3,190 research outputs found
Unbounded Viscosity Solutions of Hybrid Control Systems
We study a hybrid control system in which both discrete and continuous
controls are involved. The discrete controls act on the system at a given set
interface. The state of the system is changed discontinuously when the
trajectory hits predefined sets, namely, an autonomous jump set or a
controlled jump set where controller can choose to jump or not. At each
jump, trajectory can move to a different Euclidean space. We allow the cost
functionals to be unbounded with certain growth and hence the corresponding
value function can be unbounded. We characterize the value function as the
unique viscosity solution of the associated quasivariational inequality in a
suitable function class. We also consider the evolutionary, finite horizon
hybrid control problem with similar model and prove that the value function is
the unique viscosity solution in the continuous function class while allowing
cost functionals as well as the dynamics to be unbounded
On Singular Control Problems with State Constraints and Regime-Switching: A Viscosity Solution Approach
This paper investigates a singular stochastic control problem for a
multi-dimensional regime-switching diffusion process confined in an unbounded
domain. The objective is to maximize the total expected discounted rewards from
exerting the singular control. Such a formulation stems from application areas
such as optimal harvesting multiple species and optimal dividends payments
schemes in random environments. With the aid of weak dynamic programming
principle and an exponential transformation, we characterize the value function
to be the unique constrained viscosity solution of a certain system of coupled
nonlinear quasi-variational inequalities. Several examples are analyzed in
details to demonstrate the main results
Controlled diffusion processes
This article gives an overview of the developments in controlled diffusion
processes, emphasizing key results regarding existence of optimal controls and
their characterization via dynamic programming for a variety of cost criteria
and structural assumptions. Stochastic maximum principle and control under
partial observations (equivalently, control of nonlinear filters) are also
discussed. Several other related topics are briefly sketched.Comment: Published at http://dx.doi.org/10.1214/154957805100000131 in the
Probability Surveys (http://www.i-journals.org/ps/) by the Institute of
Mathematical Statistics (http://www.imstat.org
A Piecewise Deterministic Markov Toy Model for Traffic/Maintenance and Associated Hamilton-Jacobi Integrodifferential Systems on Networks
We study optimal control problems in infinite horizon when the dynamics
belong to a specific class of piecewise deterministic Markov processes
constrained to star-shaped networks (inspired by traffic models). We adapt the
results in [H. M. Soner. Optimal control with state-space constraint. II. SIAM
J. Control Optim., 24(6):1110.1122, 1986] to prove the regularity of the value
function and the dynamic programming principle. Extending the networks and
Krylov's ''shaking the coefficients'' method, we prove that the value function
can be seen as the solution to a linearized optimization problem set on a
convenient set of probability measures. The approach relies entirely on
viscosity arguments. As a by-product, the dual formulation guarantees that the
value function is the pointwise supremum over regular subsolutions of the
associated Hamilton-Jacobi integrodifferential system. This ensures that the
value function satisfies Perron's preconization for the (unique) candidate to
viscosity solution. Finally, we prove that the same kind of linearization can
be obtained by combining linearization for classical (unconstrained) problems
and cost penalization. The latter method works for very general near-viable
systems (possibly without further controllability) and discontinuous costs.Comment: accepted to Applied Mathematics and Optimization (01/10/2015
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