In this paper, we propose a convex programming based method to address a long-standing problem of inner-approximating backward reachable sets of state-constrained polynomial systems subject to time-varying uncertainties. The backward reachable set is a set of states, from which all trajectories starting will surely enter a target region at the end of a given time horizon without violating a set of state constraints in spite of the actions of uncertainties. It is equal to the zero sublevel set of the unique Lipschitz viscosity solution to a Hamilton–Jacobi partial differential equation (HJE). We show that inner approximations of the backward reachable set can be formed by zero sublevel sets of its viscosity supersolutions. Consequently, we reduce the inner-approximation problem to a problem of synthesizing polynomial viscosity supersolutions to this HJE. Such a polynomial solution in our method is synthesized by solving a single semidefinite program. We also prove that polynomial solutions to the formulated semidefinite program exist and can produce a convergent sequence of inner approximations to the interior of the backward reachable set in measure under appropriate assumptions. This is the main contribution of this paper. Several illustrative examples demonstrate the merits of our approach.

中文翻译：

---

具有时变不确定性的多项式系统的内逼近可达集

在本文中，我们提出了一种基于凸规划的方法来解决受时变不确定性影响的状态约束多项式系统的内逼近向后可达集的长期存在的问题。后向可达集是一组状态，从该状态开始的所有轨迹肯定会在给定的时间范围结束时进入目标区域，而不会违反一组状态约束，尽管存在不确定性的动作。它等于 Hamilton-Jacobi 偏微分方程 (HJE) 的唯一 Lipschitz 粘度解的零子水平集。我们表明后向可达集的内部近似可以由其粘度超解的零子级集形成。最后，我们将内逼近问题简化为合成此 HJE 的多项式粘度超解的问题。我们方法中的这种多项式解是通过求解单个半定程序来合成的。我们还证明了公式化的半定程序的多项式解是存在的，并且可以在适当的假设下产生向后可达集内部的收敛序列。这是本文的主要贡献。几个说明性的例子证明了我们方法的优点。我们还证明了公式化的半定程序的多项式解是存在的，并且可以在适当的假设下产生向后可达集内部的收敛序列。这是本文的主要贡献。几个说明性的例子证明了我们方法的优点。我们还证明了公式化的半定程序的多项式解是存在的，并且可以在适当的假设下产生向后可达集内部的收敛序列。这是本文的主要贡献。几个说明性的例子证明了我们方法的优点。