The great leap from calculus of variations to optimal control was a broad generalization of the kinds of variations we can consider. So generally this gives necessary conditions (and indeed the maximum principle is a necessary condition where as the HJB equation is necessary and sufficient). Very simplistically, in calculus of variations, we take a function from a space of functionals, 'perturb it a bit' (that is take its variation) and then derive conditions that function and the variations would satisfy if the function were optimal to begin with. Of these the latter approach is specifically a great generalization of ideas from calculus of variations. The field of optimal control only really took off in the 1960's due to Bellman and Pontryagin who introduced dynamic programmingand the maximum principle respectively. Here traditionally the choice of control has been to stabilize a system, drive a system from one state to another etc. Now as you observed classical control theory is concerned with transfer functions, root locus, stabilization etc. Again in this case, we can consider the control to be the shape of the curve, and the objective to minimize time. Similarly, another classic problem in calculus of variation is the Brachistochrone Problem which got much attention from the likes of Newton, Bernoullis, Leibniz etc. My background: I have mostly knowledge of applied mathematics (multivariable calculus for physics and economics, linear algebra, differential equations, PDE's), and some rudimentary knowledge of pure math (analysis 101, algebra 101, mathematical logic).Ĭalculus of variation is a special case of optimal control theory in a particular sense.Ĭonsider, Dido's iso-perimetric problem (colloquially said to be the oldest calculus of variation problem) which can be viewed as an optimal control problem, in the sense that what you get to control is the 'shape' of the curve, and your objective is to maximize the area. What do you think is the best resource for studying the calculus of variations, given my background? (or optimal control theory, depending on your answer to the above). So it is not entirely clear to me what the relation is between $(1), (2), (3)$?Īlso: Is it recommendable to study calculus of variations first, and then study Optimal Control Theory? Or is it old-fashioned to study calculus of variations in isolation? However, I understand control theory to be about dynamical systems (in engineering), and how to set a variable in order to control other variables in that system in some way ( is that correct?).
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