LQR Optimal Control — Riccati Equation & Minimum-Energy Trajectory

Linear Quadratic Regulator minimizes J = ∫(xᵀQx + uᵀRu)dt by solving the Algebraic Riccati Equation P = Q + AᵀP + PA − PBR⁻¹BᵀP. The optimal gain K = R⁻¹BᵀP drives the state to the origin along a minimum-energy path.