing model with finitely many servers and it is shown that the value function associated to the limiting controlled diffusion model solves a non-linear Neumann boundary value problem. The solution is obtained in the viscosity sense and therefore, it is hard to extract any information about the optimal control, even for the diffusion control problem.…”