## What is it about?

A decisive feature of the turnpike pheomenon is that far away from the boundary points of the time interval, the solution of the dynamic problem is close to the solution of the corresponding static problem. The notion 'the turnpike phenomenone with interior decay' refers to this situation. In this paper we show that under a cheap control condition and a strict dissipativity assumption this turnpike property holds.

## Featured Image

Photo by Alexander Grey on Unsplash

## Why is it important?

The turnpike property allows to decompose optimal control processes into three phases: An initial phase, a terminal phase and an intermediate phase between the first two phases. With increasing time interval, only the length of the intermediate phase increases. This is an important insight that helps to understand the nature of the optimal control process. Moreover, this structural information can be used to gain computational efficiency in the appproximation of the optimal controls.

## Perspectives

## Read the Original

This page is a summary of: On the turnpike property with interior decay for optimal control problems, Mathematics of Control Signals and Systems, March 2021, Springer Science + Business Media,

DOI: 10.1007/s00498-021-00280-4.

You can read the full text:

## Resources

- Related Content
### Optimal Control Problems without Terminal Constraints: The Turnpike Property with Interior Decay

We show a turnpike result for problems of optimal control with possibly nonlinear systems as well as pointwise-in-time state and control constraints. The objective functional is of integral type and contains a tracking term which penalizes the distance to a desired steady state. In the optimal control problem, only the initial state is prescribed. We assume that a cheap control condition holds that yields a bound for the optimal value of our optimal control problem in terms of the initial data. We show that the solutions to the optimal control problems on the time intervals [0, T] have a turnpike structure in the following sense: For large T the contribution to the objective functional that comes from the subinterval [T/2, T], i.e., from the second half of the time interval [0, T], is at most of the order 1/T. More generally, the result holds for subintervals of the form [r T,T], where r ∈ (0, 1/2) is a real number. Using this result inductively implies that the decay of the integral on such a subinterval in the objective function is faster than the reciprocal value of a power series in T with positive coefficients. Accordingly, the contribution to the objective value from the final part of the time interval decays rapidly with a growing time horizon. At the end of the paper we present examples for optimal control problems where our results are applicable.

- URL
### Optimal Control Problems without Terminal Constraints: The Turnpike Property with Interior Decay

We show a turnpike result for problems of optimal control with possibly nonlinear systems as well as pointwise-in-time state and control constraints. The objective functional is of integral type and contains a tracking term which penalizes the distance to a desired steady state. In the optimal control problem, only the initial state is prescribed. We assume that a cheap control condition holds that yields a bound for the optimal value of our optimal control problem in terms of the initial data. We show that the solutions to the optimal control problems on the time intervals [0, T] have a turnpike structure in the following sense: For large T the contribution to the objective functional that comes from the subinterval [T/2, T], i.e., from the second half of the time interval [0, T], is at most of the order 1/T. More generally, the result holds for subintervals of the form [r T,T], where r ∈ (0, 1/2) is a real number. Using this result inductively implies that the decay of the integral on such a subinterval in the objective function is faster than the reciprocal value of a power series in T with positive coefficients. Accordingly, the contribution to the objective value from the final part of the time interval decays rapidly with a growing time horizon. At the end of the paper we present examples for optimal control problems where our results are applicable.

## Contributors

The following have contributed to this page