Fundamental Limits of Adaptive Stabilization with an Unknown Growth Exponent

A basic question in adaptive control is how rapidly a discrete-time nonlinear plant with unknown parameters may grow while remaining stabilizable. When the nonlinear growth exponent is known and only a scalar coefficient is unknown, existing theory identifies 4 as the exact critical exponent for stabilizability. We determine how this critical exponent changes when the growth exponent is also unknown. For any positive disturbance bound, one feedback law stabilizes all plants in some neighborhood of a nominal parameter pair if and only if the nominal exponent is below $3\sqrt{3}/2$. At equality, every compact parameter set whose exponents do not exceed this value remains stabilizable. If the exponent belongs to a known finite set, the critical exponent remains 4 for every nondegenerate compact coefficient interval. Hence finite exponent sets and arbitrarily short exponent intervals can have different critical exponents. Uncertainty in the growth rate therefore changes the range of growth that feedback can stabilize, not merely the size of the uncertainty set. The critical exponent remains $3\sqrt{3}/2$ under a known positive state-dependent multiplier bounded above and away from zero. For a system whose coefficient and exponent are known functions of an unknown parameter, a compact parameter family is stabilizable when its exponents do not exceed this value. At an interior parameter point where the Jacobian of these functions has rank two and the exponent is at least this value, no compact neighborhood is stabilizable.