A Fixed-Point Characterization of the Optimal Costate in Finite-Horizon Optimal Control Problems

• 成果类型: Article
• 署名作者: Sassano, Mario; Astolfi, Alessandro
• 署名单位: University of Rome Tor Vergata; Imperial College London
• 刊物名称: IEEE TRANSACTIONS ON AUTOMATIC CONTROL
• ISSN/ISSBN: 0018-9286
• DOI: 10.1109/TAC.2020.3021403
• 发表日期: 2021
• 页码: 3562-3574
• 关键词: Optimal control; Symmetric matrices; Task analysis; Partial differential equations; CONVERGENCE; tools; ELECTRONIC MAIL; Hamiltonian dynamics; nonlinear control systems; optimal control

摘要

A fixed-point characterization of the optimal costate in finite-horizon optimal control problems for nonlinear systems is presented. It is shown that the optimal initial condition of the costate variable must be a fixed-point, for any time, of the composition of the forward and backward flows of the underlying Hamiltonian dynamics. Such an abstract property is then translated into a constructive condition by relying on a sequence of repeated Lie brackets involving the Hamiltonian dynamics and evaluated at a single point in the state space. This leads to a system of algebraic equations in the unknown initial optimal costate that allows achieving a desired degree of accuracy of the approximation while always consisting of a number of equations equal to the dimension of the state of the underlying system, regardless of the achieved accuracy. A dual characterization of the optimal terminal value of the state is also discussed, together with a few computational aspects of the proposed strategy.