A Direct Construction Method for Optimal Solutions of a Multi-Stage Minimum Control Energy Problem for Chain-of-Integrator Systems

This study addresses the multi-stage minimum control energy problem for chain-of-integrator systems, where the objective is to minimize the L2-norm of the control input over a fixed terminal time, subject to boundary conditions and a sequence of intermediate state constraints. Through rigorous variational analysis, we establish the existence and uniqueness of a global optimal solution and characterize it as a piecewise polynomial. Building on this analytical foundation, we reformulate the optimization problem into a system of linear equations, effectively bypassing the need for traditional iterative solvers. We then propose a direct construction method capable of solving this system with a linear computational complexity of O(M). Numerical experiments demonstrate the superior efficiency and robustness of the proposed method. Compared to state-of-the-art methods, our method significantly reduces the computational burden, making it highly viable for high-stake, real-time applications.