28% said no permanent address
20% said alcohol or drug use
17% said disability
14% said age
13% said need for clothing/shower facilities
...
9% said mental health concerns
So it does seems like lack of housing is the largest barrier for SF's homeless population. Additionally, mental illness is most prevalent amongst the chronically homeless, which only make up about 25% of the homeless population in SF. The vast majority of homeless people are homeless for months not years, and the many programs to aid them in finding housing and preventing homelessness in the first place have a significant measurable effect on the average number of homeless shelter entries and length of stay.
Guarantees of safety (more accurately stability) is the entire point of lyaponov analysis, and it's used on noisy systems all of the time (https://www.mathematik.hu-berlin.de/~imkeller/research/paper...). Can you point to a specific noisy system that control theory is ill suited for?