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Polynomial Turnpike Property for a Class of Infinite-Dimensional Oscillating Systems

Alexander Zuyev, Emmanuel Trélat

Abstract

We establish a polynomial turnpike estimate for an optimal control problem consisting of a system of infinitely many controlled oscillators, considered as an abstract differential equation in a Hilbert space, with a quadratic cost. Our proof relies on spectral considerations and on the construction of a Riesz basis. A concrete example is given, which involves a rotating bodybeam system. To our knowledge, this is the first example of a pointwise turnpike estimate around a steady-state that is polynomial but not exponential.

Polynomial Turnpike Property for a Class of Infinite-Dimensional Oscillating Systems

Abstract

We establish a polynomial turnpike estimate for an optimal control problem consisting of a system of infinitely many controlled oscillators, considered as an abstract differential equation in a Hilbert space, with a quadratic cost. Our proof relies on spectral considerations and on the construction of a Riesz basis. A concrete example is given, which involves a rotating bodybeam system. To our knowledge, this is the first example of a pointwise turnpike estimate around a steady-state that is polynomial but not exponential.
Paper Structure (10 sections, 17 theorems, 169 equations)

This paper contains 10 sections, 17 theorems, 169 equations.

Table of Contents

  1. Introduction
  2. Oscillating system with modal coordinates
  3. Proof of Theorem \ref{['thm_turnpike_coordinates']}
  4. Analysis of the spectrum
  5. Construction of a Riesz basis
  6. Isomorphism property of $\pi_0$ on $V_{0,\gamma}^X$
  7. Polynomial decay conditions
  8. End of the proof of Theorem \ref{['thm_turnpike_coordinates']}
  9. Application to a rotating flexible beam
  10. Conclusion

Key Result

Proposition 2.1

Let Assumptions A1 and A2 hold, and let $T>T_0$, where Then, system cs_op is exactly controllable in time $T$ in the Hilbert space ${V_{0,1}}$, and we have ${\cal R}_T (x^0)= {V_{0,1}}$ for each $x^0\in {V_{0,1}}$, i.e., ${V_{0,1}}$ is the largest Hilbert subspace of $H$ in which the system is exactly controllable in time $T$. Moreover, any solution of

Theorems & Definitions (35)

  • Remark 1
  • Proposition 2.1
  • proof
  • Remark 2
  • Theorem 2.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • ...and 25 more