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 LaSalles 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 Lure, and quasi-polynomial systems. Quadratic and convex-sum-of-square Lyapunov functions are constructed for the Lure-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.
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