On boundary optimal control of a coefficient in a nonlinear parabolic equation


  • Nataliya L. Gol’dman


parabolic equations
optimal control with final observation
conjugate problem
applications for physical-chemical processes


The work is connected with investigation of nonlinear parabolic systems arising in the mathematical modeling and control of physical-chemical processes in which inner properties of materials are subjected to changes. We consider optimal control in one of such systems that involves a boundary value problem of the third kind for a quasilinear parabolic equation with an unknown coefficient at the time derivative and, moreover, an additional equation for a time dependence of this coefficient. The optimal problem with a boundary control regime is justified for the given final observation of the sought coefficient. The exact representation for the differential of the minimization functional in terms of the solutions of the conjugate problem is obtained. The form of this conjugate problem and conditions of unique solvability in a class of smooth functions are shown. The obtained results are important for applications in various technical fields, medicine, geology, etc. Some examples of such applications are discussed.





Methods and algorithms of computational mathematics and their applications

Author Biography

Nataliya L. Gol’dman


  1. O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Ural’tseva, Linear and Quasilinear Equations of Parabolic Type (Nauka, Moscow, 1967; AMS Press, Providence, 1968).
  2. A. Friedman, Partial Differential Equations of Parabolic Type (Prentice Hall, Englewood Cliffs, 1964; Mir, Moscow, 1968).
  3. N. L. Gol’dman, “A Nonlinear Problem for a Parabolic Equation with anUnknown Coefficient at the Time Derivative and Its Applications inMathematical Models of Physical-Chemical Processes,” Vychisl. Metody Programm. 18, 247-266 (2017).
  4. N. L. Gol’dman, “Boundary Value Problems for a Quasilinear Parabolic Equation with an Unknown Coefficient,” J. Differ. Equations 266 (8), 4925-4952 (2019).
  5. J.-L. Lions, Optimal Control of Systems Governed by Partial Differential Equations. (Springer, Berlin, 1971; Mir, Moscow, 1972).
  6. N. L. Gol’dman, Inverse Stefan Problems (Kluwer, Dordrecht, 1997).
  7. N. L. Gol’dman, Inverse Stefan Problems. Theory and Methods of Solutions (Mosk. Gos. Univ., Moscow, 1999) [in Russian].
  8. S. M. Nikol’skii, Approximation of Functions of Several Variables and Imbedding Theorems (Nauka, Moscow, 1969; Springer, New York, 1975).
  9. L. V. Kantorovich and G. P. Akilov, Functional Analysis (Nauka, Moscow, 1977; Pergamon Press, New York, 1982).
  10. F. P. Vasil’ev, Optimization Methods , Vols. 1, 2 (MTsNMO, Moscow, 2011) [in Russian].
  11. A. N. Tikhonov, A. V. Goncharskii, V. V. Stepanov, and A. G. Yagola, Regularizing Algorithms and a Priori Information (Nauka, Moscow, 1983) [in Russian].
  12. S. F. Gilyazov and N. L. Gol’dman, Regularization of Ill-Posed Problems by Iteration Methods (Kluwer, Dordrecht, 2000).
  13. 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)].