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Minimum energy with infinite horizon: From stationary to non-stationary states.

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Title: Minimum energy with infinite horizon: From stationary to non-stationary states.
Authors: Acquistapace, P.1 (AUTHOR) paolo.acquistapace@unipi.it, Gozzi, F.1,2 (AUTHOR) fgozzi@luiss.it
Source: Nonlinear Analysis: Real World Applications. Feb2022, Vol. 63, pN.PAG-N.PAG. 1p.
Subjects: Operator functions, Selfadjoint operators, Algebraic equations, Linear operators, Riccati equation, Carleman theorem
Abstract: We study a non-standard infinite horizon, infinite dimensional linear–quadratic control problem arising in the physics of non-stationary states (see e.g. Bertini et al. (2004, 2005)): finding the minimum energy to drive a given stationary state x ̄ = 0 (at time t = − ∞) into an arbitrary non-stationary state x (at time t = 0). This is the opposite to what is commonly studied in the literature on null controllability (where one drives a generic state x into the equilibrium state x ̄ = 0). Consequently, the Algebraic Riccati Equation (ARE) associated with this problem is non-standard since the sign of the linear part is opposite to the usual one and since its solution is intrinsically unbounded. Hence the standard theory of AREs does not apply. The analogous finite horizon problem has been studied in the companion paper (Acquistapace and Gozzi, 2017). Here, similarly to such paper, we prove that the linear selfadjoint operator associated with the value function is a solution of the above mentioned ARE. Moreover, differently to Acquistapace and Gozzi (2017), we prove that such solution is the maximal one. The first main result (Theorem 5.8) is proved by approximating the problem with suitable auxiliary finite horizon problems (which are different from the one studied in Acquistapace and Gozzi (2017)). Finally in the special case where the involved operators commute we characterize all solutions of the ARE (Theorem 6.5) and we apply this to the Landau–Ginzburg model. [ABSTRACT FROM AUTHOR]
Copyright of Nonlinear Analysis: Real World Applications is the property of Elsevier B.V. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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DbLabel: Engineering Source
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  Data: Minimum energy with infinite horizon: From stationary to non-stationary states.
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  Data: <searchLink fieldCode="AR" term="%22Acquistapace%2C+P%2E%22">Acquistapace, P.</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> paolo.acquistapace@unipi.it</i><br /><searchLink fieldCode="AR" term="%22Gozzi%2C+F%2E%22">Gozzi, F.</searchLink><relatesTo>1,2</relatesTo> (AUTHOR)<i> fgozzi@luiss.it</i>
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  Data: We study a non-standard infinite horizon, infinite dimensional linear–quadratic control problem arising in the physics of non-stationary states (see e.g. Bertini et al. (2004, 2005)): finding the minimum energy to drive a given stationary state x ̄ = 0 (at time t = − ∞) into an arbitrary non-stationary state x (at time t = 0). This is the opposite to what is commonly studied in the literature on null controllability (where one drives a generic state x into the equilibrium state x ̄ = 0). Consequently, the Algebraic Riccati Equation (ARE) associated with this problem is non-standard since the sign of the linear part is opposite to the usual one and since its solution is intrinsically unbounded. Hence the standard theory of AREs does not apply. The analogous finite horizon problem has been studied in the companion paper (Acquistapace and Gozzi, 2017). Here, similarly to such paper, we prove that the linear selfadjoint operator associated with the value function is a solution of the above mentioned ARE. Moreover, differently to Acquistapace and Gozzi (2017), we prove that such solution is the maximal one. The first main result (Theorem 5.8) is proved by approximating the problem with suitable auxiliary finite horizon problems (which are different from the one studied in Acquistapace and Gozzi (2017)). Finally in the special case where the involved operators commute we characterize all solutions of the ARE (Theorem 6.5) and we apply this to the Landau–Ginzburg model. [ABSTRACT FROM AUTHOR]
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  Data: <i>Copyright of Nonlinear Analysis: Real World Applications is the property of Elsevier B.V. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Value: 10.1016/j.nonrwa.2021.103413
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      – Code: eng
        Text: English
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      – SubjectFull: Operator functions
        Type: general
      – SubjectFull: Selfadjoint operators
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      – SubjectFull: Algebraic equations
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      – SubjectFull: Linear operators
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      – SubjectFull: Riccati equation
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      – SubjectFull: Carleman theorem
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              Text: Feb2022
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