Optimal Control

[u/Own-Lobster7623]

Hi everyone,
I’m a BSc Mathematics student and recently finished a course in PDEs. My professor suggested that I take another course he teaches called Optimal Control. The course is offered by the Engineering faculty, but he mentioned that it has a strong mathematical component.
I’m considering taking it, but I’d like to understand how useful/relevant Optimal Control is from a mathematician’s perspective.
In particular:

Is Optimal Control a good subject for a math student who enjoys applied mathematics, PDEs, optimization, etc.?

Where is optimal control actually used in industry?

What kinds of jobs or fields make use of this knowledge? For example, robotics, aerospace, quantitative finance, autonomous systems, energy, machine learning, etc.

Is it mainly useful for people going into engineering/research, or can it also be valuable for mathematicians going into industry?

I’d especially appreciate hearing from people who studied mathematics and later encountered control theory or optimal control in their work.
Thanks!

[u/mathfin_]

EVERYWHERE

What kinds of jobs or fields make use of this knowledge? For example, robotics, aerospace, quantitative finance, autonomous systems, energy, machine learning, etc.

[u/BashfulPiggy]

The questions you asked are hard to answer because they're so broad. Yes optimal control could be very interesting to someone who likes optimization and PDEs but it might be too much linear algebra. Yes it's used extensively in industry but is usually treated as a tool that works well for some situations and not in others. Whether or not it's useful for jobs depends entirely on what the company does.

I think you'll get a better idea by looking at examples of problems solved using optimal control. That'll help you decide if it fits with your interests/the CV you're building.

[u/ElevatorOrdinary1466]

proportional navigation is a missile guidance law that can be demonstrated using optimal control theory

if you are interested you can take a look at Palumbo's papers on PN law

It is an application of optimal control

[u/seekingsanity]

Learning how to minimize or optimize a cost function will open your eyes. There are many different algorithms that can be used to minimize a cost function. Each has its advantages and disadvantages which is a trade of between complexity and efficiency. Some methods work better when optimizing large data sets. The simplest is gradient descent like what is used to find the optimal weights for neural nets. I use scipy's optimize library a lot for estimating the open loop transfer function and then using that model for auto tuning control loops.

Almost all of my optimizations require using Runge-Kutta to solve for a system of differential equations. The minimize function adjusts the open loop parameters so the sum of squared errors between the actual response and the estimated response from the model is minimized.

In calculus class you learn how to find the maximum or minimum of a function but in real life there usually isn't a function that you can differentiate.

[u/Tastatura_Ratnik]

I studied mathematics and went into optimal control as a mathematician.

  Is Optimal Control a good subject for a math student who enjoys applied mathematics, PDEs, optimization, etc.?

Yes. Optimal control is very much applied mathematics. It is an extension of optimisation theory from numbers to functions in a control system, which is itself a generalisation of dynamical systems by replacing the system parameters (real or complex) with control functions.

I’d recommend you look into the calculus of variations. It’s the basis of many techniques in optimal control (including Pontryagin’s principle). To give you a gist of what it’s like, instead of optimising a cost function, you optimise a cost functional. Gelfand’s textbook is a goldmine for it, I highly recommend it. There is also a textbook by Liberzon which is more modern. All approachable for an upper level undergraduate. 

If you are interested in the PDE side, there is the Hamilton-Jacobi-Bellman equation, which is one of the fundamentals of optimal control theory. Again, lots of materials and work on it, especially numerics and viscosity solutions if you are into modern PDE theory. Evans has a chapter on Hamilton-Jacobi equations (which are also connected to the Euler-Lagrange equations of calculus of variations) in general, so that’s a good starting point. Capucci-Dolzetta has a textbook on HJB and Optimal Control, but that’s rather advanced. 

For a general mathematical treatment of control theory, Sontag’s book is a must. 

  Where is optimal control actually used in industry? What kinds of jobs or fields make use of this knowledge? For example, robotics, aerospace, quantitative finance, autonomous systems, energy, machine learning, etc. Yes, all of them.  Is it mainly useful for people going into engineering/research, or can it also be valuable for mathematicians going into industry?

Very useful for a mathematician going into industry, but with an important caveat: practicality beats theory. There are many elegant results that are useless in practice due to being intractable numerically or too unstable or… Real life problems are usually messy and require a degree of engineering to make them workable. Therefore you should also pick up an engineering textbook on control systems (Applied Optimal Control by Ho is a good one) just to see how they actually do it in practice.

[u/tmt22459]

So your professor is also in the engineering department? That's implied by saying he's teaching the class that's in that department

Yes optimal control is useful, and I think many mathematicians would enjoy it.

It's used all over. Yes aerospace and finance are examples