How do I approach Optimal Control? Other than learning basic calculus, I don't really have an advanced background. I was curious to learn about Optimal Control (the theory that involves, bang-bang, Potryagin's Maximum Principle etc.) but any article that I start off with, mentions the following: "Consider a control system of the form..." and then goes on to defining partial differential equations. In short, I am lost. 
Can someone suggest me a path I should take to learn more about Optimal Control from the very basics?
 A: My field is mathematical programming, and I tend to look at optimal control as just optimization with ODEs in the constraint set; that is, it is the optimization of dynamic systems. I would start by studying some optimization theory (not LPs but NLPs) and getting an intuitive feel for the motivations behind stationarity and optimality conditions -- that will lead naturally into optimal control theory.
I should mention there is another facet of optimal control, related to control systems. The systems considered are discrete time (as opposed to continuous in PMP) therefore it's difference equations instead of differential equations. Examples of optimal control laws in this latter sense are Linear Quadratic Regulators (LQRs), Linear Quadratic Gaussian (LQGs), Model Predictive Control (MPC). It is this latter type of optimal control that is actually applied in industry. The Pontryagin principle, while useful for analysis, is generally intractable for real-time application to nontrivial plants.
A: I recommend Eduardo Sontag's Mathematical Control Theory without hesitation. It explains the basics of control theory, optimal control inclusive, as mathematicians see it - geared towards advanced undergrads but useful for all. The easier books to read are for and by engineers - nothing against them, I'm one - but if you want a mathematical text that gives the whole story I suggest you look at Sontag's.
A: One potential tactic would be start with Estimation Theory rather than Control Theory. I've enjoyed the approach taken in H. Vincent Poor's An Introduction to Signal Detection and Estimation.
A: It might help to understand what background you already have.  Have you taken any courses in ordinary differential equations? partial differential equations? real analysis?  What mathematics courses have you taken?  What kind of background do you have in engineering approaches to dynamical systems?  Have you taken an introductory course in linear systems?  Are you familiar with basic concepts like feedback? stability of a system?      
A: A very good little book on the subject is Analytical Methods of Optimization by D.F. Lawden,
available from Dover Press.  It covers Pontryagin's principal, Hamiltonian and Lagrangian
formulations, and should be accessible to a person with your background.
A: Another great book is "Optimal control theory: An introduction to the theory and its applications" by Peter Falb and Michael Athans, also published by Dover.
Also, I would recommend looking at the videos of the edX course "Underactuated Robotics", taught by professor Russ Tedrake of MIT. 
A: If you have a background in differential geometry, you will probably like the two books of Jurdjevic, Geometric Control Theory, and Optimal Control and Geometry: Integrable Systems, both from Cambridge Univesity Press.
A: You might want to check Lawrence Evans' notes on optimal control: An Introduction to Mathematical Optimal Control Theory [pdf].
A: Kamien and Schartwz (1991) is an extremely complete book on this subject. It can occasionally be oblique, but is otherwise quite helpful.
The Economists' Mathematical Manual contains all of Kamien and Schwartz's results (and much more) in a handy summary format.
A: Try  Singiresu S. Rao, Engineering Optimization: Theory and Practice, Wiley, 2009, 4th edition.
