# Computing Probabilistic Controlled Invariant Sets

## TL;DR

This paper introduces probabilistic controlled invariant sets (PCISs) for stochastic control systems, providing algorithms for their computation in discrete and continuous spaces, with applications demonstrated through motion planning simulations.

## Contribution

It proposes finite- and infinite-horizon PCISs, explores their relation to robust invariant sets, and develops computational algorithms for practical control system applications.

## Key findings

- Algorithms for PCIS computation are computationally tractable.
- Finite-horizon PCISs converge with space discretization.
- Infinite-horizon PCISs relate to stochastic backward reachable sets.

## Abstract

This paper investigates stochastic invariance for control systems through probabilistic controlled invariant sets (PCISs). As a natural complement to robust controlled invariant sets~(RCISs), we propose finite- and infinite-horizon PCISs, and explore their relation to RICSs. We design iterative algorithms to compute the PCIS within a given set. For systems with discrete spaces, the computations of the finite- and infinite-horizon PCISs at each iteration are based on linear programming and mixed integer linear programming, respectively. The algorithms are computationally tractable and terminate in a finite number of steps. For systems with continuous spaces, we show how to discretize the spaces and prove the convergence of the approximation when computing the finite-horizon PCISs. In addition, it is shown that an infinite-horizon PCIS can be computed by the stochastic backward reachable set from the RCIS contained in it. These PCIS algorithms are applicable to practical control systems. Simulations are given to illustrate the effectiveness of the theoretical results for motion planning.