september 18

\(\def\Real{\mathbb{R}}\def\Comp{\mathbb{C}}\def\Rat{\mathbb{Q}}
\def\Field{\mathbb{F}}
\def\Fun{\mathbf{Fun}}
\def\e{\mathbf{e}}
\def\f{\mathbf{f}}\)

Remarks
  • What operators can appear as state transition (fundamental) ones, \(\Phi(t)\)?
  • For LTVs, any operator \(\Phi\) with \(\det(\Phi)>0\) can be a fundamental solution; for LTIs the situation is far more complicated (say, the diagonal matrix with \(-1, -2\) on the diagonal cannot be a fundamental solution.

  • Dual evolution.

Stability of linear systems

Examples.
Lyapunov stability, asymptotic stability, global asymptotic stability – definitions.

Global asymptotic stability for LTI.

Hurwitz operators. Multiple eigenvalues and Jordan normal forms.

Lyapunov’s direct method

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