We study the well-known minimum-energy control of the double integrator, along with the simultaneous minimization of the total variation in the control variable. We derive the optimality conditions and obtain the unique optimal solution to the combined problem, where the initial and terminal boundary points are specified. We study the problem from a multi-objective optimal control viewpoint, constructing the Pareto front. We show that the unique asymptotic optimal control function, for the minimization of the total variation alone, is piecewise constant with one switching at the midpoint of the time horizon. For any instance of the boundary conditions of the problem, we prove that the asymptotic optimal total variation is exactly 2/3 of the total variation of the minimum-energy control. We illustrate the results for a particular instance of the problem and include a link to a video which animates the solutions while moving along the Pareto front.

中文翻译：

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具有最小总变差的双积分器的优化控制

我们研究了众所周知的双积分器的最小能量控制，以及控制变量中总变化的同时最小化。我们推导出最优条件并获得组合问题的唯一最优解，其中指定了初始边界点和终止边界点。我们从多目标最优控制的角度研究问题，构建帕累托前沿。我们表明，独特的渐近最优控制函数，仅用于最小化总变化，是分段常数，在时间范围的中点进行一次切换。对于问题边界条件的任何实例，我们证明渐近最优总变差恰好是最小能量控制总变差的 2/3。