Linearly Solvable Stochastic Control Lyapunov Functions
Abstract
This paper presents a new method for synthesizing stochastic control Lyapunov functions for a class of nonlinear stochastic control systems. The technique relies on a transformation of the classical nonlinear Hamilton--Jacobi--Bellman partial differential equation to a linear partial differential equation for a class of problems with a particular constraint on the stochastic forcing. This linear partial differential equation can then be relaxed to a linear differential inclusion, allowing for relaxed solutions to be generated using sum of squares programming. The resulting relaxed solutions are in fact viscosity super-/subsolutions, and by the maximum principle are pointwise upper and lower bounds to the underlying value function, even for coarse polynomial approximations. Furthermore, the pointwise upper bound is shown to be a stochastic control Lyapunov function, yielding a method for generating nonlinear controllers with pointwise bounded distance from the optimal cost when using the optimal controller. These approximate solutions may be computed with nonincreasing error via a hierarchy of semidefinite optimization problems. Finally, this paper develops a priori bounds on trajectory suboptimality when using these approximate value functions and demonstrates that these methods, and bounds, can be applied to a more general class of nonlinear systems not obeying the constraint on stochastic forcing. Simulated examples illustrate the methodology.
Additional Information
© 2016 Society for Industrial and Applied Mathematics. Received by the editors February 1, 2016; accepted for publication (in revised form) August 31, 2016; published electronically December 6, 2016. A short version of this work appeared as [24].Attached Files
Published - 16m105767x.pdf
Submitted - 1410.0405v3.pdf
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Additional details
- Eprint ID
- 74003
- Resolver ID
- CaltechAUTHORS:20170203-081725763
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2017-02-03Created from EPrint's datestamp field
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2021-11-11Created from EPrint's last_modified field