Sparse successive approximation for nonlinear H₂ and H_∞ optimal control problems under residual errors.

Area of Science:

• Control Theory
• Applied Mathematics
• Numerical Analysis

Background:

• Successive approximation techniques are crucial for solving nonlinear Hamilton-Jacobi-Bellman (HJB) and Hamilton-Jacobi-Isaacs (HJI) equations in optimal control problems (OCPs).
• Existing methods face challenges with convergence due to residual errors and computational complexity in nonlinear H2 and H-infinity OCPs.

Purpose of the Study:

• To propose a unified formulation and a novel sparse successive approximation method for infinite-horizon nonlinear H2 and H-infinity OCPs.
• To address convergence issues caused by residual errors and reduce the computational burden of existing numerical methods.

Main Methods:

• A unified formulation for HJB/HJI partial differential equations (PDEs) in nonlinear H2 and H-infinity OCPs.
• Transformation of nonlinear PDEs into sequences of linear PDEs solvable point-wise via initial value problems.
• Incorporation of extra constraints to ensure convergence despite residual errors.
• Utilization of sparse grid-based collocation points and basis functions for efficient numerical implementation.

Main Results:

• The proposed method transforms complex nonlinear PDEs into a series of simpler linear PDEs.
• Convergence is guaranteed even in the presence of residual errors through added constraints.
• Efficient numerical implementation is achieved using sparse grid techniques.
• Simulations demonstrate the effectiveness of the proposed method.

Conclusions:

• The sparse successive approximation method offers an effective and computationally efficient approach for solving nonlinear H2 and H-infinity OCPs.
• The unified formulation and convergence guarantees make the method robust for practical applications.
• The use of sparse grids significantly enhances the numerical tractability of solving HJB/HJI equations.