The paper presents a nonlinear state-feedback con-
trol design approach for long-time average cost control, where the
control effort is assumed to be expensive. The approach is based
on sum-of-squares and semi-definite programming techniques. It
is applicable to dynamical systems whose right-hand side is a
polynomial function in the state variables and the controls. The
key idea, first described but not implemented in (Chernyshenko
et
al.
Phil. Trans. R. Soc. A, 372, 2014), is that the difficult problem
of optimizing a cost function involving long-time averages is
replaced by an optimization of the upper bound of the same
average. As such, controller design requires the simultaneous
optimization of both the control law and a tunable function,
similar to a Lyapunov function. The present paper introduces
a method resolving the well-known inherent non-convexity of
this kind of optimization. The method is based on the formal
assumption that the control is expensive, from which it follows
that the optimal control is small. The resulting asymptotic
optimization problems are convex. The derivation of all the
polynomial coefficients in the controller is given in terms of
the solvability conditions of state-dependent linear and bilinear
inequalities. The proposed approach is applied to the problem
of designing a full-information feedback controller that mitigates
vortex shedding in the wake of a circular cylinder in the laminar
regime via rotary oscillations. Control results on a reduced-order
model of the actuated wake and in direct numerical simulation
are reported.