Optimal control of a Vlasov–Poisson plasma by an external magnetic field

Abstract

The aim of various technical applications (for example fusion research) is to control a plasma by magnetic fields in a desired fashion. In our model the plasma is described by the Vlasov–Poisson system that is equipped with an external magnetic field. We will prove that this model satisfies some basic properties that are necessary for calculus of variations. After that, we will analyze an optimal control problem with a tracking type cost functional with respect to the following topics: necessary conditions of first order for local optimality, derivation of an optimality system, sufficient conditions of second order for local optimality, uniqueness of the optimal control under certain conditions.

This is a preview of subscription content, log in to check access.

References

  1. 1.

    Batt, J.: Global symmetric solutions of the initial value problem in Stellar dynamics. J. Differ. Equ. 25, 342–364 (1977)

    Article  Google Scholar 

  2. 2.

    Dragomir, S.S.: Some Gronwall Type Inequalities and Applications. Nova Science Publishers, Hauppauge (2003)

    Google Scholar 

  3. 3.

    Evans, L.C.: Partial Differential Equations, Grad. Stud. in Math. 19, 2nd edn. Amer. Math. Soc., Providence (2010)

    Google Scholar 

  4. 4.

    Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order, vol. 3. Springer, Berlin (2001)

    Google Scholar 

  5. 5.

    Kurth, R.: Das Anfangswertproblem der Stellardynamik. Z. Astrophys. 30, 213–229 (1952)

    MathSciNet  MATH  Google Scholar 

  6. 6.

    Kreuter, M.: Sobolev spaces of vector-valued functions, Master Thesis, Ulm University

  7. 7.

    Lieb, E.H., Loss, M.: Analysis, vol. 14, 2nd edn. Amer. Math. Soc., Providence (2001)

    Google Scholar 

  8. 8.

    Lions, P.-L., Perthame, B.: Propagation of moments and regularity for the 3-dimensional Vlasov–Poisson system. Invent. Math. 105, 415–430 (1991)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Pettis, B.J.: On integration in vector spaces. Trans. Am. Math. Soc. 44, 277–304 (1938)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Pfaffelmoser, K.: Global classical solutions of the Vlasov–Poisson system in three dimensions for general initial data. J. Differ. Equ. 95, 281–303 (1992)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Rein, G.: Collisionless kinetic equations from astrophysics: the Vlasov-Poisson system. In: Dafermos, C.M., Feireisl, E. (eds.) Handbook of Differential Equations: Evolutionary Equations, vol. 3. Elsevier, Amsterdam (2007)

    Google Scholar 

  12. 12.

    Schaeffer, J.: Global existence of smooth solutions to the Vlasov–Poisson system in three dimensions. Commun. Part. Differ. Equ. 16, 1313–1335 (1991)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Stein, E.: Singular Integrals and Differentiability Properties of Functions. Princeton Univ. Press, Princeton (1970)

    Google Scholar 

  14. 14.

    Yosida, K.: Functional Analysis, vol. 6. Springer, Berlin (1980)

    Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Patrik Knopf.

Additional information

Communicated by L. Ambrosio.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Knopf, P. Optimal control of a Vlasov–Poisson plasma by an external magnetic field. Calc. Var. 57, 134 (2018). https://doi.org/10.1007/s00526-018-1407-x

Download citation

Mathematics Subject Classification

  • 49J20
  • 35Q83