This paper studies the problem of enforcing safety of a stochastic dynamical
system over a finite-time horizon. We use stochastic control barrier functions
as a means to quantify the probability that a system exits a given safe region
of the state space in finite time. A barrier certificate condition that bounds
the expected value of the barrier function over the time horizon is recast as a
sum-of-squares optimization problem for efficient numerical computation. Unlike
prior works, the proposed certificate condition includes a state-dependent
upper bound on the evolution of the expectation. We present formulations for
both continuous-time and discrete-time systems. Moreover, for systems for which
the drift dynamics are affine-in-control, we propose a method for synthesizing
polynomial state feedback controllers that achieve a specified probability of
safety. Several case studies are presented which benchmark and illustrate the
performance of our verification and control method in the continuous-time and
discrete-time domains.

研究 stochastic dynamical system 在 finite-time horizon 下的安全性问题，提出基于 stochastic control barrier functions 的 barrier certificate condition 以量化系统在有限时间内退出给定的安全区域的概率，并将该条件表述为 sum-of-squares 优化问题进行高效的数值计算，在保证 expected value 的发展的状态相关的上限的前提下，提供了连续时间和离散时间系统的解决方案，针对控制的仿射动力学系统，提出了合成多项式状态反馈控制器以实现指定的安全概率，使用实例研究连续时间和离散时间领域中验证和控制方法的性能。