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The paper presents a nonlinear state-feedback control design approach forlong-time average cost control, where the control effort is assumed to beexpensive. The approach is based on sum-of-squares and semi-definiteprogramming techniques. It is applicable to dynamical systems whose right-handside is a polynomial function in the state variables and the controls. The keyidea, first described but not implemented in (Chernyshenko et al., Phil. Trans.R. Soc. A, 372, 2014), is that the difficult problem of optimizing a costfunction involving long-time averages is replaced by an optimization of theupper bound of the same average. As such, controller design requires thesimultaneous optimization of both the control law and a tunable function,similar to a Lyapunov function. The present paper introduces a method resolvingthe well-known inherent non-convexity of this kind of optimization. The methodis based on the formal assumption that the control is expensive, from which itfollows that the optimal control is small. The resulting asymptoticoptimization problems are convex. The derivation of all the polynomialcoefficients in the controller is given in terms of the solvability conditionsof state-dependent linear and bilinear inequalities. The proposed approach isapplied to the problem of designing a full-information feedback controller thatmitigates vortex shedding in the wake of a circular cylinder in the laminarregime via rotary oscillations. Control results on a reduced-order model of theactuated wake and in direct numerical simulation are reported.

Expensive control of long-time averages using sum of squares and its application to a laminar wake flow