Optimal parametric excitation in a resonant circuit

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An L-C resonant circuit with parametric excitation is dealt with when the circuit corresponds to the differential equation of the form : ẍ＋[ω²＋u(t)]x=0 where function u(t) is controllable under the condition u min≤u(t)≤u max. The objective is to find the optimal trajectories so that an initial phase point moves toward a terminal in a minimum time. The terminal which is used here is a circle with its center at the origin. Using both maximum principle of Pontryagin and transversality condition, synthesis is made on the phase plane for outside and inside of the circle. According to the results obtained herein the control function u(t) takes either minimum value or maximum value. From the synthesis on the phase plane it is known that the function u(t) changes its value four times when the argument on the phase plane increases by 2π. In other words it is most effective to induce oscillation of 1/2 subharmonic type in order to make the phase point move toward a circle in a minimum time.

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