# Stability Theory

In mathematics, stability theory addresses the stability of solutions of differential equations and of trajectories of dynamical systems under small perturbations of initial conditions. The heat equation, for example, is a stable partial differential equation because small perturbations of initial data lead to small variations in temperature at a later time as a result of the maximum principle.

In partial differential equations one may measure the distances between functions using Lp norms or the sup norm, while in differential geometry one may measure the distance between spaces using the Gromovâ€“Hausdorff distance.

In dynamical systems, an orbit is called Lyapunov stable if the forward orbit of any point is in a small enough neighborhood or it stays in a small (but perhaps, larger) neighborhood. Various criteria have been developed to prove stability or instability of an orbit. Under favorable circumstances, the question may be reduced to a well-studied problem involving eigenvalues of matrices. A more general method involves Lyapunov functions. In practice, any one of a number of different stability criteria are applied.

from  Stability Theory - Wikipedia

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## The Institute for Systems Theory and Automatic Control MATLAB Apps

Intermediate
App
Theory

The Institute for Systems Theory and Automatic Control offers 5 Matlab Apps on the topics of the Nyquist Criterion, Robustness and Stability, Loopshaping, Controllability and Observability...

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## Respect the Unstable

71 min
Beginner
Video
Theory

An understanding of fundamental limitations is an essential element in all engineering. Shannon's early results on channel capacity have always had center court in signal processing...

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## Gain and Phase Margins Explained!

13 min
Beginner
Video
Theory

In this video I explain gain and phase margins. If you are confused by this topic I hope this video will help tie all of the concepts together that go into understanding what gain and phase...

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## Stability of Closed Loop Control Systems

11 min
Beginner
Video
Theory

This video explains why we need design tools like the Routh-Hurwitz Criterion, Bode Plots, Nyquist Plots, and Root Locus. This is an introduction into the difficulties of determining the...

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## Stability and Eigenvalues [Control Bootcamp]

19 min
Beginner
Video
Theory

Here we discuss the stability of a linear system (in continuous-time or discrete-time) in terms of eigenvalues. Later, we will actively modify these eigenvalues, and hence the dynamics...

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## Introduction to System Stability and Control

11 min
Beginner
Video
Theory

This video attempts to provide an intuitive understanding of concepts like stability and stability margin. I briefly describe both of these topics with examples and explain how you can...

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