General control problem with nonlinear performance index with integral and a scrap value

Question

I have an optimal control problem with a following general form for the performance index

$J = \phi_{1}[x(t_f)] + \sqrt {\phi_{2}[x(t_f)] + \int_{t_0}^{t_f}L[x(t)] dt},$

subject to

$\dot{x}(t)=g(x(t),u(t))$, $x(t_0)=x_0$ (fixed), $x(t_f)$ (free)

where $x(t)$ is the state variable, $0 \leq u(t)\leq 1$ is the control variable, $\phi_{1}[x(t_f)]$ and $\phi_{2}[x(t_f)]$ are some functions of the scrap value (final state).

How should I start solving this optimal control problem? Can I write directly the Hamiltonian for this problem or do I have to transform the performance index into some other form before I can write down the Hamiltonian for this problem?

If so, how would you suggest to transform it?

Answer 1

If the $\phi_1$ is positive definite and you seek a minimizer then define

$$ J' := \sqrt{\phi_1(x_f)^2+\phi_2(x_f)+\int_{t_0}^{t_f} L(x)dt}. $$ By the triangle inequality $J' \leq J$ and $J'$ is minimized whenever $J'^2$ is minimized. The Hamiltonian is now standard.

Answer 2

You can also reformulate your problem by adding a new DE:

$$\dot{x}_J=L(x),\quad x_J(0)=0.$$

In this way, your $J$ can be written as $$J=\psi(x(t_f))$$ with $\psi(x(t_f))=\phi_1(x(t_f))+\sqrt{\phi_2(x(t_f))+x_J(x(t_f))}$. Such cost functional is said to be in the Mayer form.