keywords: Linear Quadratic Regulator (LQR), optimal feedback, stochastic optimality principle
We review and extend the idea of Linear Quadratic Regulator (LQR) by suppressing the time dependence, assuming that our solution is smooth and the gradient is zero. We develop necessary and sufficient conditions for optimal state feedback solution using the Stochastic Optimality Principle. Furthermore, we obtain the Stochastic Hamilton-Jacobi-Bellman (SHJB) equation, using conditional expectations. Finally, we observed that if no perturbation is present, then the SHJB equation reduces to the deterministic Hamilton-Jacobi-Bellman (HJB) equation.