* $`\forall (x,p) \in \mathrm{R}^n \times \mathrm{R}^n`$, $`u \mapsto H(x,p,u)`$ has a unique maximum denoted $`\varphi(x,p)`$ (or $`u[x,p]`$ to recall the fact that it is the control law in feedback form),
* $`\varphi`$ is smooth, that is at least $`C^1`$.
* $`\varphi`$ is smooth, that is at least $`C^1`$.
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@@ -116,3 +116,156 @@ Under these assumptions, the maximization condition (3) is equivalent to the fir
since $`\frac{\partial H}{\partial u}(z, \varphi(z))=0`$. This leads to the remarkable fact that under our assumptions, equations (2) and (3) are equivalent to the Hamiltonian differential equation
```math
\dot{z}(t) = \vec{h}(z(t)),
```
where $`\vec{h}(z) := (\nabla_p h(z), -\nabla_x h(z))`$ is the **symplectic gradient** or **Hamiltonian system** associated to $`h`$.
where $`[t] := (x(t),p(t),u(t))`$. If $`p^0 = 0`$, then $`p = 0`$ by the third equation and so $`(p, p^0) = (0,0)`$ which is not. Hence, any *extremal* $`(x, p, p^0, u)`$ given by the PMP is said to be *normal*, that is $`p^0 = -1`$ (an extremal is said *abnormal* when $`p^0=0`$).
**_Remark:_** We do not consider the transversality condition when the target $`x_f`$ is fixed. We can retrieve simply the Lagrange multiplier by the relation $`p(t_f)=\lambda`$.
**_Remark:_** The maximization condition,
```math
H[t] = \max_{w \in \mathrm{R}} H(x(t), p(t), w),
```
is equivalent here to the condition
```math
\nabla_u H[t] = 0
```
by concavity.
Solving $`\nabla_u H[t] = 0`$, the control satisfies $`u(t) = u[x(t), p(t)] := p(t)`$ where we have introduced the smooth function on $`\mathrm{R} \times \mathrm{R}`$:
```math
u[x,p] = p.
```
**_Remark:_** Plugging the control law in feedback form into the pseudo-Hamiltonian gives the (maximized) Hamiltonian:
The unknown of this BVP is the initial covector $`p(0)`$. Indeed, fixing $`p_0:=p(0)`$, then according to the [Cauchy-Lipschitz theorem](https://en.wikipedia.org/wiki/Picard–Lindelöf_theorem), there exists a unique maximal solution denoted
with $`\pi_x(x,p) := x`$. Solving $`S(p_0) = 0`$ is what we call the *indirect simple shooting method*.
**_Remark:_** Note that thanks to the PMP, we have replaced the research of u (which is a function of time) by the research of an element of $`\mathrm{R}`$: the covector $`p_0`$. The prize of such a drastic reduction is to work in the *cotangent space*, that is the trajectory $`x`$ is lifted in a bigger space and adjoined with a covector $`p`$: this makes the simple shooting method to be qualified of *indirect*. It is important to note that in the indirect methods we work with $`z=(x,p)`$ and not only with the trajectory $`x`$.