Stochastic Reachability with Constraints
This chapter is concerned with leveraging the concept of stochastic reachability such that it may capture state and time constraints, dynamics of the target set, randomisation of the time horizon and so on. We formulate state-constrained stochastic reachability for a stochastic hybrid process with no controllers. The idea is that we specialise the stochastic reachability problem asking that the desired trajectories to satisfy additional conditions (e.g. to avoid an obstacle or to go through a particular set). Analytic solutions are provided using the infinitesimal generator of the process. Concretely, the reach set probabilities in the case of state-constrained reachability are solutions for a Dirichlet boundary value problem defined using this generator. The complexity of this problem is due also to the particular expression of this integro-differential operator. To overcome this difficulty, we propose to consider a viewpoint approach and to solve the problem for different abstraction levels or for different components, which can be defined in the structure of a stochastic hybrid system.
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