On a nonlinear parabolic problem with a boundary control and on its applications


  • N.L. Gol’dman


quasilinear parabolic equations
boundary value problem of the first kind
variational problems
final observation
boundary control
conjugate problems
unique solvability
mathematical models of thermodestruction


We consider the optimal control in a system consisting of the boundary value problem of the first kind for a quasilinear parabolic equation with an unknown coefficient and an additional equation describing a time dependence of this coefficient. Two variational problems with a boundary control regime are substantiated for the given final observations. Some conditions of continuity and differentiability of the corresponding minimization functionals are formulated and proved. An exact representation for the differentials in terms of the solutions of the conjugate problems is obtained. The form of these conjugate problems and their unique solvability in a class of smooth functions are shown. This study is connected with modeling and control of physical-chemical processes in which the inner properties of materials are subjected to changes.





Methods and algorithms of computational mathematics and their applications

Author Biography

N.L. Gol’dman


  1. N. L. Goldman, “On Some Statements of Nonlinear Parabolic Problems with Boundary Conditions of the First Kind and on Methods of Their Approximate Solution,” Vychisl. Metody Programm. 19, 314-326 (2018).
  2. O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Uraltseva, Linear and Quasi-linear Equations of Parabolic Type (Nauka, Moscow, 1967; AMS Press, Providence, 1968).
  3. A. Friedman, Partial Differential Equations of Parabolic Type (Prentice-Hall, Englewood Cliffs, 1964; Mir, Moscow, 1968).
  4. N. L. Gol’dman, Inverse Stefan Problems (Kluwer, Dordrecht, 1997).
  5. N. L. Gol’dman, Inverse Stefan Problems. Theory and Methods of Solution (Mosk. Gos. Univ., Moscow, 1999) [in Russian].
  6. S. M. Nikol’skii, Approximation of Functions of Several Variables and Imbedding Theorems (Nauka, Moscow, 1969; Springer, New York, 1975).
  7. J.-L. Lions, Optimal Control of Systems Governed by Partial Differential Equations (Springer, Berlin, 1971; Mir, Moscow, 1972).
  8. L. V. Kantorovich and G. P. Akilov, Functional Analysis (Nauka, Moscow, 1977; Pergamon, New York, 1982).
  9. F. P. Vasil’ev, Optimization Methods , Vols. 1, 2 (MTsNMO, Moscow, 2011) [in Russian].
  10. C. Ciliberto, “Formule di Maggiorazione e Teoremi di Esistenza per Soluzioni delle Equazioni Paraboliche in Due Varabili,” Ricerche Mat. 3, 40-75 (1954).
  11. S. N. Kruzhkov, “A Priori Estimate for the Derivative of a Solution to a Parabolic Equation,” Vestn. Mosk. Univ., Ser. 1: Mat. Mekh., No. 2, 41-48 (1967).
  12. A. V. Goncharsky and A. G. Yagola, “The Uniform Approximation of a Monotonic Solution of Ill-Posed Problems,” Dokl. Akad. Nauk SSSR 184 (4), 771-773 (1969).
  13. V. A. Morozov, N. L. Gol’dman, and M. K. Samarin, “Method of Descriptive Regularization and Quality of Approximate Solutions,” Inzh. Fiz. Zh. 33 (6), 1117-1124 (1977) [J. Eng. Phys. 33 (6), 1503-1508 (1977)].
  14. A. N. Tikhonov, A. V. Goncharskii, V. V. Stepanov, and A. G. Yagola, Regularizing Algorithms and a Priori Information (Nauka, Moscow, 1983) [in Russian].
  15. V. V. Vasin and A. L. Ageev, Ill-Posed Problems with a Priori Information (Nauka, Yekaterinburg, 1993; VSP, Utrecht, 1995).
  16. S. F. Gilyazov and N. L. Gol’dman, Regularization of Ill-Posed Problems by Iteration Methods (Kluwer, Dordrecht, 2000).
  17. A. K. Alekseev, “On the Restoration of the Heating History of a Plate Made of a Thermodestructible Material from the Density Profile in the Final State,” Teplofiz. Vysok. Temp. 31 (6), 975-979 (1993) [High Temp. 31 (6), 897-901 (1993)].