4. In the Cartesian coordinate system, $A(1,2)$, $B(3,0)$, and $P$ is any point on the circle $(x-3)^{2}+(y-2)^{2}=1$. Let
$$
\overrightarrow{O P}=\lambda \overrightarrow{O A}+\mu \overrightarrow{O B}(\lambda, \mu \in \mathbf{R}) \text {. }
$$

Then the minimum value of $11 \lambda+9 \mu$ is
4. 12 .

Let $P(3+\cos \theta, 2+\sin \theta)(\theta \in[0,2 \pi))$.
Then by the given conditions,
$$
\lambda+3 \mu=3+\cos \theta, 2 \lambda=2+\sin \theta \text {. }
$$

Thus, $11 \lambda+9 \mu=3(\lambda+3 \mu)+4(2 \lambda)$
$$
\begin{array}{l}
=3(3+\cos \theta)+4(2+\sin \theta) \\
=17+3 \cos \theta+4 \sin \theta \\
=17+5 \sin (\theta+\varphi) \\
\geqslant 17-5=12 .
\end{array}
$$

Clearly, the equality in the above expression can be achieved.
Therefore, the minimum value of $11 \lambda+9 \mu$ is 12.