Control Theoretical Inquiry

In summary, the conversation discussed a hypothesis about gradient ascent methods and their application to quantum control. The main question was whether there is a way to optimize control variables using the value of the cost function, and whether this would be more efficient. The potential limitations of this approach were also discussed.
  • #1
Kreizhn
743
1
Hey everyone,

I have a hypothesis that I would like to confirm. I won't bore anyone with the nitty gritty details, so I will try to be as general as possible.

I'm doing a project on gradient ascent methods and their application to quantum control. The quantum part isn't important as my question is mathematical in nature, though a small caveat will appear and I'll make that clear.

Essentially, I'm trying to find an optimal control that will drive an operator [itex] \rho(t), \rho(0) =\rho_0 [/itex] to an operator [itex] \tau [/itex] in time T such that it optimizes their mutual inner product, say
[tex] C = \langle \tau , \rho(T) \rangle [/itex]
The gradient ascent method says that we should find the gradient of C, and then proceed in the direction in which the gradient is maximal. This is very useful from a numerical standpoint, and that is the context with which I will be using it.

I was asked during a seminar whether, in the event that we could directly calculate C, there was any way of formulating an optimal control just using the value of C, and if this could be potentially more efficient. Incidentally, this is where the quantum caveat occurs, in that there's no guarantee we can calculate C.

Thus my question comes down to this. Under the assumption that we can calculate the cost function directly, can I then find an algorithm to optimize my control variables?

I suspect not, since the inner product naively represents the overlap of the two operators. Hence calculating the cost function may tell us how close we are to a solution, but in an iterative numerical process, does not tell us "in which direction" to update our control variables.

Any thoughts on this?
 
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  • #2
I cannot see any changes in ##C##. If ##\rho## and ##\tau## are given, so is their angle ##C##. As I understand it, the control operator represents a path from ##\rho## and ##\tau## within the operator space, say ##\gamma(s)\,.## Thus you get angles ##C(s)=\langle \rho, \gamma(s) \rangle## which you could optimize.

However, a more specific answer depends on a more specific description.
 

1. What is control theoretical inquiry?

Control theoretical inquiry is a scientific approach that uses mathematical models and techniques to analyze and understand the behavior of complex systems. It involves studying how a system responds to various inputs and how to manipulate those inputs to achieve desired outcomes.

2. How is control theoretical inquiry used in real-world applications?

Control theoretical inquiry has a wide range of applications in fields such as engineering, economics, and biology. It is used to design and optimize control systems for processes such as manufacturing, transportation, and energy management. It is also used to study and predict the behavior of economic systems and biological systems, such as neural networks.

3. What are the key components of control theoretical inquiry?

The key components of control theoretical inquiry include a mathematical model of the system, a set of control objectives, and a control mechanism that manipulates the inputs to achieve those objectives. The model is used to analyze the behavior of the system and predict how it will respond to different inputs, while the control mechanism adjusts the inputs in real-time to achieve the desired outcomes.

4. What are the advantages of using control theoretical inquiry?

Control theoretical inquiry allows for a systematic and quantitative approach to understanding and controlling complex systems. It provides a framework for designing and optimizing control systems, which can lead to improved performance, efficiency, and stability. It also allows for the prediction and analysis of system behavior, which can help identify potential issues and improve decision-making.

5. What are some limitations of control theoretical inquiry?

One limitation of control theoretical inquiry is that it relies on mathematical models, which may not accurately represent the complexity of real-world systems. Additionally, the assumptions and simplifications made in these models may not always hold true in practice. Another limitation is that control systems can become unstable if the model or inputs are not accurately calibrated, which can lead to unintended consequences.

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