TY - GEN
T1 - Relaxing the Hamilton Jacobi Bellman Equation to Construct Inner and Outer Bounds on Reachable Sets
AU - Jones, Morgan
AU - Peet, Matthew M.
N1 - Funding Information:
ACKNOWLEDGEMENTS This work was supported by the National Science Foundation under grants No. 1538374 and 1739990.
Publisher Copyright:
© 2019 IEEE.
PY - 2019/12
Y1 - 2019/12
N2 - We consider the problem of overbounding and underbounding both the backward and forward reachable set for a given polynomial vector field, nonlinear in both state and input, with a given semialgebriac set of initial conditions and with inputs constrained pointwise to lie in a semialgebraic set. Specifically, we represent the forward reachable set using the value function which gives the optimal cost to go of an optimal control problems and if smooth satisfies the Hamilton-JacobiBellman PDE. We then show that there exist polynomial upper and lower bounds to this value function and furthermore, these polynomial sub-value and super-value functions provide provable upper and lower bounds to the forward reachable set. Finally, by minimizing the distance between these sub-value and super-value functions in the L1-norm, we are able to construct inner and outer bounds for the reachable set and show numerically on several examples that for relatively small degree, the Hausdorff distance between these bounds is negligible.
AB - We consider the problem of overbounding and underbounding both the backward and forward reachable set for a given polynomial vector field, nonlinear in both state and input, with a given semialgebriac set of initial conditions and with inputs constrained pointwise to lie in a semialgebraic set. Specifically, we represent the forward reachable set using the value function which gives the optimal cost to go of an optimal control problems and if smooth satisfies the Hamilton-JacobiBellman PDE. We then show that there exist polynomial upper and lower bounds to this value function and furthermore, these polynomial sub-value and super-value functions provide provable upper and lower bounds to the forward reachable set. Finally, by minimizing the distance between these sub-value and super-value functions in the L1-norm, we are able to construct inner and outer bounds for the reachable set and show numerically on several examples that for relatively small degree, the Hausdorff distance between these bounds is negligible.
UR - http://www.scopus.com/inward/record.url?scp=85082444944&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85082444944&partnerID=8YFLogxK
U2 - 10.1109/CDC40024.2019.9029193
DO - 10.1109/CDC40024.2019.9029193
M3 - Conference contribution
AN - SCOPUS:85082444944
T3 - Proceedings of the IEEE Conference on Decision and Control
SP - 2397
EP - 2404
BT - 2019 IEEE 58th Conference on Decision and Control, CDC 2019
PB - Institute of Electrical and Electronics Engineers Inc.
T2 - 58th IEEE Conference on Decision and Control, CDC 2019
Y2 - 11 December 2019 through 13 December 2019
ER -