Pontryagin's Minimum Principle

Notation

In what follows we will be making use of the following notation.

$\Psi_T(x(T))= \frac{\partial \Psi(x)}{\partial T}|_{x=x(T)} \,$

$\Psi_x(x(T))=\begin{bmatrix} \frac{\partial \Psi(x)}{\partial x_1}|_{x=x(T)} & \cdots & \frac{\partial \Psi(x)}{\partial x_n} |_{x=x(T)} \end{bmatrix}$

$H_x(x^*,u^*,\lambda^*,t)=\begin{bmatrix} \frac{\partial H}{\partial x_1}|_{x=x^*,u=u^*,\lambda=\lambda^*} & \cdots & \frac{\partial H}{\partial x_n}|_{x=x^*,u=u^*,\lambda=\lambda^*} \end{bmatrix}$

$L_x(x^*,u^*)=\begin{bmatrix} \frac{\partial L}{\partial x_1}|_{x=x^*,u=u^*} & \cdots & \frac{\partial L}{\partial x_n}|_{x=x^*,u=u^*} \end{bmatrix}$

$f_x(x^*,u^*)=\begin{bmatrix} \frac{\partial f_1}{\partial x_1}|_{x=x^*,u=u^*} & \cdots & \frac{\partial f_1}{\partial x_n}|_{x=x^*,u=u^*} \\ \vdots & \ddots & \vdots \\ \frac{\partial f_n}{\partial x_1}|_{x=x^*,u=u^*} & \ldots & \frac{\partial f_n}{\partial x_n}|_{x=x^*,u=u^*} \end{bmatrix}$

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Pontryagin's Minimum Principle - Notation