Results 1 
3 of
3
Linear hamilton jacobi bellman equations in high dimensions
 in Conference on Decision and Control (CDC), 2014, arXiv preprint arXiv:1404.1089
"... provides the globally optimal solution to large classes of control problems. Unfortunately, this generality comes at a price, the calculation of such solutions is typically intractible for systems with more than moderate state space size due to the curse of dimensionality. This work combines recent ..."
Abstract

Cited by 5 (3 self)
 Add to MetaCart
(Show Context)
provides the globally optimal solution to large classes of control problems. Unfortunately, this generality comes at a price, the calculation of such solutions is typically intractible for systems with more than moderate state space size due to the curse of dimensionality. This work combines recent results in the structure of the HJB, and its reduction to a linear Partial Differential Equation (PDE), with methods based on low rank tensor representations, known as a separated representations, to address the curse of dimensionality. The result is an algorithm to solve optimal control problems which scales linearly with the number of states in a system, and is applicable to systems that are nonlinear with stochastic forcing in finitehorizon, average cost, and firstexit settings. The method is demonstrated on inverted pendulum, VTOL aircraft, and quadcopter models, with system dimension two, six, and twelve respectively. I.
Optimal Controller Synthesis for Nonlinear Dynamical Systems
"... Abstract — This work presents a novel method for synthesizing optimal Control Lyapunov functions for nonlinear, stochastic systems. The technique relies on solutions to the linear Hamilton Jacobi Bellman (HJB) equation, a transformation of the classical nonlinear HJB partial differential equation ..."
Abstract
 Add to MetaCart
(Show Context)
Abstract — This work presents a novel method for synthesizing optimal Control Lyapunov functions for nonlinear, stochastic systems. The technique relies on solutions to the linear Hamilton Jacobi Bellman (HJB) equation, a transformation of the classical nonlinear HJB partial differential equation to a linear partial differential equation, possible when a particular structural constraint on the stochastic forcing on the system is satisfied. The linear partial differential equation is viewed as a set of constraints which are in turn relaxed to a linear differential inclusion. This allows for the optimization of a candidate polynomial solution using sum of squares programming. The resulting polynomials are in fact viscosity solutions of the HJB, allowing for the well developed results in the theory of viscosity solutions to be applied to these numerically generated solutions. It is shown how the viscosity solutions may be made arbitrarily close to the optimal solution via a hierarchy of semidefinite optimization problems. Furthermore, this work develops apriori bounds on trajectory suboptimality when using these approximate value functions. I.