  This paper presents a novel scalable framework to solve the optimization of a
nonlinear system with differential algebraic equation (DAE) constraints that
enforce the asymptotic stability of the underlying dynamic model with respect
to certain disturbances. Existing solution approaches to analogous
DAE-constrained problems are based on discretization of DAE system into a large
set of nonlinear algebraic equations representing the time-marching schemes.
These approaches are not scalable to large size models. The proposed framework,
based on LaSalle's invariance principle, uses convex Lyapunov functions to
develop a novel stability certificate which consists of a limited number of
algebraic constraints. We develop specific algorithms for two major types of
nonlinearities, namely Lur'e, and quasi-polynomial systems. Quadratic and
convex-sum-of-square Lyapunov functions are constructed for the Lur'e-type and
quasi-polynomial systems respectively. A numerical experiment is performed on a
3-generator power network to obtain a solution for
transient-stability-constrained optimal power flow.
