Description of PDE and ODE Control. Examples main Page

[djoroya]

https://deustotech.github.io/dycon-platform-documentation/projects/03-examples

[azaharmonge]

Description of "Control of ODE"

%----------------------------------------TEX CODE ------------------------------------------------------- Given the ODE

\begin{align} \left { \begin{array}{c} x'(t) = f(t, x(t), u(t)), \ \ t \in [0,T], \ x(0) = x_0, \end{array} \right. \nonumber \end{align} with $x(t)$ and $u(t)$ being the state and control variables respectively. The main goal is to find the control $u(t)$ that optimizes a certain functional $J(x(t),u(t))$.

Summarizing, control of ODEs is crucial when the main interest is not to find the solution $x(t)$ of the ODE, but instead to optimize a certain quantity $J(x(t),u(t))$ with respect to the control variable $u(t)$ and subject to the ODE.

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[azaharmonge]

Description of "Control of PDEs"

%------------------------------------------ TEX CODE ------------------------------------------------------- Control of PDEs can be approached by the DyCon Toolbox for all evolutionary PDEs that can be written in a semidiscrete form leading to the following finite dimensional system:

\begin{align} \left { \begin{array}{c} \frac{\partial x(t)}{\partial t} = L(t, x(t), u(t)), \ \ t \in [0,T], \ x(0) = x_0, \end{array} \right. \nonumber \end{align} with $x(t)$ and $u(t)$ being the state and control variables respectively. The target is to find the control $u(t)$ that optimizes the functional $J(x(t),u(t))$ subject to the semidiscrete PDE.
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