How Does Infinite horizon continuous time optimization Work?

Imagine you're saving up for your favorite candy bar, but instead of stopping after just one or two bars, you keep saving forever, always looking ahead to the next treat. That’s like infinite horizon continuous time optimization.

You’re trying to make the best choice right now by thinking about all the choices that will come later, and how they’ll add up over a very long time, maybe even forever. It's like planning every step of a never-ending game where you want to win the most points possible.

Think of it as a smart planner who checks your progress all the time, not just at certain moments. Instead of looking at your savings once a week, they’re checking on you every second, helping you decide whether to save more or spend now, always with an eye on the future.

This kind of planning is used in things like managing money, growing plants, or even controlling robots, wherever you want something to be as good as it can possibly be, forever. Imagine you're saving up for your favorite candy bar, but instead of stopping after just one or two bars, you keep saving forever, always looking ahead to the next treat. That’s like infinite horizon continuous time optimization.

You’re trying to make the best choice right now by thinking about all the choices that will come later, and how they’ll add up over a very long time, maybe even forever. It's like planning every step of a never-ending game where you want to win the most points possible.