Differential Equations (Control Optimization Problem)

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SUMMARY

The discussion focuses on solving a control optimization problem involving differential equations defined by the system of equations: \(y_{1}'=y_1+y_{2}\) and \(y_{2}'=y_2+u\). The objective is to determine a control function \(u\) that satisfies the initial conditions \(y_{1}(0)=y_{2}(0)=0\) and the terminal conditions \(y_{1}(1)=1\) and \(y_{2}(1)=0\). Participants suggest reviewing optimal control theory resources, including Wikipedia's Optimal Control page and a Berkeley course PDF, to gain a deeper understanding of potential control models such as bang-bang or quadratic control.

PREREQUISITES
  • Understanding of differential equations and their solutions
  • Familiarity with optimal control theory concepts
  • Knowledge of control models, specifically bang-bang and quadratic controls
  • Proficiency in mathematical notation and analysis
NEXT STEPS
  • Study the principles of Optimal Control Theory using the provided Wikipedia link
  • Review the Berkeley course PDF on control systems for advanced insights
  • Explore the characteristics and applications of bang-bang control strategies
  • Investigate quadratic control methods and their implementation in differential equations
USEFUL FOR

This discussion is beneficial for graduate students in control theory, researchers in optimal control applications, and practitioners looking to solve complex control optimization problems using differential equations.

Alexandru999
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TL;DR
Differential Equations
\begin{equation}
y_{1}{}'=y_1{}+y_{2}

\end{equation}

\begin{equation}
y_{2}{}'=y_2{}+u
\end{equation}

build a control
\begin{equation}

u \epsilon L^{2} (0,1)
\end{equation}

for the care of the appropriate system solution
\begin{equation}
y_{1}(0)=y_{2}(0)=0
\end{equation}


satisfy \begin{equation}
y_{1}(1)=1 ,y_{2}(1)=0
\end{equation}
Please kindly if you can help me
Discipline is Optimal ControlHELP! i need to find control u

I am not cost functional, how to solve?
 
Last edited by a moderator:
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Welcome to the PF. :smile:

Schoolwork questions generally go in the Homework Help forums, but this is a complex enough question that it can probably stay here in the DE forum for now. You mentioned in our PM discussion that this is for a Master's degree level control class. Here are two of the links we were discussing as background:

https://en.wikipedia.org/wiki/Optimal_control

https://math.berkeley.edu/~evans/control.course.pdf

Can you give any more information about this question? What kind of control model are you expecting? Bang-bang, quadratic, etc.?
 
berkeman said:
Welcome to the PF. :smile:

Schoolwork questions generally go in the Homework Help forums, but this is a complex enough question that it can probably stay here in the DE forum for now. You mentioned in our PM discussion that this is for a Master's degree level control class. Here are two of the links we were discussing as background:

https://en.wikipedia.org/wiki/Optimal_control

https://math.berkeley.edu/~evans/control.course.pdf

Can you give any more information about this question? What kind of control model are you expecting? Bang-bang, quadratic, etc.?

RequirementBuild a control u for which The solution corresponding to a system with \begin{equation}
y_{1}(0)=y_{2}(0)=0
\end{equation} satisfy \begin{equation}
y_{1}(1)=1 ,y_{2}(1)=0
\end{equation}HELP !
for the system
above
 

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