A Highly Efficient Numerical Algorithm for Maximum Principle Based on Pseudospectral Methods

Pseudospectral methods represent a numerical approach that transforms continuous optimal control problems into nonlinear programming problems, particularly suitable for solving trajectory optimization problems satisfying the maximum principle. This maximum principle-based numerical algorithm employing pseudospectral methods discretizes both state and control variables at specific collocation points, achieving high-precision solutions through orthogonal polynomial approximations.

The algorithm's core advantage lies in its global interpolation strategy, which enables higher solution accuracy with fewer discrete nodes compared to traditional direct shooting or multiple shooting methods. Implementation typically requires integration with professional nonlinear programming solvers like SNOPT to handle the large-scale optimization problems resulting from discretization. The code structure generally involves defining collocation points using Legendre or Chebyshev polynomials, constructing the Jacobian matrix for constraint derivatives, and implementing boundary condition handling.

Notably, this algorithm demonstrates three key characteristics in engineering applications: first, strong capability in handling control variable inequality constraints through proper constraint formulation; second, fast convergence rates, particularly for smooth problems due to spectral accuracy; third, good adaptability to singular control problems through appropriate node distribution. Users must pre-install professional optimization solvers like SNOPT since the discretized nonlinear programming problems require efficient solver support. The implementation typically involves setting up path constraints using inequality functions and configuring solver tolerance parameters.

In complex optimal control problems such as aerospace trajectory optimization and robotic path planning, these pseudospectral-based numerical algorithms perform exceptionally well. Their mathematical foundation involves unifying the solution of state equations and costate equations into a single nonlinear programming problem, ensuring solution accuracy through spectral precision discretization. Key implementation aspects include proper scaling of variables, handling of differential-algebraic equations, and post-processing for continuous solution reconstruction from discrete data.