The turnpike property in nonlinear optimal control—A geometric approach

Sakamoto, Noboru; Zuazua, Enrique

doi:10.1016/j.automatica.2021.109939

Cited by 22 publications

(35 citation statements)

References 39 publications

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“…The proofs of theorem 1.1 and related lemmas remains unchanged, except the proof of lemma 3.4. Indeed in this case, since f is not C 2 , we cannot use condition (3) to conclude from (26). However, for any y ∈ R and for any λ ∈ R,…”

Section: Remark 1 the Thesis Of Theorem 11 Holds With Nonsmooth Nonli...mentioning

confidence: 98%

“…In (26), because of (20), λ / ∈ {0, ±1} and G(−1) = 0. Hence, (32) together with (26) leads to a contradiction.…”

Section: Remark 1 the Thesis Of Theorem 11 Holds With Nonsmooth Nonli...mentioning

confidence: 99%

“…The issue of uniqueness of the minimizer for elliptic problems is of primary importance when studying the turnpike property for the corresponding time-evolution control problem (see, [28,24,27,26]). Indeed, the existence of multiple global minimizers for the steady problem generates multiple potential attractors for the time-evolution problem.…”

Section: Introductionmentioning

confidence: 99%

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Nonuniqueness of minimizers for semilinear optimal control problems

Pighin

2020

Preprint

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A counterexample to uniqueness of global minimizers of semilinear optimal control problems is given. The lack of uniqueness occurs for a special choice of the state-target in the cost functional. Our arguments show also that, for some state-targets, there exist local minimizers, which are not global. When this occurs, gradient-type algorithms may be trapped by the local minimizers, thus missing the global ones. Furthermore, the issue of convexity of quadratic functional in optimal control is analyzed in an abstract setting. As a Corollary of the nonuniqueness of the minimizers, a nonuniqueness result for a coupled elliptic system is deduced. Numerical simulations have been performed illustrating the theoretical results. We also discuss the possible impact of the multiplicity of minimizers on the turnpike property in long time horizons.

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Section: Remark 1 the Thesis Of Theorem 11 Holds With Nonsmooth Nonli...mentioning

confidence: 98%

“…In (26), because of (20), λ / ∈ {0, ±1} and G(−1) = 0. Hence, (32) together with (26) leads to a contradiction.…”

Section: Remark 1 the Thesis Of Theorem 11 Holds With Nonsmooth Nonli...mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Nonuniqueness of minimizers for semilinear optimal control problems

Pighin

2020

Preprint

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“…The estimate is obtained, by seeing the problem as an optimal control problem, thus writing the Optimality Condition as a first order Pontryagin system. In this context, we prove the hyperbolicity of the Pontryagin system around steady optima, to apply the Stable Manifold Theorem (see [15, Corollary page 115] and [18]). Our conclusions fit in the general framework of Control Theory and, in particular, of stabilization, turnpike and controllability (see e.g.…”

Section: Introductionmentioning

confidence: 99%

“…In Proposition 4, the Optimality Conditions are deduced in the form of Euler-Lagrange equations or equivalently as a state-adjoint state Pontryagin system. In Proposition 5 the asymptotic behaviour of the optima is analyzed in the spirit of stabilization and turnpike theory (see [16,22,18]). The Lojasiewicz inequality is employed to show that, in any condition, the optima stabilize towards a steady configuration.…”

Section: Introductionmentioning

confidence: 99%

Rotor imbalance suppression by optimal control

Gnuffi¹,

Pighin²,

Sakamoto³

2019

Preprint

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An imbalanced rotor is considered. A system of moving balancing masses is given. We determine the optimal movement of the balancing masses to minimize the imbalance on the rotor. The optimal movement is given by an open-loop control solving an optimal control problem posed in infinite time. By methods of the Calculus of Variations, the existence of the optimum is proved and the corresponding optimality conditions have been derived. By Lojasiewicz inequality, we proved the convergence of the optima towards a steady configuration, as time t → +∞. This guarantees that the optimal control stabilizes the system. In case the imbalance is below a computed threshold, the convergence occurs exponentially fast. This is proved by the Stable Manifold Theorem applied to the Pontryagin optimality system. Numerical simulations have been performed, validating the theoretical results.

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Rotor imbalance suppression by optimal control

Gnuffi

Pighin

Sakamoto

2021

Optim Control Appl Methods

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An imbalanced rotor is considered. A system of moving balancing masses is given. We determine the optimal movement of the balancing masses to minimize the imbalance on the rotor. The optimal movement is given by an open-loop control solving an optimal control problem posed in infinite time. By methods of the Calculus of Variations, the existence of the optimum is proved and the corresponding optimality conditions have been derived. Asymptotic behavior of the control system is studied rigorously. By Łojasiewicz inequality, convergence of the optima as time t → +∞ towards a steady configuration is ensured. An explicit estimate of the convergence rate is given. This guarantees that the optimal control stabilizes the system. In case the imbalance is below a computed threshold, the convergence occurs exponentially fast. This is proved by the Stable Manifold Theorem applied to the Pontryagin optimality system. Moreover, a closed-loop control strategy based on Reinforcement Learning is proposed.Numerical simulations have been performed, validating the theoretical results.

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The turnpike property in nonlinear optimal control—A geometric approach

Cited by 22 publications

References 39 publications

Nonuniqueness of minimizers for semilinear optimal control problems

Nonuniqueness of minimizers for semilinear optimal control problems

Rotor imbalance suppression by optimal control

Rotor imbalance suppression by optimal control

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