Your controls clearly show variable expression for the genes you're plotting so to get them to stick together on the left, you would need to set cluster_columns=F and manually order your matrix:

mat <- mat[,c(paste0("C", 1:9), paste0("P", 1:9))]

Alternatively, you can split your data into two matrices of controls and patients and then use grid or cowplot or a similar package to plot the two heatmaps together.