@@ -66,7 +69,7 @@ The *indirect simple shooting method* consists in solving this equation.
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In order to solve the shooting equations, we need to compute the control law $u[\cdot]$, the Hamiltonian system $`\vec{H}`$ (or $`\vec{h}`$), we need an [integrator method](https://en.wikipedia.org/wiki/Numerical_methods_for_ordinary_differential_equations) to compute the *exponential map* $\exp(t \vec{H})$ defined by
In order to solve the shooting equations, we need to compute the control law $`u[\cdot]`$, the Hamiltonian system $`\vec{H}`$ (or $`\vec{h}`$), we need an [integrator method](https://en.wikipedia.org/wiki/Numerical_methods_for_ordinary_differential_equations) to compute the *exponential map* $`\exp(t \vec{H})`$ defined by
```math
\exp(t \vec{H})(x_0, p_0) := z(t, x_0, p_0),
```
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@@ -77,3 +80,12 @@ and we need a [Newton-like](https://en.wikipedia.org/wiki/Newton%27s_method) sol
**_Remark:_**
It is important to understand that if $`(p_0^*, \lambda^*)`$ is solution of $`S=0`$, then the control $`u(\cdot) := u[z(\cdot, x_0, p_0^*)]`$ is a candidate as a solution of the optimal control problem. It is only a candidate and not a solution of the OCP since the PMP gives necessary conditions of optimality. We would have to go further to check whether the control is locally or globally optimal.
Figure: Illustration of the shooting method in the state-time space. The blue trajectory reaches the target in red. The shooting method consists in finding the right impulse to reach the target.