An algorithm for quasi-linear control problems in the economics of renewable resources: The steady state and end state for the infinite and long-term horizon

Entry and leaving arcs of turnpikes: their exact computation in the calculus of variations

“…As a matter of fact, one of us had already noticed this in the previous paper 19 . There, a model of renewable resource exploitation in an open-access fishery 20 , more detailed and general than 17 , is studied.…”

“…In this note, we intend to settle the question of the entry and leaving arcs of the turnpike in the generic hyperbolic situation for variational problems. The key point was suggested in 19 but not led to its natural consequence there. In short, and loosely speaking, our statement can be summarized as follows (for autonomous problems in ℝ): assume 𝑃 = (𝑥 𝑃 , ẋ𝑃 ) is a turnpike for a problem with initial condition 𝑥 0 and free terminal condition, and let 𝑇 (𝑥, ẋ) = 0 denote the equation giving the transversality condition.…”

“…{ 𝐹 (𝑥, 𝑢) − 𝑢𝐹 𝑢 (𝑥, 𝑢) = 𝐶(𝑥 𝑃 , 0) 𝐹 𝑢 (𝑥, 𝑢) = 0 which, after simplifying, becomes (see 19 , where this system of equations appeared for the first time):…”

A modified backward integration method for optimal control problems with degenerate equilibrium points

“…We have considered two possible alternative methods: a naive use of the backward integration method 7 and a version of the shooting method which we have successfully used in several other contexts. [23][24][25] We point out in passing that the degenerate singularity we are studying is of the "simplest" type, as there are only three invariant curves passing through it. The drawbacks of these methods will only increase in more complicated cases.…”

“…However, this hyperbolic saddle, for some special family of parameters becomes a degenerate singularity, which upon further inspection is of the type we can solve. Using our technique, we explicitly compute a solution and compare the values we obtain to those found using a different method (which has been proved useful for the nondegenerate case), [23][24][25] which is unable to approach the equilibrium point efficiently.…”