Clopen Loop

 In this note, I explain how one can use the Laplacian operator to test controllability properties of a nonlinear control system without drift. For example, your car. Unlike their fully actuated dynamical system cousins, every control engineer thinks of only one question when they see a car:  How does one generate three degree of motion with two degrees of control? Can it be parallel parked? The difficulty of this problem exponentially sky rockets if you are trying to park in LA especially if you have Sir 405 honking behind you. But that is not a problem I propose to solve. Let us first introduce the classical Laplacian operator. Some (just me) call it the Thor's hammer of the applied math world. But you do not need to be Thor (or Laplace) to wield it. So what is the Laplacian?  It takes a function $h$ and maps it to it's sum of second order derivatives, \[ h \mapsto \Delta h := \sum_{i =1}^m \frac{\partial^2 h }{\partial x^2_i} \]The problem that we are interested...

 Since I work in control theory, maybe this advice is most relevant to folks from the same field. These are mostly things I learnt during the job seasons over the last two years. It eventually ended me not getting any offers. So take it with a pinch of salt, or a dab of hot sauce, or a sprinkle of third reviewer or...ok I'll stop. My stats: I applied to around 60 places, including to controls, robotics and applied math positions. Got 8 zoom interviews. And one campus interview.  This was my second year trying. So here goes. 1. Apply early during your postdoc phase. Did you just start your postdoc? Yes, you are ready! Are you in your last year of PhD?? Yes, you are ready!  Do not wait till you have finished 2 or 3 years of postdoc. A common feeling among postdocs is “I haven’t done enough.” "my citation number is still in double digits." My experience has been that age from PhD graduation is a worse factor. In fact, if you wait for too long, it’s likely goi...