# Pontryagin's maximum principle problem, minimising an integral

I am really stuck on the following: Use Pontryagin's maximum principle to show that $\pi$ is the minimum value of the integral $\frac{1}{2} \int _{0} ^{1} u^{2} + v^{2} \,dt$ subject to constraints $x(0) = y(0) = z(0) = x(1) = y(1) = 0$ and $z(1) = 1$ where $\dot{x} = u$, $\dot{y} = v$ and $\dot{z} = uy - vx$.

The problem is symmetric about the z-axis and so without loss of generality can have $v(0) = 0$.

My Hamiltonian is $H = \lambda u + \mu v + \nu (uy - vx) - \frac{1}{2}(u^{2} - v^{2})$ and so I get that $\nu$ is constant and that $u = \lambda + \nu y$ and $v = \mu - \nu x$. I have tried messing around with taking derivatives in order to further track down the adjoint variables however have found nothing that gives any help.

From the assumption about $v(0) = 0$ can deduce that $\mu(0) = 0$.

Any help?

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The first step is always to apply the minimum principle and the transversality conditions. Something special also happens because there is no explicit time dependence in your problem. –  Dominique May 20 '12 at 18:12