Cycloid: Brachistochrone and Tautochrone

Outline
00:00 - Intro
02:55 - Brachistochrone problem
20:52 - Beginning of the calculus of variations
32:00 - Principle of least action
42:37 - Maximum principle
1:02:35 - Dynamic programming
1:11:12 - Linear quadratic control
1:16:37 - Beyond optimal control: games, nonsmooth analysis, MPC, RL
1:28:40 - Outro

Links
300 years of optimal control: https://tinyurl.com/2s3t8se4
Brachistochrone: https://tinyurl.com/mwmv38ew
Acta Eruditorum, 1696: https://tinyurl.com/55yf5v49
Acta Eruditorum, 1697: https://tinyurl.com/2a7msaaj
Bernoulli family: https://tinyurl.com/y2vx2xdn
Leibniz–Newton calculus controversy: https://tinyurl.com/3974fdhd
Calculus of variations: https://tinyurl.com/3vvz8tuf
Beginning of the Calculus of Variations: https://tinyurl.com/mv6btxfn
Lagrangian mechanics: https://tinyurl.com/ycx5fv46
Euler–Lagrange equation: https://tinyurl.com/53yybvyx
Hamiltonian mechanics: https://tinyurl.com/yfrd8zhz
Hamilton–Jacobi equation: https://tinyurl.com/46m9cuvs
Pontryagin: https://tinyurl.com/35ehxnex
Pontryagin’s autobiography:  https://ega-math.narod.ru/LSP/book.htm
Discovery of the Maximum Principle: https://tinyurl.com/3s43nv4t
Maximum Principle: https://tinyurl.com/4f7352t4
Goddard problem: https://tinyurl.com/5n8swp2m
Hamilton–Jacobi–Bellman equation: https://tinyurl.com/4uemn5y4
Kalman filter: https://tinyurl.com/39zx5yry
Clarke: https://tinyurl.com/yj2tzcjb
MPC: https://tinyurl.com/4sf5pzvy
RL: https://tinyurl.com/ee5ne7sz
AlphaGo: https://tinyurl.com/ydrf8jsc

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Acknowledgments and sponsors
This episode was supported by the National Centre of Competence in Research on «Dependable, ubiquitous automation» and the IFAC Activity fund. The podcast benefits from the help of an incredibly talented and passionate team. Special thanks to L. Seward, E. Cahard, F. Banis, F. Dörfler, J. Lygeros, ETH studio and mirrorlake . Music was composed by A New Element.