Problem $17$: Hamilton Jacobi Bellman Equation $-$ Finite Horizon

Consider the following problem to find the closed loop controller u : R$2\backslash$times $[0,$ tf $]!$ R:

min

u

x$1($tf $)+$ x$2($tf $)$

s$.$t$.$ x$1=$ x$23$u$,$ x$1(0)=$ x$10$

x$2=$ u$3,$ x$2(0)=$ x$20$

u $0.$

Determine whether there exists an affine $($in x$)$ time$-$varying value function for this problem.