On the Optimal Impulse Control Problem for Degenerate Diffusions

In this paper, we give a characterization of the optimal cost of an impulse control problem as the maximum solution of a quasi-variational inequality without assuming nondegeneracy. An estimate of the velocity of uniform convergence of the sequence of stopping time problems associated with the impulse control problem is given

On the Optimal Stopping Time Problem for Degenerate Diffusions

In this paper we give a characterization of the optimal cost of a stopping time problem as the maximum solution of a variational inequality without coercivity. Some properties of continuity for the optimal cost are also given

Stochastic Hybrid Control

The objective of this paper is to study the stochastic version of a previous paper of the authors, in which hybrid control for deterministic systems was considered. The modelling is quite similar to the deterministic case. We have a system whose state is composed of a continuous part and a discrete part. They are affected by a continuous type control and an impulse control. The dynamics is moreover perturbed by noise, also a continuous and a discrete noise process. The Markovian character of the state process is preserved. We develop the model and show how the dynamic programming approach leads to some involved quasi-variational inequality

Optimal Stochastic Scheduling of Power Generation Systems with Scheduling Delays and Large Cost Differentials

The optimal scheduling or unit commitment of power generation systems to meet a random demand involves the solution of a class of dynamic programming inequalities for the optimal cost and control law. We study the behavior of this optimality system in terms of two parameters: (i) a scheduling delay, e.g., the startup time of a generation unit; and (ii) the relative magnitudes of the costs (operating or starting) of different units. In the first case we show that under reasonable assumptions the optimality system has a solution for all values of the delay, and, as the delay approaches zero, that the solutions converge uniformly to those of the corresponding system with no delays. In the second case we show that as the cost of operating or starting a given machine increases relative to the costs of the other machines, there is a point beyond which the expensive machine is not used, except in extreme situations. We give a formula for the relative costs that characterize this point. Moreover, we show that as the relative cost of the expensive machine goes to infinity the optimal cost of the system including the expensive machine approaches the optimal cost of the system without the machine

Singular Ergodic Control for Multidimensional Gaussian-Poisson Processes

Singular control for multidimensional Gaussian-Poisson processes with a long-run (or ergodic) and a discounted criteria are discussed. The dynamic programming yields the corresponding Hamilton-Jacobi-Bellman equations, which are discussed. Full details on the proofs and further extensions are left for coming works

On Some Optimal Stopping Problems with Constraint

We consider the optimal stopping problem of a Markov process {xt : t ≤ 0} when the controller is allowed to stop only at the arrival times of a signal, that is, at a sequence of instants {τn : n ≤ 1} independent of {xt : t ≤ 0}. We solve in detail this problem for general Markov–Feller processes with compact state space when the interarrival times of the signal are independent identically distributed random variables. In addition, we discuss several extensions to other signals and to other cases of state spaces. These results generalize the works of several authors where {xt : t ≤ 0} was a diffusion process and where the signal arrives at the jump times of a Poisson process

On Some Ergodic Impulse Control Problems with Constraint

This paper studies the impulse control of a general Markov process under the average (or ergodic) cost when the impulse instants are restricted to be the arrival times of an exogenous process, and this restriction is referred to as a constraint. A detailed setting is described, a characterization of the optimal cost is obtained as a solution of an HJB equation, and an optimal impulse control is identified

Impulse Control of Stochastic Navier-Stokes Equations

In this paper we study stopping time and impulse control problems for stochastic Navier-Stokes equation. Exploiting a local monotonicity property of the nonlinearity, we establish existence and uniqueness of strong solutions in two dimensions which gives a Markov-Feller process. The variational inequality associated with the stopping time problem and the quasi-variational inequality associated with the impulse control problem are resolved in a weak sense, using semigroup approach with a convergence uniform over path

Stochastic 2-D Navier-Stokes Equation

In this paper we prove the existence and uniqueness of strong solutions for the stochastic Navier-Stokes equation in bounded and unbounded domains. These solutions are stochastic analogs of the classical Lions-Prodi solutions to the deterministic Navier-Stokes equation. Local monotonicity of the nonlinearity is exploited to obtain the solutions in a given probability space and this signi cantly improves the earlier techniques for obtaining strong solutions, which depended on pathwise solutions to the Navier-Stokes martingale problem where the probability space is also obtained as a part of the solution