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M  "Zettelkasten/Permanent Notes/20250829114522-hybrid-systems.md"

A  "Zettelkasten/Permanent Notes/20250911165736-switched-systems.md"

A  "Zettelkasten/Permanent Notes/20250911170650-lipschitz-continuous.md"

M  "Zettelkasten/Permanent Notes/Literature Notes/LIT-20250911143337-multiple-lyapunov-functions-and-other-analysis-tools-for-swtiched-and-hybrid-systems.md"

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Switched Systems

Switched systems are those that mix continuous and discrete dynamics. They are systems that are 'multimodal'. This means that they can have different continuous dynamic modes.

I'm borrowing form multiple-lyapunov-functions-and-other-analysis-tools-for-swtiched-and-hybrid-systems, but here's a short description of how they work.

A prototypical switched system is as follows:

\dot{x}(t)=f_i ( x(t)), \quad i \in Q \simeq {1,...,N}

with two conditions:

Each f_i is globally Lipschitz Continuous
The i's are picked in a way that there are finite switches in finite time.

There's also this thing called a continuous switched system. A continuous switched system is one that does not change continuous states when a switch occurs. That is to say when switching from i to i':

f_i(x(t_i),t_i) = f_{i'}(x(t_{i'}),t_{i'})

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Switched Systems

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