Continuous-Time Infinite-Horizon Linear-Quadratic Regulator

For a linear time-invariant continuous-time system $$\dot{x}(t)=\underset{f(x(t),u(t),t)}{\underbrace{Ax(t)+Bu(t)}}$$ and a quadratic total cost $$J(x(t_0),u(\tau ),t_0)={\int }_{t_0}^{\infty }\underset{l(x(\tau ),u(\tau ),\tau )}{\underbrace{x(\tau {)}^{\top }Qx(\tau )+u(\tau {)}^{\top }Ru(\tau )}}d\tau ,$$ where $Q⪰0$ and $R\succ 0$, of its trajectory $(x(\tau ),u(\tau ))$, $\tau \in (t_0,\infty )$ the optimal controller can be derived based on the assumption that the value function takes the form $$V(x(t),t)=x(t{)}^{\top }Sx(t),S\succ 0.$$ When substituted into the Hamilton-Jacobi-Bellman Equation along with the system's dynamics we attain $$0=\underset{u(t)}{\min }\left\{x(t{)}^{\top }Qx(t)+u(t{)}^{\top }Ru(t)+2x(t{)}^{\top }S\left(Ax(t)+Bu(t)\right)\right\}.\text{\hspace{1em}(1)}$$ To find the minimum, we may take the gradient of its argument (which is by design quadratic and convex) with respect to $u(t)$, set it to zero and find the solution (optimal control input) $$u^{\ast }(t)=-R^{-1}{B}^{\top }Sx(t).$$

The input can then be substituted back into (1). As the equation must hold for all $x(t)$, through basic manipulations we then attain the continuous-time algebraic Riccati equation (CARE) $$0=Q-S{B}^{\top }{R}^{-1}BS+SA+A^{\top }S,S\succ 0.$$