# A Two-Dimensional Dynamical System - Part 2

**Calculus**Level 5

A two-dimensional dynamical systems is characterized by the equation

\[ x(k+1) = A x(k) + B u \]

where

\[ x(k) = \begin{bmatrix} x_1(k) \\ x_2(k) \end{bmatrix} \]

is the state vector. In addition, matrix \( A \) and vector \( B \) are given by

\[ A = \begin{bmatrix} 0.9 && -0.1 \\ 0.3 && 0.5 \end{bmatrix} \]

\[ B = \begin{bmatrix} b_1 \\ b_2 \end{bmatrix} \]

Now, if \( u = 1 \), and the initial state \( x(0) \) is given by

\[ x(0) = \begin{bmatrix} 5 \\ 10 \end{bmatrix} \]

And, in addition,

\[ \lim_{k \to \infty} x(k) = \begin{bmatrix} 5 \\ 7 \end{bmatrix} \]

then, what must vector \( B \) be ?