Optimal Transport Over a Linear Dynamical System
Optimal Transport Over a Linear Dynamical System
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DOI：
10.1109/tac.2016.2602103
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发表时间：
2017-05-01
影响因子：
6.8
通讯作者：
Pavon, Michele
中科院分区：
文献类型：
--
作者：
Chen, Yongxin;Georgiou, Tryphon T.;Pavon, Michele
我们考虑使用最小能量控制在有限时间内将线性系统状态向量的初始概率密度转向最终概率密度的问题。在动力学对应于积分器（(x) over dot (t) = u(t)）的情况下，这相当于 Monge-Kantorovich 最优质量传递 (OMT) 问题。总的来说，我们表明该问题可以再次简化为解决 OMT 问题，并且它具有唯一的解决方案。同时，我们研究了具有白噪声扰动的线性随机系统的状态密度的最优控制；已知这对应于薛定谔电桥。当白噪声强度趋于零时，密度流收敛于确定性动力学的密度流，并且可以作为计算其确定性对应物的解的方法。在这两种情况下，解决方案都可以用高斯初始和最终状态密度的封闭形式表示。
We consider the problem of steering an initial probability density for the state vector of a linear system to a final one, in finite time, using minimum energy control. In the case where the dynamics correspond to an integrator ((x) over dot (t) = u(t)) this amounts to a Monge-Kantorovich Optimal Mass Transport (OMT) problem. In general, we show that the problem can again be reduced to solving an OMT problem and that it has a unique solution. In parallel, we study the optimal steering of the state-density of a linear stochastic system with white noise disturbance; this is known to correspond to a Schrodinger bridge. As the white noise intensity tends to zero, the flow of densities converges to that of the deterministic dynamics and can serve as a way to compute the solution of its deterministic counterpart. The solution can be expressed in closed-form for Gaussian initial and final state densities in both cases.