18. Consider the time-invariant, deterministic, continuous-time linear system: 1(t) = ax(t) + bu(t), where u(t) is a scalar control, and x(t) is a scalar state. The initial state is x(0) = x0. The...

18. Consider the time-invariant, deterministic, continuous-time linear system: 1(t) = ax(t) + bu(t), where u(t) is a scalar control, and x(t) is a scalar state. The initial state is x(0) = x0. The finite time horizon is T. The goal is to choose the control input over the time interval [0,7] to minimize the cost function: px2(T) + T (qx2(t)+ru2(0)dt, where p > 0, q > 0,r > 0. Find the differential equation to be satisfied by s(t), and the boundary condition to be satisfied by s(T), so that V(x, t) := s(t)x2 is the optimal cost-to-go from state x at time t. [Hint: Use the Hamilton-Jacobi-Bellman (HJB) equation. Substitute V(x, t) into it, and find out what differential equation for s(t) would make it satisfy the HJB equation.]