  In this paper, we present passivity based convergence analysis of continuous
time primal-dual gradient method for convex optimization problems. We first
show that a convex optimization problem with only affine equality constraints
admit a Brayton Moser formulation. This observation leads to a new passivity
property derived from a Krasovskii type storage function. Secondly, the
inequality constraints are modeled as a state dependent switching system. Using
hybrid methods, it is shown that each switching mode is passive and the
passivity of the system is preserved under arbitrary switching. Finally, the
two systems, (i) one derived from the Brayton Moser formulation and (ii) the
state dependent switching system, are interconnected in a power conserving way.
The resulting trajectories of the overall system are shown to converge
asymptotically, to the optimal solution of the convex optimization problem. The
proposed methodology is applied to an energy management problem in buildings
and simulations are provided for corroboration.
