An Analytical Approach to Mixed-Constrained Quadratic Optimal Control Problems

Abstract

This study investigates the analytical solution of quadratic optimal control problems (OCPs) constrained by ordinary differential equations (ODEs) with real and coefficients. The formulation is based on the application of first-order optimality conditions to the Hamiltonian function, which yield a coupled system of first-order differential equations representing the necessary conditions for optimality. The resulting system is solved analytically using the method of eigenvalue decomposition and state transformation to determine the optimal state, control, and adjoint variables. The analytical procedure is illustrated through two examples of quadratic OCPs, confirming the effectiveness and accuracy of the developed method in deriving exact optimal solutions.