This paper is concerned with the data-driven optimal control of nonlinear systems. We present a convex formulation of the optimal control problem with a discounted cost function. We consider optimal control problems with both positive and negative discount factors. The convex approach relies on lifting nonlinear system dynamics in the space of densities using the linear Perron–Frobenius operator. This lifting leads to an infinite-dimensional convex optimization formulation of the optimal control problem. The data-driven approximation of the optimization problem relies on the approximation of the Koopman operator and its dual: the Perron–Frobenius operator, using a polynomial basis function. We write the approximate finite-dimensional optimization problem as a polynomial optimization which is then solved efficiently using a sum-of-squares-based optimization framework. Simulation results demonstrate the efficacy of the developed data-driven optimal control framework.

本文关注非线性系统的数据驱动优化控制。我们提出了具有折扣成本函数的最优控制问题的凸公式。我们考虑具有正贴现因子和负贴现因子的最优控制问题。凸方法依赖于使用线性 Perron-Frobenius 算子提升密度空间中的非线性系统动力学。这种提升导致最优控制问题的无限维凸优化公式。优化问题的数据驱动逼近依赖于 Koopman 算子及其对偶的逼近：Perron–Frobenius 算子，使用多项式基函数。我们将近似有限维优化问题写成多项式优化，然后使用基于平方和的优化框架有效求解。仿真结果证明了所开发的数据驱动最优控制框架的有效性。