Cone-bounded nonlinearities and mean-square bounds--Quadratic regulation bounds

Deterministic and stochastic control problems are examined for a broad class of nonlinear dynamic systems with quadratic cost criteria. The systems considered are partially observed continuous-time stochastic process, and are incrementally conic in the sense that, when modeled by Itô differential equations, they contain drift coefficients that are jointly linear in state and control to within a uniformly Lipschitz residual. For deterministic problems, upper bounds are derived on the cost incurred by linear state-feedback control laws, and lower bounds are established on the performance attainable by any control law. For stochastic problems, upper bounds are found on the performance of simple, separated estimator-controller designs involving linear feedback of an easily-implemented, suboptimum estimate of the state, while a lower bound is determined on the performance attainable by any causal control law in a broad admissible class. The bounds share similar structural properties and computational requirements with the exact solution to linear regulator problems. The stochastic bounds owe much of their simplicity to the use of the control-law independent bounds on estimation performance derived in earlier companion papers. All the bounds coverge with vanishing nonlinearity (vanishing Lipschitz constants) to the optimum performance for the nominal linear system. Consequently, the bounds are asymptotically tight and the simple designs studied are asymptotically optimal with vanishing nonlinearity.