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Constrained LQR problems in elliptic distributed control systems with point observations—convergence results

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Abstract

In this paper constrained LQR problems in distributed control systems governed by the elliptic equation with point observations are studied. A variational inequality approach coupled with potential theory in a Banach space setting is adopted. First the admissible control set is extended to be bounded by two functions, and feedback characterization of the optimal control in terms of the optimal state is derived; then two numerical algorithms proposed in [5] are modified, and the strong convergence and uniform convergence in Banach space are proved. This verifies that the numerical algorithm is insensitive to the partition number of the boundary. Since our control variables are truncated below and above by two functions inL p and in our numerical computation only the layer density not the control variable is assumed to be piecewise smooth, uniform convergence guarantees a better convergence. Finally numerical computation for an example is carried out and confirms the analysis.

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References

  1. Armijo L (1966) Minimization of functionals having continuous partial derivatives. Pacific J Math 16: 1–3

    MATH  MathSciNet  Google Scholar 

  2. Casas E (1986) Control of an elliptic problem with pointwise state constraints. SIAM J Control Optim 24: 1309–1318

    Article  MATH  MathSciNet  Google Scholar 

  3. Casas E (1993) Boundary control of semilinear elliptic equations with pointwise state constraints. SIAM J Control Optim 31: 993–1006

    Article  MATH  MathSciNet  Google Scholar 

  4. Chen G, Zhou J (1992) Boundary Element Methods. Academic Press, London

    MATH  Google Scholar 

  5. Ding Z, Ji L, Zhou J (1996) Constrained LQR problems in elliptic distributed control systems with point observations. SIAM J Control Optim 74: 264–294

    Article  MathSciNet  Google Scholar 

  6. Ding Z, Zhou J (1995) Constrained LQR problems governed by the potential equation on Lipschitz domain with point observations. J Math Pures Appl 74: 1–28

    MathSciNet  Google Scholar 

  7. Dunn JC (1981) Global and asymptotic convergence rate estimates for a class of projected gradient processes. SIAM J Control Optim 19: 368–400

    Article  MATH  MathSciNet  Google Scholar 

  8. Fabes EB, Jodeit M Jr, Riviere NM (1978) Potential techniques for boundary value problems onC 1-domains. Acta Math 141: 165–186

    Article  MATH  MathSciNet  Google Scholar 

  9. Gilbarg D, Trudinger NS (1983) Elliptic Partial Differential Equations of Second Order. Springer-Verlag, New York

    MATH  Google Scholar 

  10. Grisvard P (1985) Elliptic Problems in Nonsmooth Domains. Pitman, Boston

    MATH  Google Scholar 

  11. Goldstein AA (1965) On steepest descent. SIAM J Control Ser A 3: 147–151

    Article  Google Scholar 

  12. Goldstein AA (1974) On gradient projection. Proc 12th Allerton Conf Circuit and System Theory, University of Illinois, pp. 38–40

  13. Hsiao G, MacCamy RC (1973) Solutions of boundary value problems by integral equations of the first kind. SIAM Rev 15: 687–705

    Article  MATH  MathSciNet  Google Scholar 

  14. Ji L, Chen G (1993) Point observation in linear quadratic elliptic distributed control systems. Proc AMS Summer Conf Control and Identification of PDEs, SIAM, Philadelphia, PA, pp 155–170

    Google Scholar 

  15. Kenig CE (1985) Recent Progress on Boundary-Value Problems on Lipschitz Domains. Proceedings of Symposia in Pure Mathematics, Vol 43. American Mathematical Society, Providence, RI, pp 175–205

    Google Scholar 

  16. Lee CE, Taylor HE (1994) Fiber-optic Fabry-Perot temperature sensor using a low-coherence light source. J Lightwave Technol 9: 129–134

    Article  Google Scholar 

  17. Lions JL (1968) Contrôle optimal des sysèms gouvernés par deséquations aux dèrivèes partielles. Dunod, Gauthier-Villars, Paris

    Google Scholar 

  18. Lions JL, Magenes E (1970) Nonhomogeneous Boundary Value Problems and Applications, Vol 1, Springer-Verlag, New York

    Google Scholar 

  19. Verchota G (1982) Layer potentials and boundary value problems for Laplace’s equation on Lipschitz domains. PhD thesis, University of Minnesota

  20. Verchota G (1984) Layer potentials and regularity for the Dirichlet problems for Laplace’s equation in Lipschitz domains. J Funct Anal 59: 572–611

    Article  MATH  MathSciNet  Google Scholar 

  21. Wang MC (1995) Fiber-optic Fabry—Perot temperature and dynamic sensor system using a low-soherence LED light source. PhD dissertation, Department of Electrical Engineering, Texas A&M University

  22. Wlodarczyk MT, He G (1994) A fiber-optic combustion pressure sensor system for automotive engine control. Sensors 11(6): 35–42

    Google Scholar 

  23. Zamani NG, Chuang JM (1987) Optimal control of current in a cathodic protection system: a numerical investigation. Optimal Control Appl Methods 8: 339–350

    Article  MATH  MathSciNet  Google Scholar 

  24. Zamani NG, Porter JF, Mufti AA (1986) A survey of computational efforts in the field of corrosive engineering. J Numer Methods Engrg 23: 1295–1311

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Texas A&M University, 77843, College Station, TX, USA

    Z. Ding & J. Zhou

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  1. Z. Ding
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  2. J. Zhou
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Communicated by I. Lasiecka

This research was supported in part by NSF Grant DMS-9404380 and by an IRI Award of Texas A&M University. The current address of the first author is the Department of Mathematical Science, University of Nevada at Las Vegas, Las Vegas, NV 89154, USA.

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Ding, Z., Zhou, J. Constrained LQR problems in elliptic distributed control systems with point observations—convergence results. Appl Math Optim 36, 173–201 (1997). https://doi.org/10.1007/BF02683342

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