  In this paper, we generalize the chance optimization problems and introduce
constrained volume optimization where enables us to obtain convex formulation
for challenging problems in systems and control. We show that many different
problems can be cast as a particular cases of this framework. In constrained
volume optimization, we aim at maximizing the volume of a semialgebraic set
under some semialgebraic constraints. Building on the theory of measures and
moments, a sequence of semidefinite programs are provided, whose sequence of
optimal values is shown to converge to the optimal value of the original
problem. We show that different problems in the area of systems and control
that are known to be nonconvex can be reformulated as special cases of this
framework. Particularly, in this work, we address the problems of probabilistic
control of uncertain systems as well as inner approximation of region of
attraction and invariant sets of polynomial systems. Numerical examples are
presented to illustrate the computational performance of the proposed approach.
