Covariance Steering for Nonlinear Control-affine Systems

Abstract

We consider the covariance steering problem for nonlinear control-affine systems. Our objective is to find an optimal control strategy to steer the state of a system from an initial distribution to a target one whose mean and covariance are given. Due to the nonlinearity, the existing techniques for linear covariance steering problems are not directly applicable. By leveraging the celebrated Girsanov theorem [1], we formulate the problem into an optimization over the space path distributions. We then adopt a generalized proximal gradient algorithm to solve this optimization, where each update requires solving a linear covariance steering problem. Our algorithm is guaranteed to converge to a local optimal solution with a sublinear rate. In addition, each iteration of the algorithm can be achieved in closed form, and thus the computational complexity of it is insensitive to the resolution of time-discretization.

Maximum entropy optimal density control of discrete-time linear systems and Schrödinger bridges

“…A substantial literature is devoted to reveal the relationships between stochastic optimal control and Schrödinger bridges in a continuous-time setting [12]- [17]. For the optimal density control of nonlinear systems and non-Gaussian densities, see e.g., [18]- [20]. Also for the discrete time and space setting, the counterpart of the density control and Schrödinger bridge problems is explored in [21].…”