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This paper addresses the optimal controller problem for a polynomial system over linear observations with respect to different Bolza-Meyer criteria, where: 1) the integral control and state energy terms are quadratic and the nonintegral term is of the first degree or 2) the control energy term is quadratic and the state energy terms are of the first degree. The optimal solutions are obtained as sliding mode controllers, each consisting of a sliding mode filter and a sliding mode regulator, whereas the conventional feedback polynomial-quadratic controller fails to provide a causal solution. Performance of the obtained optimal controllers is verified in the illustrative example against the conventional LQG controller that is optimal for the quadratic Bolza-Meyer criterion. The simulation results confirm an advantage in favor of the designed sliding mode controllers.