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Beginning with a Lagrangian, we derived an approximate relativistic
orbit equation which describes relativistic corrections to Keplerian
orbits. The critical angular moment to guarantee the existence of periodic
orbits is determined. An approximate relativistic Kepler’s elliptic orbit is
illustrated by numerical simulation via a second-order perturbation method of averaging.
This paper considers the optimal control problem for time-delay bilinear systems affected by sinusoidal disturbances with known frequency and measurable amplitude and phase. Firstly, using the differential homeomorphism, a time-delay bilinear system affected by sinusoidal disturbances is changed to a time-delay pseudo linear system through the coordinate transformation. Then the system with time-delay in control variable is transformed to a linear controllable system without delay using model transformation. At last based on the theory of linear quadratic optimal control, an optimal control law which is used to eliminate the influence of the disturbances is derived from a Riccati equation and Matrix equations. The simulation results show the effectiveness of the method.