  This paper proposes a semidefinite programming based method for estimating
moments of a stochastic hybrid system (SHS). For polynomial SHSs -- which
consist of polynomial continuous vector fields, reset maps, and transition
intensities -- the dynamics of moments evolve according to a system of linear
ordinary differential equations. However, it is generally not possible to solve
the system exactly since time evolution of a specific moment may depend upon
moments of order higher than it. One way to overcome this problem is to employ
so-called moment closure methods that give point approximations to moments, but
these are limited in that accuracy of the estimations is unknown. We find lower
and upper bounds on a moment of interest via a semidefinite program that
includes linear constraints obtained from moment dynamics, along with
semidefinite constraints that arise from the non-negativity of moment matrices.
These bounds are further shown to improve as the size of semidefinite program
is increased. The key insight in the method is a reduction from stochastic
hybrid systems with multiple discrete modes to a single-mode hybrid system with
algebraic constraints. We further extend the scope of the proposed method to a
class of non-polynomial SHSs which can be recast to polynomial SHSs via
augmentation of additional states. Finally, we illustrate the applicability of
results via examples of SHSs drawn from different disciplines.

Optimization and Control