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{ \large \textbf{\textsc{Math 5588 -- Homework 12 (Due Tuesday May 2)}}}
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\begin{enumerate}
\item Suppose $H(p)$ is convex and superlinear. Show that $L = H^*$ is also convex and superlinear. [Hint: For convexity, you need to show that $L(\lambda x + (1-\lambda)y) \leq \lambda L(x) + (1-\lambda)L(y)$ for all $x,y$ and $0 < \lambda < 1$, assuming the same is true for $H$. Write the left hand above using the definition of $L$ and use the fact that $\max\{f + g\} \leq \max f + \max g$. For superlinearity, show that for every $M>0$
\[\lim_{|q|\to \infty} \frac{L(q)}{|q|} \geq M.\]
To do this, write
\[L(q) = \max_{p\in \R} \{pq - H(p)\}\]
and choose $p=\frac{q}{|q|}M$ to obtain
\[L(q) \geq M|q| - H(Mq/|q|).\]
Proceed from here with the rest of the proof.]
\item Suppose that $u-\phi$ has a local maximum at $x\in \R$, which means that for some $r>0$
\[u(y) - \phi(y) \leq u(x)-\phi(x) \ \ \text{ for all } |x-y|<r.\]
Assume also that $u$ is bounded, that is for some $C>0$ we have $|u(y)|\leq C$ for all $y\in\R$. Show that there exists another test function $\psi$ such that $\psi'(x)=\phi'(x)$ and
\[u(y) - \psi(y) \leq u(x)-\psi(x) \ \ \text{ for all } y \in \R.\]
Thus, we can turn the local max (or min) in the definition of viscosity solutions into a global max (or min) without loss of generality. [Hint: Define
\[\psi(y) = \phi(y) + K(x-y)^2\]
for a carefully chosen constant $K$.]
\item Consider the following optimal control problem: The state $x(t) \in \R^n$ obeys the dynamics
\[\dot{x}(s) = \alpha(s) \text{ for } t < s \leq T \text{ and } x(t)=x\]
where $\alpha:[0,T]\to \R^n$ is the control, and the cost is
\[C_{x,t}(\alpha) = \int_t^T |\alpha(s)|^2 \, ds + x(T)^2.\]
Find the Hamilton-Jacobi-Bellman equation for the value function
\[u(x,t) = \min_{\alpha:[0,T]\to \R^n} C_{x,t}(\alpha).\] 
\item Repeat problem 3 for the dynamics
\[\dot{x}(s) = x(s) + \alpha(s)\]
where $\alpha:[0,T] \to \R^n$.
\item Repeat problem 3 for the dynamics
\[\dot{x}(s) = \mathbf{1}\alpha(s)\]
where $\mathbf{1}=(1,\dots,1)\in \R^n$ and $\alpha:[0,T] \to \R$.
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