Automatic Control and Computer Sciences

, Volume 44, Issue 4, pp 179–190

Parametric optimal control problems with weighted L1-norm in the cost function

  • O. I. Kostyukova
  • E. A. Kostina
  • N. M. Fedortsova


In the paper, an optimal control problem with weighted L1-norm in the cost function is studied. The problem is considered as a parametric problem where L1-norm weight ratio is treated as a parameter. We analyze the dependence of solution to the mentioned optimization problem on values of the parameter. A theorem that describes properties of the solution under small parameter perturbations is proved. Differential properties of the solution are investigated. Under assumption that a solution to unperturbed problem is known, rules for construction of solutions to perturbed optimization problems are given.

Key words

optimal control problems differentiability of solution with respect to parameter weight ratio 


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  1. 1.
    Stadler, G., Elliptic Optimal Control Problems with L 1-Control Cost and Applications for the Placement of Control Devices, Computational Optimization and Applications, 2009, vol. 44, no. 2, pp. 159–181.MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Figueiredo, M.A.T., Nowak, R.D., and Wright, S.J., Gradient Projection for Sparse Reconstruction: Applications to Compressed Sensing and Other Inverse Problems, IEEE Journal of Selected Topics in Signal Processing, 2007, vol. 4, pp. 586–597.CrossRefGoogle Scholar
  3. 3.
    Fornasier, M. and Rauhut, H., Recovery Algorithms for Vector-Valued Data with Joint Sparsity Constraints, SIAM Journal on Numerical Analysis, 2008, vol. 46, no. 2, pp. 577–613.MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Daubechies, I., Fornasier, M., and Loris, I., Accelerated Projected Gradient Methods for Linear Inverse Problems with Sparsity Constraints, Journal of Fourier Analysis and Applications, 2008, vol. 14, nos. 5–6, pp. 764–792.MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Kostyukova, O.I. and Kurdina, M.A., Asymptotic Properties of Solutions of Parametric Optimal Control Problems with Varying Index of the Singular Arcs, Differential Equaltions, 2008, vol. 44, no. 11, pp. 1510–1522.MathSciNetGoogle Scholar
  6. 6.
    Bliss, G., Lectures on the Calculus of Variations, The University of Chicago Press, 1963.Google Scholar
  7. 7.
    Krawczyk, D. and Rudnicki, M., Regularization Parameter Selection in Discrete Ill-posed Problems—The Use of the u-Curve, Int. J. Appl. Math. Comput. Sci.. 2007, vol. 17, no. 2, pp. 157–164.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Allerton Press, Inc. 2010

Authors and Affiliations

  • O. I. Kostyukova
    • 1
  • E. A. Kostina
    • 2
  • N. M. Fedortsova
    • 3
  1. 1.Institute of MathematicsBelarus Academy of SciencesMinskBelarus
  2. 2.Marburg University, GermanyMarburgGermany
  3. 3.Belarusian State University of Computer Science and RadioelectronicsMinskBelarus

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