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Linear-quadratic regulator with intermediate points for degenerate equations with unbounded operator

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Abstract

We study a linear-quadratic optimal control problem in a Hilbert space where the state equation is unsolvable with respect to the derivative and contains an unbounded operator. The performance index is the sum of the integral of a quadratic form with respect to the control and the state variable on a finite or infinite time interval and also quadratic forms with respect to the differences between the state variable values at fixed points and given values.

The optimal control is presented in the feedback form using the implicit differential operator Riccati equation or the operator equation of the Riccati type under a special symmetry condition for the solution. The minimal value of the minimized functional is calculated. Some illustrative examples are also given.

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References

  1. D. J. Bender and A. J. Laub. The linear-quadratic optimal regulator for descriptor systems. IEEE Trans. Automat. Control 32 (1987), No. 8, 672-688.

    Article  MathSciNet  MATH  Google Scholar 

  2. Yu. I. Berdyshev. On a problem of a sequential optimization without decomposition with respect to time [in Russian]. Kibernetika (1987), No. 4, 32-35.

  3. A. V. Dmitruk and A. M. Kaganovich. Maximum principle for optimal control problems with intermediate constraints [in Russian]. In: Nelineinaja dinamika i upravlenie (Ed. S. V. Emeljanov, C. K. Korovin), Fizmatlit, Moscow 6 (2008), 101-136.

  4. A. Favini and L. Pandolfi. A quadratic regulator problem related to identification problems and singular systems. SIAM J. Control Optim. 47 (2008), No. 2, 768-790. (http://calvino.polito.it/~lucipan/ricerca_html/03angl-6.pdf)

  5. A. Favini and A. Yagi. Degenerate differential equations in banach spaces. Marcel Dekker, Inc., New York, Basel, Hong Kong (1999).

    MATH  Google Scholar 

  6. T. Kato. Perturbations theory for linear operators [in Russian]. Mir, Moscow (1972).

    Google Scholar 

  7. G. A. Kurina. On operator Riccati equation unresolved with respect to derivative [in Russian]. Differencial. Uravnen. 22 (1986), No. 10, 1826-1829.

    MathSciNet  MATH  Google Scholar 

  8. G. A. Kurina. Control of a descriptor system in an infinite interval. Comput. System Sci. Int. 32 (1994), No. 6, 30-35.

    MathSciNet  Google Scholar 

  9. G. A. Kurina. On some linear-quadratic optimal control problems for descriptor systems. Research Reports in Mathematics. Department of Mathematics, Stockholm University (2006), No. 1. (http://www.math.su.se/reports/2006/1)

  10. G. A. Kurina and R. März. On linear-quadratic optimal control problems for time-varying descriptor systems. SIAM J. Control Optim. 42 (2004), No. 6, 2062-2077.

    Article  MathSciNet  MATH  Google Scholar 

  11. _____, Feedback solutions of optimal control problems with DAE constraints. SIAM J. Control Optim. 46 (2007), No. 4, 1277-1298.

  12. G. A. Kurina, E. V. Smirnova, and Y. Zhou. Optimal control problem in a feedback form for linear-quadratic problem with intermediate points [in Russian]. Trudy matematicheskogo fakulteta, Nauchnaya kniga, Voronezh (2007), 121-127.

  13. I. Lasiecka. Optimal control problems and Riccati equations for systems with unbounded controls and partially analytic generators-Applications to boundary and point control problems. Chapter in Functional Analytic Methods for Evolution Equations, Springer Verlag 1855 (2004), 313-371.

    Article  MathSciNet  Google Scholar 

  14. I. Lasiecka and R. Triggiani. Differential and algebraic Riccati equations with application to boundary /point control problems: Continuous theory and approximation theory. Lecture Notes in Control and Information Sciences 164 (1991).

  15. E. B. Lee and L. Markus. Foundations of optimal control theory. John Willey & Sons, Inc., New York, London, Sydney (1967).

    MATH  Google Scholar 

  16. D. L. Lukes and D. L. Russel. The quadratic criterion for distributed systems. SIAM J. Control 7 (1969), No. 1, 101-121.

    Article  MathSciNet  MATH  Google Scholar 

  17. A. S. Matveev and V. A. Yakubovich. Optimal control systems: Ordinary differential equations. Special problems [in Russian]. Saint Petersburg University (2003).

  18. A. G. Rutkas. On classification and properties of solutions of the equation Ax' + Bx = f(t) [in Russian]. Differencial. Uravnen. 25 (1989), No. 7, 1150-1155.

    MathSciNet  MATH  Google Scholar 

  19. E. R. Smolyakov. Unknown pages of optimal control history. Editorial URSS, Moscow (2002) (in Russian).

    Google Scholar 

  20. L. A. Vlasenko. Evolutionary models with implicit and degenerate differential equations [in Russian]. Sistemnye tehnologii, Dnepropetrovsk (2006).

    Google Scholar 

  21. J. Zabczyk. Mathematical control theory: An introduction. Birkhäuser, Boston-Basel-Berlin (1992).

    MATH  Google Scholar 

  22. Y. Zhou, M. Egerstedt, and C. Martin. Control theoretic splines with deterministic and random data. Proc. 44th IEEE Conf. Decision and Control and the European Control Conf., Seville, Spain (2005), 362-367.

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Authors and Affiliations

  1. Università degli Studi di Bologna, Bologna, Italy

    A. Favini

  2. Voronezh State University, Voronezh Institute of Law and Economics, Voronezh, Russia

    G. Kurina

Authors
  1. A. Favini
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  2. G. Kurina
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Correspondence to A. Favini.

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Favini, A., Kurina, G. Linear-quadratic regulator with intermediate points for degenerate equations with unbounded operator. J Dyn Control Syst 19, 95–121 (2013). https://doi.org/10.1007/s10883-013-9166-7

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  • DOI: https://doi.org/10.1007/s10883-013-9166-7

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