20,246 research outputs found
user's guide to viscosity solutions of second order partial differential equations
The notion of viscosity solutions of scalar fully nonlinear partial
differential equations of second order provides a framework in which startling
comparison and uniqueness theorems, existence theorems, and theorems about
continuous dependence may now be proved by very efficient and striking
arguments. The range of important applications of these results is enormous.
This article is a self-contained exposition of the basic theory of viscosity
solutions.Comment: 67 page
Nonexistence of positive supersolutions of elliptic equations via the maximum principle
We introduce a new method for proving the nonexistence of positive
supersolutions of elliptic inequalities in unbounded domains of .
The simplicity and robustness of our maximum principle-based argument provides
for its applicability to many elliptic inequalities and systems, including
quasilinear operators such as the -Laplacian, and nondivergence form fully
nonlinear operators such as Bellman-Isaacs operators. Our method gives new and
optimal results in terms of the nonlinear functions appearing in the
inequalities, and applies to inequalities holding in the whole space as well as
exterior domains and cone-like domains.Comment: revised version, 32 page
Semidefinite Relaxations for Stochastic Optimal Control Policies
Recent results in the study of the Hamilton Jacobi Bellman (HJB) equation
have led to the discovery of a formulation of the value function as a linear
Partial Differential Equation (PDE) for stochastic nonlinear systems with a
mild constraint on their disturbances. This has yielded promising directions
for research in the planning and control of nonlinear systems. This work
proposes a new method obtaining approximate solutions to these linear
stochastic optimal control (SOC) problems. A candidate polynomial with variable
coefficients is proposed as the solution to the SOC problem. A Sum of Squares
(SOS) relaxation is then taken to the partial differential constraints, leading
to a hierarchy of semidefinite relaxations with improving sub-optimality gap.
The resulting approximate solutions are shown to be guaranteed over- and
under-approximations for the optimal value function.Comment: Preprint. Accepted to American Controls Conference (ACC) 2014 in
Portland, Oregon. 7 pages, colo
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