Long Time versus Steady State Optimal Control

Porretta, Alessio; Zuazua, Enrique

doi:10.1137/130907239

Cited by 130 publications

(187 citation statements)

References 21 publications

Supporting

Mentioning

177

Contrasting

Order By: Relevance

“…This property, known in the optimal control literature as the turnpike property, will play a key role in our analysis. Interestingly, the turnpike property, which is a classical property in optimal control [52,18,39,14] has recently attracted renewed attention, not least because of its importance for Receding Horizon Control [16,19,20,50,42].…”

Section: Introductionmentioning

confidence: 99%

Approximation Properties of Receding Horizon Optimal Control

Grüne

2016

Jahresber. Dtsch. Math. Ver.

View full text Add to dashboard Cite

In this survey, receding horizon control is presented as a method for obtaining approximately optimal solutions to infinite horizon optimal control problems by iteratively solving a sequence of finite horizon optimal control problems. We investigate conditions under which we can obtain mathematically rigorous approximation results for this approach. A key ingredient of our analysis is the so-called turnpike property of optimal control problems.

show abstract

Section: Introductionmentioning

confidence: 99%

Approximation Properties of Receding Horizon Optimal Control

Grüne

2016

Jahresber. Dtsch. Math. Ver.

View full text Add to dashboard Cite

show abstract

“…Moreover, under an exponential turnpike assumption, cf. [5,12], the trajectories converge to a neighborhood of x e and there exists at least one time horizon for which the closed loop trajectory is also approximately optimal in a finite horizon sense. Since (approximate) optimality in an infinite horizon averaged sense is in fact a rather weak optimality concept (as the trajectory may be far from optimal on any finite time interval) the latter is important because it tells us that the closed loop trajectory when initialized away form the optimal steady state approaches this equilibrium in an approximately optimal way.…”

mentioning

confidence: 98%

Asymptotic stability and transient optimality of economic MPC without terminal conditions

Grüne

Stieler

2014

Journal of Process Control

118

View full text Add to dashboard Cite

We consider an economic nonlinear model predictive control scheme without terminal constraints or costs. We give conditions based on dissipativity and controllability properties under which the closed loop is practically asymptotically stable. Under the same conditions we prove approximate transient optimality of the closed loop on finite time intervals. Two numerical examples illustrate our theoretical findings.

show abstract

“…Dissipativity, see [8,9], and related properties, see, e.g, [19,Condition 2.2], are already well known for establishing non-exponential turnpike for continuous time problems without assuming convexity or concavity. For finite and infinite dimensional linear quadratic continuous time problems, the recent paper [23] establishes exponential turnpike theorems via the use of Riccati equations (for an earlier Riccati approach to turnpike-like results see also [3]). While this approach yields similar results to ours in the linear quadratic case, our approach in this paper applies to general nonlinear nonquadratic problems and entirely avoids the use of Riccati equations.…”

mentioning

confidence: 99%

“…Hence, C P grows linearly with the distance of the points in X 0 from x e . (ii) We consider Theorem 6.2 remarkable since many known turnpike theorems (both exponential ones like [23,Theorem 2.3] and non exponential ones like [9, Theorem 4.2]), when specialized to the linear quadratic case require controllability of (A, B) instead of the weaker stabilizability our theorem requires. The equivalence statement of our theorem moreover shows that one cannot further weaken this assumption.…”

mentioning

confidence: 99%

An Exponential Turnpike Theorem for Dissipative Discrete Time Optimal Control Problems

Damm¹,

Grüne²,

Stieler³

et al. 2014

SIAM J. Control Optim.

144

151

View full text Add to dashboard Cite

Abstract. We investigate the exponential turnpike property for finite horizon undiscounted discrete time optimal control problems without any terminal constraints. Considering a class of strictly dissipative systems we derive a boundedness condition for an auxiliary optimal value function which implies the exponential turnpike property. Two theorems illustrate how this boundedness condition can be concluded from structural properties like controllability and stabilizability of the control system under consideration.Key words. turnpike property, optimal control, dissipativity, stabilizability, controllability, model predictive control AMS subject classifications. 49K30, 49K21, 93B051. Introduction. An optimal trajectory of a control problem is said to have the turnpike property if it first approaches an equilibrium state, stays close to it for a while and finally turns away from it again. The name turnpike property is motivated by the analogy of the behavior of the optimal trajectories to the strategy of driving from a point A to B on a road system consisting of highways ("turnpikes") and smaller roads. When the distance from A to B is sufficiently long, it is typically time optimal to first drive from A to the nearest highway (= move to the equilibrium), drive on the highway towards the nearest exit to B (= stay near the equilibrium) and then exit in order to reach B via smaller roads (= turn away from the equilibrium).The turnpike property has been studied at least since the work of von Neumann in 1945 [22] and Dorfman, Samuelson and Solow in 1958 [11, p. 331]. Since then it has been observed in many optimal control problems. There is a vast amount of literature on sufficient conditions for this phenomenon to hold, see, e.g., [9, Section 4.4] or [28], particularly in economics, see, e.g., [21] and the references therein. However, only very few references treat the case of exponential turnpike which we consider in this paper for nonlinear undiscounted discrete time optimal control problems without terminal constraints. Our main motivation for studying this property is its recently discovered importance for obtaining convergence results in economic model predictive control (MPC) without terminal constraints, see [13]. The particular interest in exponentially fast versions of the turnpike property is triggered by the fact that, compared to slower turnpike properties, the exponential turnpike property allows to conclude additional qualitative properties of the MPC closed loop solution, like trajectory convergence and approximate finite time optimal transient behavior, for details see Section 3, below. While some exponential turnpike theorems can be found in the literature, our approach extends these results in various ways, e.g., by assuming only strict

show abstract

Long Time versus Steady State Optimal Control

Cited by 130 publications

References 21 publications

Approximation Properties of Receding Horizon Optimal Control

Approximation Properties of Receding Horizon Optimal Control

Asymptotic stability and transient optimality of economic MPC without terminal conditions

An Exponential Turnpike Theorem for Dissipative Discrete Time Optimal Control Problems

Contact Info

Product

Resources

About