What is common between Lagrangian/Hamiltonian in control theory/optimization and those functions in physics?

Question

In physics, Lagrangian is function which defined by kinetic energy and potential energy, however in optimization(also Economics or other Engineering), there are no such energy concepts. Also in Lagrangian in physics is defined by q and q' but Lagrangian in optimization is defined by x and lamda, adjoint variable(which means not x'). It looks like optimization theory adopts more abstract defintion of Lagrangian than physics, however, I failed to understand how Lagrangian in physics can be translated into Lagrangian optimization. I want more (pure) mathematical explanation, which does not borrow terms or concepts from physics or other applied subjects.

Answer 1

This all comes from the optimization of "functionals". A functional takes a function as an input and gives a real number as an output, such as a definite integral. To optimize a functional, you need to find a "function" that minimizes the functional's value. This is done using "calculus of variations".

Minimization of functionals comes up in many science and engineering diciplines, such as economics, control theory, etc., which might not be related to the classical mechanics.

However, it turns out that nature also "wants" to minimize certain functionals. If you select the so-called action functional as the integral of the Lagrangian and try to minimize its value, you get Newton's laws of motion.

If you happen to have some constraints, then you need to introduce the "Lagrange multipliers" into your functional (the $\lambda$ you have seen). This has nothing to do with the $q$ and $q'$ in the Lagrangian mechanics however, because those are the coordinates and the optimization is unconstrained. So, actually $x = \begin{bmatrix}q \\ q^\prime \end{bmatrix}$ is the state vector in the classical mechanics case (also in optimal control theory). If the states happen to be in this form and the functional is the integration of some function of the states, then the solution to Euler-Lagrange equation minimizes the functional.