september 16

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

linear time dependent systems

Generalizing the notion of state transition matrix, we can address also the time-dependent systems
\[
\dot{x}(t)=A(t)x(t).
\]
Here, we dove the equation on \([t_1,t_2]\) with \(x(t_1)=x\); the value of solution at \(t_2\) depends on \(x\) linearly. Denote this operator as
\[
\Phi(t_2\leftarrow t_1).
\]

Clearly,
\[
\Phi(t_3\leftarrow t_2)\Phi(t_2\leftarrow t_1)=\Phi(t_3\leftarrow t_1),
\]
and
\[
x(t)=\Phi(t\leftarrow 0)x(0)+\int_0^t \Phi(t\leftarrow s)Bu(s)ds.
\]
It remains to find \(\Phi\). Peano-Baker series (resulting from Picard iterations) give an answer…

Relevant notions and facts – existence and uniqueness of solutions of ODEs.

Remarks
  • What operators can appear as state transition (fundamental) ones, \(\Phi(t)\)?
  • Dual evolution.

Stability of linear systems

Examples.

Lyapunov stability, asymptotic stability, global asymptotic stability.

Global asymptotic stability for LTI. Hurwitz operators.

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