@@ -66,7 +69,7 @@ The *indirect simple shooting method* consists in solving this equation.
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@@ -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
```math
\exp(t \vec{H})(x_0, p_0) := z(t, x_0, p_0),
\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
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@@ -77,3 +80,12 @@ and we need a [Newton-like](https://en.wikipedia.org/wiki/Newton%27s_method) sol
**_Remark:_**
**_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.
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.
