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A boundary control problem for a Kirchhoff plate equation with a structural damping is considered. A distinctive feature- of this problem is the lack of strong coercivity with respect to control variable. It is shown that the optimal control admits a pointwise state feedback synthesis via a solution of nonstandard Riccati equation. The novelty of the problem with respect to the literature is that both: the associated Riccati equation and the feedback control operator, are nonstandard.