### Optimal Transport Over a Linear Dynamical System

Yongxin Chen, Tryphon T. Georgiou, Michele Pavon
2017 IEEE Transactions on Automatic Control
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 (ẋ(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
more » ... f a linear stochastic system with white noise disturbance; this is known to correspond to a Schrödinger 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. When the state distribution represents density of particles whose position obeysẋ(t) = u(t) (i.e., A(t) ≡ 0, B(t) ≡ I, and = 0) the problem reduces to the classical Optimal Mass Transport (OMT) problem 2 with quadratic cost [3], [7]. Thus, the above problem, for = 0, represents a generalization of OMT to deal with particles obeying known "prior" non-trivial dynamics while being steered between two end-point distributions -we refer to this as the problem of OMT with prior dynamics (OMT-wpd). The problem of OMT-wpd was first introduced in our previous work [10] for the case where B(t) ≡ I. The difference of course to the classical OMT is that, here, the linear dynamics are arbitrary and may facilitate or hinder transport. Applications are envisioned in the steering of particle beams through timevarying potential, the steering of swarms (UAV's, large collection of microsatelites, etc.), as well as in the modeling of the flow and collective motion of particles, clouds, platoons, flocking of insects, birds, fish, etc. between end-point distributions [11] , and the interpolation/morphing of distributions [12] .