Transversality Condition w/ Right End Point Free & Int. Constraints

In summary, the conversation discusses the calculus of variations and the transversality condition for optimization problems with free end points and integral constraints. The speaker is unsure about the necessary transversality condition for problems with free end points and integral constraints and is seeking more information on isoperimetric problems with free endpoints.
  • #1
Mosis
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Hi, I've been learning about the calculus of variations this term, and we just covered the transversality condition for an optimization problem with the right end point free, as well as the first necessary condition on the augmented Lagrangian for a problem with integral constraints. I'm wondering what the necessary transversality condition is for a problem with integral contraints with the right end point free. I can't find this information in my textbook or anywhere on the internet, and I have two hunches, though I'm not overwhelmingly confident in either.
 
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  • #2


to put it another way, I wish to know some information about isoperimetric problems with free endpoints, namely the transversality condition at the free end point
 

1. What is the transversality condition with right end point free and intermediate constraints?

The transversality condition with right end point free and intermediate constraints is a mathematical concept used in optimal control theory. It states that the optimal solution to a dynamic optimization problem must satisfy a boundary condition at the right end point of the time interval, as well as any intermediate constraints that may exist.

2. Why is the transversality condition important in optimal control theory?

The transversality condition is important because it helps to ensure that the optimal solution to a dynamic optimization problem is well-behaved and satisfies all necessary constraints. Without this condition, the solution may not be unique or may not be feasible in practice.

3. How is the transversality condition with right end point free and intermediate constraints applied in real-world problems?

The transversality condition is commonly used in economics, engineering, and other fields to model and optimize real-world systems. For example, it may be used to determine the optimal path for a rocket to travel from one point to another, taking into account constraints such as fuel consumption and atmospheric conditions.

4. Are there any variations of the transversality condition?

Yes, there are several variations of the transversality condition, including the left end point condition, which applies a boundary condition at the left end point of the time interval, and the mixed conditions, which apply boundary conditions at both the left and right end points. These variations may be used depending on the specific problem being solved.

5. How is the transversality condition mathematically expressed?

The transversality condition is typically expressed as a first-order necessary condition for optimality, using the Pontryagin Maximum Principle. It involves taking the partial derivative of the Hamiltonian function with respect to the state variables and setting it equal to zero at the right end point of the time interval. This condition is then combined with any intermediate constraints to determine the optimal solution.

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