This will just be a short post. It's basically a flag to myself in that I am going to promise to deliver something here and hopefully that will keep me to my promise. I've started to teach myself optimal control for ordinary differential equations. That is, looking at solving systems of ODEs which involve some "control" that is involved in some other quantity that we wish to optimise in some way.
For example, say we have an exponentially growing tumour (#cells is x) being treated with some drug (concentration u) such that the governing equation is
x'(t) = ax(t) - u(t), x(0)=0.
Say that drug is also harmful to healthy cells as well. Then we might wish to minimise both the tumour size at some endtime (T) and also the effects over time of the drug. eg:
min { x(T) + \int_0^T u^2 dt }.
So basically I'm looking into optimal control theory for such problems. It doesn't look too horrific actually. I tried looking at this years ago when I did my postdoc (my boss had published some papers on optimal treatment of tumours) but it was over my head. Now I think I can actually understand some of the material that's available on the subject and I think I can write about it myself.
The plan is to write some short notes on the topic that are understandable and make them available here soon (next couple of weeks). Then y'all can let me know what you think. yay :-)
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