Advertisement

Lipschitz continuity of the value function in mixed-integer optimal control problems

  • Martin Gugat
  • Falk M. HanteEmail author
Original Article

Abstract

We study the optimal value function for control problems on Banach spaces that involve both continuous and discrete control decisions. For problems involving semilinear dynamics subject to mixed control inequality constraints, one can show that the optimal value depends locally Lipschitz continuously on perturbations of the initial data and the costs under rather natural assumptions. We prove a similar result for perturbations of the initial data, the constraints and the costs for problems involving linear dynamics, convex costs and convex constraints under a Slater-type constraint qualification. We show by an example that these results are in a sense sharp.

Keywords

Parametric optimal control Parametric switching control Parametric optimization Sensitivity Mixed-integer optimal control problems Optimal value function Lipschitz continuity 

Notes

Acknowledgements

This work was supported by the DFG Grant CRC/Transregio 154, projects C03 and A03. The authors thank the reviewers and the associate editor for the constructive suggestions.

References

  1. 1.
    Barbu V, Da Prato G (1983) Hamilton–Jacobi equations in Hilbert spaces. Research Notes in Mathematics, vol 86. Pitman (Advanced Publishing Program), BostonGoogle Scholar
  2. 2.
    Bensoussan A, Da Prato G, Delfour MC, Mitter SK (1992) Representation and control of infinite-dimensional systems. Systems & Control: Foundations & Applications. Birkhäuser, BostonGoogle Scholar
  3. 3.
    Bonnans JF, Shapiro A (2000) Springer series in operations research. Perturbation Analysis of Optimization Problems. Springer, New YorkCrossRefGoogle Scholar
  4. 4.
    Borwein JM, Zhuang D (1986) On fan’s minimax theorem. Math Progam 26:232–234MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cannarsa P, Frankowska H (1992) Value function and optimality conditions for semilinear control problems. Appl Math Optim 26(2):139–169MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Ekeland I, Turnbull T (1983) Infinite-dimensional optimization and convexity. Chicago Lectures in Mathematics. University of Chicago Press, ChicagoGoogle Scholar
  7. 7.
    Gauvin J, Dubeau F (1982) Differential properties of the marginal function in mathematical programming. Math Program Stud 19:101–119MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Gerdts M (2006) A variable time transformation method for mixed-integer optimal control problems. Opt Control Appl Meth 27(3):169–182MathSciNetCrossRefGoogle Scholar
  9. 9.
    Gugat M (1994) One-sided derivatives for the value function in convex parametric programming. Optimization 28(3–4):301–314MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Gugat M (1997) Parametric disjunctive programming: one-sided differentiability of the value function. J Optim Theory Appl 92(2):285–310MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Hante FM, Sager S (2013) Relaxation methods for mixed-integer optimal control of partial differential equations. Comput Optim Appl 55(1):197–225MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Iftime OV, Demetriou MA (2009) Optimal control of switched distributed parameter systems with spatially scheduled actuators. Autom J IFAC 45(2):312–323MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Mordukhovich BS, Nam NM, Yen ND (2009) Subgradients of marginal functions in parametric mathematical programming. Math Program 116(1–2, Ser. B):369–396MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Pazy A (1983) Applied mathematical sciences series. Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New YorkCrossRefGoogle Scholar
  15. 15.
    Rüffler F, Hante FM (2016) Optimal switching for hybrid semilinear evolutions. Nonlinear Anal Hybrid Syst 22:215–227MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Sager S (2009) Reformulations and algorithms for the optimization of switching decisions in nonlinear optimal control. J Process Control 19(8):1238–1247CrossRefGoogle Scholar
  17. 17.
    Vasudevan R, Gonzalez H, Bajcsy R, Sastry SS (2013) Consistent approximations for the optimal control of constrained switched systems–part 1: a conceptual algorithm. SIAM J Control Optim 51(6):4463–4483MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Vasudevan R, Gonzalez H, Bajcsy R, Sastry SS (2013) Consistent approximations for the optimal control of constrained switched systems–part 2: an implementable algorithm. SIAM J Control Optim 51(6):4484–4503MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Williams AC (1989) Marginal values in mixed integer linear programming. Math Program 44(1, (Ser. A)):67–75MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Zhu F, Antsaklis PJ (2015) Optimal control of hybrid switched systems: a brief survey. Discrete Event Dyn Syst 25(3):345–364MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Zuazua E (2011) Switching control. J Eur Math Soc 13:85–117MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag London 2016

Authors and Affiliations

  1. 1.Lehrstuhl für Angewandte Mathematik 2, Department MathematikFriedrich-Alexander Universität Erlangen-NürnbergErlangenGermany

Personalised recommendations