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.