Relaxing The Hamilton Jacobi Bellman Equation To Construct Inner And Outer Bounds On Reachable Sets
Relaxing The Hamilton Jacobi Bellman Equation To Construct Inner And Outer Bounds On Reachable Sets
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DOI：
10.1109/cdc40024.2019.9029193
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发表时间：
2019-03
期刊：
影响因子：
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通讯作者：
Morgan Jones;M. Peet
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文献类型：
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作者：
Morgan Jones;M. Peet
我们考虑的问题上界和下界的向后和向前可达集为一个给定的多项式向量场，非线性的状态和输入，与一个给定的半代数集的初始条件和输入约束逐点躺在一个半代数集。具体来说，我们表示的前向可达集使用的“价值函数”，它给出了最优控制问题的最优成本去，如果光滑满足Hamilton-Jacobi-Bellman偏微分方程。然后，我们表明，存在多项式的上限和下限，这个值函数，而且，这些多项式的“子值”和“超值”功能提供可证明的上限和下限的前向可达集。最后，通过最小化这些“子值”和“超值”之间的距离在L1-范数的功能，我们能够构造的内部和外部边界的可达集和数值上显示的几个例子，对于相对较小的程度，这些边界之间的Hausdorff距离是可以忽略的。
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.