Volume 3 (2), December 2020, Pages 207-222

Samir Z. Guliyev

High Performance Computing Research Advance Center, Department of General and Applied Mathematics, Azerbaijan State Oil and Industry University, Baku, Azerbaijan; Institute of Control Systems of Azerbaijan National Academy of Sciences, Baku, Azerbaijan, This email address is being protected from spambots. You need JavaScript enabled to view it.


The paper studies the problem of synthesis of control of lumped sources for an object with distributed parameters based on discrete observation of the phase state at specific object points. We propose an approach in which the whole phase space at the observed points is preliminarily divided in some way into given subsets (zones). The synthesized controls are selected from the class of piecewise-constant functions, and their current values are determined by a subset of the phase space containing the population of current states of the object at the observed points, at which controls take constant values. Such synthesized controls are called zonal. We give a numerical technique for obtaining optimal values of zonal controls using efficient first-order optimization methods. To this purpose, we derive formulas for the gradient of the objective function in the space of zonal controls.


Synthesis of Control, Zonal Control, Feedback, Distributed System, Heat Equation, the Gradient of Functional.





Aida-Zade, K. R., Kuliev, S. Z. (2008). A class of inverse problems for discontinuous systems. Cybernetics and Systems Analysis, 44(6), 915-924.

Aida-Zade, K. R., Kuliev, S. Z. (2011). Numerical solution of nonlinear inverse coefficient problems for ordinary differential equations. Computational Mathematics and Mathematical Physics, 51(5), 803-815.

Aida-Zade, K. R., Kuliev, S. Z. (2012). On numerical solution of one class of inverse problems for discontinuous dynamic systems. Automation and Remote Control, 73(5), 786-796.

Arthur, E., Bryson, Yu-Chi Ho. (1975) Applied optimal control: Optimization, estimation and control. CRC Press; 1st edt., 1975.

Bazaraa, M. S., Sherali, H. D., & Shetty, C. M. (2013). Nonlinear programming: theory and algorithms. John Wiley & Sons.

Butkovsky, A.G. (1975) Methods for controlling systems with distributed parameters. Moscow: Nauka. (in Russian)

Deng, Y. (2012). Applied parallel computing. World Scientific.

Dongarra, J., Foster, I., Fox, G., Gropp, W., Kennedy, K., Torczon, L., & White, A. (2003). Sourcebook of parallel computing (Vol. 3003). San Francisco eCA CA: Morgan Kaufmann Publishers.

Egorov, A.I. (2004) Foundations of control theory. Moscow: Fizmatlit. (in Russian)

Fursikov, A.V. (1999) Optimal control of distributed systems: Theory and applications. American Mathematical Society.

Guliyev, S. Z. (2013). Synthesis of control in nonlinear systems with different types of feedback and strategies of control. Journal of Automation and Information Sciences, 45(7), 74-86.

Guliyev, S. Z. (2018). Numerical solution of a zonal feedback control problem for the heating process. IFAC-PapersOnLine, 51(30), 251-256.

Guliyev, S.Z., Aida-zade, K.R. (2005) Optimization of location and operation modes of oilfield wells. Computational Technologies SB RAS, 10 (4), 52-62. (in Russian)

Guliyev, S.Z., Aida-zade, K.R. (2016) Hydraulic resistance coefficient identification in pipelines. Automation and Remote Control, 77 (7), 1225–1239.

Itkis, U. (1976). Control systems of variable structure. Halsted Press.

Kuliev, S. Z. (2011). Synthesis of zonal controls of nonlinear systems under discrete observations. Automatic Control and Computer Sciences, 45(6), 338-345.

Lions, J.L. (1971) Optimal control of systems governed by partial differential equations. Springer-Verlag (Berlin).

Lurie, K.A. (1993) Applied optimal control theory of distributed systems. Springer US.

Moiseev, N.N. (1971) Numerical methods in the theory of optimal systems. Moscow: Nauka. (in Russian)

Nocedal, J., Wright, S. (2006). Numerical optimization. Springer Science & Business Media.

Panteleev, A.V. Letova, T.A. (2015) Optimization methods in examples and problems. Saint-Petersburg: Lan Publishing. (in Russian)

Rapoport, E.Ya. (2009) Optimal control of systems with distributed parameters. Moscow: Higher school. (in Russian)

Sirazetdinov, T.K. (1977) Optimization of systems with distributed parameters. Moscow: Nauka. (in Russian)

Stewart, D. E., Anitescu, M. (2010). Optimal control of systems with discontinuous differential equations. Numerische Mathematik, 114(4), 653-695.

Vasiliev, F.P. (2002) Optimization methods. Moscow: Factorial Press, 2002. (in Russian)