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\noindent
Math 555 \hfill Winter 2015\\
{\bf Homework 4} \hfill {\bf Due Friday, January 30}    

\begin{enumerate}
\item
We introduce a notion of differentiability 
for real valued functions on $\C$ by defining a corresponding function on $\R^2$.
Define the linear mapping
	\( \Theta: \R^2 \rightarrow \C \)
by
	\( \Theta(x_1, x_2)  =  x_1  +  \sfi x_2 \).
Then, given $f: \C \to \R$, define a corresponding mapping
	\( \tilde{ f } : \R^2 \rightarrow \R \)
by 
	\( \tilde{ f }  =   f  \circ  \Theta.\)
We say that the mapping 
	\( f \)
is \textit{differentiable in the real sense} if 
	\( \tilde{ f } \)
is differentiable, and 
	\( f \)
is \textit{twice differentiable in the real sense} if 
	\( \tilde{ f } \)
is twice differentiable.  
The chain rule gives
	\( f '(\zeta)  =  \Theta \nabla \tilde{ f }(\Theta^{-1}\zeta) \) 
and
	\( f''(\zeta)\delta  =  \Theta \nabla ^2 
		\tilde{ f }  (\Theta^{-1}\zeta)\Theta^{-1}\delta . \)
		
Differentiability in the real sense is the only notion of differentiability 
we use for mappings from $\C$ to $\R$, 
so when we say $f$ is differentiable it means
$f$ is differentiable in the real sense.
The directional derivative of $ f $ in the direction $\delta$ is given by
	\( f' (\zeta; \delta) = \RE( \overline{f' (\zeta)}\delta) , \)
and the second derivative is given by
	\( f'' (\zeta; \om, \delta)  = \RE(\bar{\om} f'' (\zeta)\delta) . \)
The function $f$ is quadratic if $f''(\zeta)$ is constant in $\zeta$.
For example, the function 
	\( r_2(\zeta) = \abs{\zeta}^2/2 \)
is quadratic with 
	\( r_2'(\zeta) = \zeta \)
and
	\( r_2''(\zeta) = I . \)

\begin{enumerate}
\item
Suppose $f:\C\to\R$ is differentiable on $\C$, show that
\[
f(y)=f(x)+f'(x;y-x)+o(|y-x|).
\]
\item
This notion of differentiability is easily extended to mappings
$f:\Cn\to\R$. Describe how this is done, and define a notion of the gradient,
$\nabla f(x)$, that works with this notion of differentiability
(hint: you should get $f'(x;d)=\RE({\nabla f(x)}^H{d})=\RE\ip{\nabla f(x)}{d}$).
Then use these definitions to show that
\[
f(y)=f(x)+f'(x;y-x)+o(|y-x|).
\]
\item
We have the following theorem from the course notes:
\smallskip

\noindent
{\bf Theorem}{\it\ 
Suppose that the autonomous DE $x'=f(x)$ (with $f\in C^1$)
has a Lyapunov function $V$ for which
\[
\RE(\nabla V(x)^Hf(x))<0\quad\mbox{whenever $x\ne 0$ with $\RE(\nabla V(x)^Hf(x))=0$ when $x=0$}.
\]
Then every solution to DE is asymptotically stable to zero.
}\smallskip

\noindent
What goes wrong in the non-autonomous case $x'=f(t,x)$? Are there reasonable additional
conditions that one might impose on $f(t,x)$ for which one can prove a non-autonomous
version of the theorem stated above? If so, state and prove such a result.
\end{enumerate}
\item
Prove the following lemma.
\smallskip

\noindent
{\bf Lemma}{\it\ 
The mapping $\map{\cH}{\Cnn}{\Cnn}$ given by
$\cH(M)=\half(M+M^*)$ is the orthogonal projection of 
$\Cnn$ onto the subspace $\Hn$ in the Frobenius inner
product.
Similarly, the mapping $\map{\cS}{\Rnn}{\Rnn}$ given by
$\cS(M)=\half(M+M^T)$ is the orthogonal projection of
$\Rnn$ onto the subspace $\Sn$.
}\smallskip

\item
Let $M\in\Hn$ and show that $M\preceq \gamma I$ if and only if 
$\lambda\le \gamma$ for all $\lambda\in\sigma(M)$.
\item
Prove the following theorem.

\noindent
{\bf Theorem}{\it\ 
[Exponential Stability of (CLH)]
Let $A\in\Cnn$ be such that $\alpha(A)<0$. Then, given $\alpha(A)<\gamma<0$,
there exists $0\prec P\in\Hn$
such that  
\[
PA+A^HP\preceq 2\gamma P.
\]
Moreover, for every
solution $x(t)$ to (CLH),
\[
|x(t)|\le \sqrt{\kappa(P)}\, |x(t_0)|e^{\gamma (t-t_0)}\qquad\forall\ t\ge t_0,
\]
where $\kappa(P)$ is the condition number for the matrix $P$.
}\smallskip

\item
Let $0<\epsilon < < 1$ and consider the matrix
\[
A=\begin{bmatrix}-\eps &\eps^{-2}\\ 0&-\eps\end{bmatrix}.
\]
\begin{enumerate}
\item
Obtain a closed form expression for the solution $x(t)$ for the IVP
$x'=Ax$ with $x(0)=\delta (1,1)^T$ for $\delta >0$, and use
Matlab to graph this solution for $\eps=0.1$ and $\delta=1$.
\item
Use this $A$ to show that the stability bound in the theorem of problem (4)
is sharp, i.e. one cannot do better than $\sqrt{\kappa(P)}$ and obtain an
exponential stability bound (hint: consider the solution near time $t=\eps^{-1}-\eps^2$).
\end{enumerate}
\end{enumerate}




\end{document}