Optimality-Induced Stabilization in Constrained Networked Optimization

Stability of networked stochastic systems is traditionally achieved through explicitly engineered control policies that enforce negative Lyapunov drift. In contrast, many operational infrastructures are governed by economic optimization objectives rather than stabilizing design constraints. This paper identifies a structural mechanism under which stability emerges as a necessary consequence of optimality in constrained dynamic optimization. We consider controlled Markov systems with convex capacity regions induced by shared resource constraints and stage costs composed of a convex, coercive congestion potential and a control cost. We prove that if the exogenous load vector lies strictly in the interior of the capacity region, then every optimal stationary policy for the discounted problem satisfies a Foster–Lyapunov drift inequality outside a compact set. Consequently, the induced state process is positive recurrent. Stability is not imposed—it is structurally enforced by convex capacity geometry and coercive cost growth. Complementing this interior result, we establish a sharp capacity boundary theorem: if the load lies outside the convex capacity region, no admissible policy can render the system positive recurrent. The convex throughput polytope therefore characterizes the stabilizable load set exactly. A multi-terminal transportation network with shared polyhedral constraints illustrates the geometric phase transition predicted by the theory. The results reveal an intrinsic alignment between economic optimality and stochastic stabilization in convex networked systems.