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Constrained Optimal Control for a Class of Semilinear Infinite Dimensional Systems

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Abstract

The aim of this paper is to investigate a constrained optimal control problem governed by a class of semilinear infinite dimensional systems. For a state-quadratic cost functional and a closed convex set of admissible controls, the existence of an optimal control is proven, then this control is characterized for several cases of constraints. An algorithm is developed in order to compute the optimal control. The results are illustrated through simulations of a transport equation and a wave equation.

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References

  1. Ahmed NU, Ding X. A semilinear Mckean-Vlasov stochastic evolution equation in Hilbert space. Stoch Proc Appl 1995;60:65–85.

    Article  MathSciNet  MATH  Google Scholar 

  2. Ahmed NU, Xiang X. Necessary conditions of optiMality for infinite dimensional uncertain systems. Math Probl Eng 1995;1(3):179–191.

    Article  MATH  Google Scholar 

  3. Ahmed NU, Xiang X. Nonlinear boundary control of semilinear parabolic systems. SIAM J Control Optim 1996;34(2):473–490.

    Article  MathSciNet  MATH  Google Scholar 

  4. Ball JM, Marsden JE, Slemrod M. Controllability for distributed bilinear systems. SIAM J Contr Opt 1982;20(4):575–597.

    Article  MathSciNet  MATH  Google Scholar 

  5. Barbu V. Analysis and control of nonlinear infinite dimensional systems. Mathematics in science and engineering, Vol. 190. Academic press; 1993.

  6. Cannarsa P, Frankowska H, function Value. OptiMality condition for semilinear control problems. II Parabolic case. Appl Math Optim 1992;26:139–169.

    Article  MathSciNet  MATH  Google Scholar 

  7. Deng D, Wei W. Existence and stability analysis for nonlinear optimal control problems with 1-mean equicontinuous controls. J Ind Manam Optim 2015;11(4):1409–1422.

    Article  MathSciNet  MATH  Google Scholar 

  8. Dorn B. Flows in infinite networks - A semigroup approach. Tübingen: Phd thesis, Eberhard Karls Universität; 2008.

    MATH  Google Scholar 

  9. Fattorini HO, Frankowska H. Necessary conditions for infinite-dimensional control problems. Math Control Signal Systems 1991;4:41–67.

    Article  MathSciNet  MATH  Google Scholar 

  10. Kantorovich L, Akilov G. 1982. Functional Analysis, 2nd. Pergamon Press.

  11. Li X, Yong J. Optimal control theory for infinite dimensional systems. Systems & Control: Foundations & Applications. Birkhäuser Basel; 1995.

  12. Nowakowski A. Shape optimization of control problems described by wave equations. Control Cybern 2008;37(4):1045–1055.

    MathSciNet  MATH  Google Scholar 

  13. Raymond JP, Zidani H. Optimal control problem governed by a semilinear parabolic equation. Syst. Model. and Optim., part two, pp 211–217. US: Springer; 1996.

    Google Scholar 

  14. Trang TTH, Phat VN. finite-time stabilisation and \(h_{\infty }\) control of nonlinear delay systems via output feedback. J Ind Manam Optim 2016;12(1): 303–315.

    Article  MATH  Google Scholar 

  15. Yu J, Liu Z, Peng D, Xu D, Zhou Y. Existence and stability analysis of optimal control. Optim Contr Appl Meth 2014;35:721–729.

    Article  MathSciNet  MATH  Google Scholar 

Download references

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Correspondence to El Hassan Zerrik.

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Zerrik, E.H., Boukhari, N.E. Constrained Optimal Control for a Class of Semilinear Infinite Dimensional Systems. J Dyn Control Syst 24, 65–81 (2018). https://doi.org/10.1007/s10883-016-9358-z

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  • DOI: https://doi.org/10.1007/s10883-016-9358-z

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