\(\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}}\)

#### matrix exponential

which is instrumental in finding solutions to linear systems of differential equations.

Computing the matrix exponentials can be done by just taking power series (not advised),

or by using the Jordan normal forms

(easy – block by block),

or by using Laplace transforms – requires finding the resolvent of a matrix,

\[

(sE-A)^{-1},

\]

or by using Cayley-Hamilton: works for distinct eigenvalues…

#### solving LTI state models

Once the matrix exponential (state transition matrix) is found, one can solve

\[

\dot{x}=Ax+Bu; y=Cx+Du.

\]

Indeed,

\[

x(t)=\Phi(t)x(0)+\int_0^t\Phi(t-s)Bu(s)ds.

\]

#### 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.

\]

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