Application of Hamilton–Jacobi–Bellman Equation/Pontryagin’s Principle for Constrained Optimal Control
本文用Valentine变换将带约束的最优控制问题转为无约束问题，证明值函数唯一性，并基于庞特里亚金原理开发求解器，在无人机和自主水下航行器的三维避障路径规划中验证了方法。
Abstract This article applies novel results for infinite- and finite-horizon optimal control problems with nonlinear dynamics and constraints. We use the Valentine transformation to convert a constrained optimal control problem into an unconstrained one and show uniqueness of the value function to the corresponding Hamilton–Jacobi–Bellman (HJB) equation. From there, we show how to approximate the solution of the initial (in)finite-horizon problem with a family of solutions that is $$\varGamma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Γ</mml:mi></mml:math> -convergent. Optimal solutions are efficiently obtained via a solver based on Pontryagin’s Principle (PP). The proposed methodology is demonstrated on the path planning problem using the full nonlinear dynamics of an unmanned aerial vehicle (UAV) and autonomous underwater vehicle (AUV) involving state constraints in 3D environments with obstacles.