  The two main contributions of this paper are a proof of concept of the recent
novel idea in the area of long-time average cost control, and a new method of
overcoming the well-known difficulty of non-convexity of simultaneous
optimization of a control law and an additional tunable function. A
recently-proposed method of obtaining rigorous bounds of long-time average cost
is first outlined for the uncontrolled system with polynomials of system state
on the right-hand side. In this method the polynomial constraints are relaxed
to be sum-of-squares and formulated as semi-definite programs. It was proposed
to use the upper bound of long-time average cost as the objective function
instead of the time-average cost itself in controller design. In the present
paper this suggestion is implemented for a particular system and is shown to
give good results. Designing the optimal controller by this method requires
optimising simultaneously both the control law and a tunable function similar
to the Lyapunov function. The new approach proposed and implemented in this
paper for overcoming the inherent non-convexity of this optimisation is based
on a formal assumption that the amplitude of control is small. By expanding the
tunable function and the bound in the small parameter, the long-time average
cost is reduced by minimizing the respective bound in each term of the series.
The derivation of all the polynomial coefficients in controller is given in
terms of the solvability conditions of state-dependent linear and bilinear
inequalities. The resultant sum-of-squares problems are solved in sequence,
thus avoiding the non-convexity in optimization. The proposed approach is
implemented for a simple model of oscillatory vortex shedding behind a
cylinder.
