# A Two-Dimensional Dynamical System

Calculus Level pending

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

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

$B = \begin{bmatrix} 2 \\ 1 \end{bmatrix}$

Now, if $$u = 5$$, and the initial state $$x(0)$$ is given by

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

what will be the final state which is given by this limit:

$\lim_{k \to \infty} x(k)$

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