Lagrangian Adaptation

The stochastic Lagrangian adaptation scheme proposed in this work is derived: First, a time-varying constrained optimization problem is formulated. Given that the resulting necessary conditions may lead to a triviality, these conditions are replaced by associated conditions where the control cost gradient is replaced with its projection onto the tangent space of parameters giving rise to indistinguishable closed-loop dynamics. Next, the SDEs of consistent approximations to the solutions of the constrained optimization problem are derived and the existence of unique strong solutions established. Finally, these solutions are shown to make the generated AML estimates strongly consistent from which optimal long-run LQ performance follows. This chapter concludes with the computation of the Regret Rate of the proposed scheme.