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The two main contributions of this paper are a proof of concept of the recentnovel idea in the area of long-time average cost control, and a new method ofovercoming the well-known difficulty of non-convexity of simultaneousoptimization of a control law and an additional tunable function. Arecently-proposed method of obtaining rigorous bounds of long-time average costis first outlined for the uncontrolled system with polynomials of system stateon the right-hand side. In this method the polynomial constraints are relaxedto be sum-of-squares and formulated as semi-definite programs. It was proposedto use the upper bound of long-time average cost as the objective functioninstead of the time-average cost itself in controller design. In the presentpaper this suggestion is implemented for a particular system and is shown togive good results. Designing the optimal controller by this method requiresoptimising simultaneously both the control law and a tunable function similarto the Lyapunov function. The new approach proposed and implemented in thispaper for overcoming the inherent non-convexity of this optimisation is basedon a formal assumption that the amplitude of control is small. By expanding thetunable function and the bound in the small parameter, the long-time averagecost is reduced by minimizing the respective bound in each term of the series.The derivation of all the polynomial coefficients in controller is given interms of the solvability conditions of state-dependent linear and bilinearinequalities. The resultant sum-of-squares problems are solved in sequence,thus avoiding the non-convexity in optimization. The proposed approach isimplemented for a simple model of oscillatory vortex shedding behind acylinder.

Long-time Average Cost Control of Polynomial Systems: A Sum-of-squares-based Small-feedback Approach