Can polymyxins work to treat pseudomonas aeruginosa?

Yes. Polymyxins are a very old class of antibiotics. They tend to have kidney and nervous system toxicity, and because they are very old the exact effective doses of these drugsis not well known. Still given problems with bacterial antibiotic resistance, sometimes these agents are used when a highly resistant bacteria is encountered. Colistin the one most often used can be given via inhalation or iv.
Yes. Polymyxin b or colistin are used for pseudomonas and acinetobacter infections that are resistant to all other antibiotics. As these drugs have significant renal toxicity they are reserved for infections resistant to less toxic antibiotics.