# The interval turnpike property for adjoints

@article{Faulwasser2020TheIT, title={The interval turnpike property for adjoints}, author={Timm Faulwasser and Lars Grune and Jukka-Pekka Humaloja and Manuel Schaller}, journal={arXiv: Optimization and Control}, year={2020} }

In this work we derive an interval turnpike result for adjoints of finite- and infinite-dimensional nonlinear optimal control problems under the assumption of an interval turnpike on states and controls. We consider stabilizable dynamics governed by a generator of a semigroup with finite-dimensional unstable part satisfying a spectral decomposition condition and show the desired turnpike property under continuity assumptions on the first-order optimality conditions. We further give stronger…

## 8 Citations

### Inferring the adjoint turnpike property from the primal turnpike property

- Mathematics, Economics2021 60th IEEE Conference on Decision and Control (CDC)
- 2021

This paper investigates an interval turnpike result for the adjoints/costates of finite- and infinite-dimensional nonlinear optimal control problems under the assumption of an interval turnpike on…

### Abstract nonlinear sensitivity and turnpike analysis and an application to semilinear parabolic PDEs

- Mathematics, Computer ScienceESAIM: Control, Optimisation and Calculus of Variations
- 2021

An exponential turnpike result is proved and it is shown that perturbations of the extremal equation’s dynamics, e.g., discretization errors decay exponentially in time, can be used for very efficient discretized schemes in a Model Predictive Controller, where only a part of the solution needs to be computed accurately.

### Linear turnpike theorem

- Mathematics, Economics
- 2020

The turnpike phenomenon stipulates that the solution of an optimal control problem in large time, remains essentially close to a steady-state of the dynamics, itself being the optimal solution of an…

### Turnpike Properties in Optimal Control: An Overview of Discrete-Time and Continuous-Time Results

- EconomicsArXiv
- 2020

The present chapter provides an introductory overview of discrete-time and continuous-time results in finite and infinite-dimensions and comments on dissipativity-based approaches and finite-horizon results, which enable the exploitation of turnpike properties for the numerical solution of problems with long and infinite horizons.

### Optimal Control of Port-Hamiltonian Descriptor Systems with Minimal Energy Supply

- MathematicsSIAM Journal on Control and Optimization
- 2022

We consider the singular optimal control problem of minimizing the energy supply of linear dissipative port-Hamiltonian descriptor systems subject to control and terminal state constraints. To this…

### Primal or Dual Terminal Constraints in Economic MPC? - Comparison and Insights

- BusinessArXiv
- 2020

This chapter compares different formulations for Economic nonlinear Model Predictive Control (EMPC) which are all based on an established dissipativity assumption on the underlying Optimal Control Problem (OCP), and suggests a conceptual framework for approximation of the minimal stabilizing horizon length.

## References

SHOWING 1-10 OF 40 REFERENCES

### Steady-State and Periodic Exponential Turnpike Property for Optimal Control Problems in Hilbert Spaces

- MathematicsSIAM J. Control. Optim.
- 2018

This work designs an appropriate dichotomy transformation, based on solutions of the algebraic Riccati and Lyapunov equations, that establishes the steady-state (or periodic) exponential turnpike property of optimal control problems in Hilbert spaces.

### An Exponential Turnpike Theorem for Dissipative Discrete Time Optimal Control Problems

- MathematicsSIAM J. Control. Optim.
- 2014

Two theorems illustrate how this boundedness condition can be concluded from structural properties like controllability and stabilizability of the control system under consideration of the class of strictly dissipative systems under consideration.

### On the relation between turnpike properties and dissipativity for continuous time linear quadratic optimal control problems

- MathematicsMathematical Control & Related Fields
- 2021

The paper is devoted to analyze the connection between turnpike phenomena and strict dissipativity properties for continuous-time finite dimensional linear quadratic optimal control problems. We…

### The turnpike property in nonlinear optimal control — A geometric approach

- Economics, Mathematics2019 IEEE 58th Conference on Decision and Control (CDC)
- 2019

First, it is shown that a turnpike-like property appears in general dynamical systems with hyperbolic equilibrium and then, apply it to optimal control problems to obtain sufficient conditions for the turnpikes occurs.

### Exponential sensitivity and turnpike analysis for linear quadratic optimal control of general evolution equations

- Mathematics
- 2020

### Integral and measure-turnpike properties for infinite-dimensional optimal control systems

- Mathematics, EconomicsMath. Control. Signals Syst.
- 2018

It is proved that strict strong duality, which is a classical notion in optimization, implies strict dissipativity, and measure-turnpike, and the property of the time average of the distance from any optimal solution to the turnpike set converges to zero, as the time horizon tends to infinity.

### On turnpike and dissipativity properties of continuous-time optimal control problems

- Mathematics, EconomicsAutom.
- 2017

### On the Turnpike Phenomenon for Optimal Boundary Control Problems with Hyperbolic Systems

- MathematicsSIAM J. Control. Optim.
- 2019

It is shown that for optimal boundary control problems with integer constraints for the controls the turnpike phenomenon occurs, and a numerical verification is given for a control problem in gas pipeline operations.

### Long Time versus Steady State Optimal Control

- MathematicsSIAM J. Control. Optim.
- 2013

This paper analyzes the convergence of optimal control problems for an evolution equation in a finite time-horizon toward the limit steady state ones as T tends to infinity and shows that the optimal controls and states exponentially converge in the transient time.