$$\int\limits_{0}^{1}x(t)\sqrt{1+u^2(t)}dt\to inf, \; \dot{x}(t)=u(t), \; u\in PC^1[0,1], \; x(t)\geq 0, \; |u(t)|\leq k, \; x(\pm 1)=R$$
I'm having trouble finding an optimisation. I decided to use Pontryagin's maximum principle. I defined the cost function
$$J[u] = \int\limits_{0}^{1}x(t)\sqrt{1+u^2(t)}dt$$
The Pontryagin maximum principle states that the optimal control $u^*(t)$ must satisfy the following condition $$H(x^*(t), u^*(t), \lambda(t)) = \max_{u} H(x^*(t), u, \lambda(t))$$ where $H$ is the Hamiltonian and $\lambda(t)$ is the conjugate variable associated with the state $x(t)$
For my case we can define $H(x, u, \lambda) = x\sqrt{1+u^2} + \lambda u$. The equation of state can be written as $\dot{x} = \frac{\partial H}{\partial \lambda} = u$. The equation of the conjugate variable is $\dot{\lambda} = -\frac{\partial H}{\partial x} = -\sqrt{1+u^2}$. So we have a system of four equations: two equations of state ($x$ and $u$) and two equations of the conjugate variable ($\lambda$ and $u$)
I can't seem to solve this system, maybe there is a simpler way? Can someone demonstrate it?