March  2019, 24(3): 1243-1258. doi: 10.3934/dcdsb.2019014

Quasi-optimal control with a general quadratic criterion in a special norm for systems described by parabolic-hyperbolic equations with non-local boundary conditions

1. 

Department of Mathematical Modelling of Economic System, Igor Sikorsky Kyiv Polytechnic Institute, 37, Peremohy ave., 03056, Kyiv, Ukraine

2. 

Faculty of Computer Science and Cybernetics, Taras Shevchenko National University of Kyiv, Volodymyrska Str. 60, 01033, Kyiv, Ukraine

* Corresponding author: I. O. Pyshnograiev

Received  November 2017 Revised  March 2018 Published  January 2019

In this work, we consider a dynamical system generated by a parabolic-hyperbolic equation with non-local boundary conditions. The optimal control problem for this system is studied using a notion of quasi-optimal solution. Existence and uniqueness of quasi-optimal control are proved.

Citation: Volodymyr O. Kapustyan, Ivan O. Pyshnograiev, Olena A. Kapustian. Quasi-optimal control with a general quadratic criterion in a special norm for systems described by parabolic-hyperbolic equations with non-local boundary conditions. Discrete & Continuous Dynamical Systems - B, 2019, 24 (3) : 1243-1258. doi: 10.3934/dcdsb.2019014
References:
[1]

A. I. Egorov, Optimal Control for Linear Systems, Kyiv, Naukova dumka, 1988.Google Scholar

[2]

V. O. Kapustyan and I. O. Pyshnograiev, The conditions of existence and uniqueness of the solution of a parabolic-hyperbolic equation with nonlocal boundary conditions (Ukrainian), Science News NTUU ``KPI", 4 (2012), 72-86. Google Scholar

[3]

V. O. Kapustyan and I. O. Pyshnograiev, Distributed control with the general quadratic criterion in a special norm for systems described by parabolic-hyperbolic equations with nonlocal boundary conditions, Cybernetics and Systems Analysis, 51 (2015), 438-447. doi: 10.1007/s10559-015-9735-8. Google Scholar

[4]

V. O. Kapustyan and I. O. Pyshnograiev, Approximate optimal control for parabolic-hyperbolic equations with nonlocal boundary conditions and general quadratic quality criterion, Advances in Dynamical Systems and Control. Springer International Publishing, 69 (2016), 387-401. Google Scholar

[5]

V. O. KapustyanO. A. Kapustian and O. K. Mazur, Problem of optimal control for the Poisson equation with nonlocal boundary conditions, Journal of Mathematical Sciences, 201 (2014), 325-334. doi: 10.1007/s10958-014-1992-y. Google Scholar

[6]

V. O. KapustyanO. V. KapustyanO. A. Kapustian and O. K. Mazur, The optimal control problem for parabolic equation with nonlocal boundary conditions in circular sector, Continuous and Distributed Systems II. Springer International Publishing, 30 (2015), 297-314. doi: 10.1007/978-3-319-19075-4_18. Google Scholar

[7]

V. S. Mel'nik and M. Z. Zgurovsky, Nonlinear Analysis and Control of Physical Processes and Fields, Berlin, Springer, 2004. doi: 10.1007/978-3-642-18770-4. Google Scholar

[8]

M. Yu. Romanovsky and Yu. M. Romanovsky, Introduction to Econophysics. Statistical and Dynamic Models, Moscow, IKI, 2012.Google Scholar

[9]

P. N. Vabishchevich and A. A. Samarskii, Solving the problems of the dynamics of an incompressible fluid with alternating viscosity, Zhurnal Vychislitel'noi Matematiki i Matematicheskoi Fiziki, 12 (2000), 1813-1822. Google Scholar

[10]

F. P. Vasil'ev, Numerical Methods of Solving Extremal Problems, Nauka, Moscow, 1980. Google Scholar

[11]

L. A. Zolina, On a boundary-value problem for a model equation of hyperbolas-parabolic type, Zhurnal Vychislitel'noi Matematiki i Matematicheskoi Fiziki, 6 (1966), 991-1001. Google Scholar

show all references

References:
[1]

A. I. Egorov, Optimal Control for Linear Systems, Kyiv, Naukova dumka, 1988.Google Scholar

[2]

V. O. Kapustyan and I. O. Pyshnograiev, The conditions of existence and uniqueness of the solution of a parabolic-hyperbolic equation with nonlocal boundary conditions (Ukrainian), Science News NTUU ``KPI", 4 (2012), 72-86. Google Scholar

[3]

V. O. Kapustyan and I. O. Pyshnograiev, Distributed control with the general quadratic criterion in a special norm for systems described by parabolic-hyperbolic equations with nonlocal boundary conditions, Cybernetics and Systems Analysis, 51 (2015), 438-447. doi: 10.1007/s10559-015-9735-8. Google Scholar

[4]

V. O. Kapustyan and I. O. Pyshnograiev, Approximate optimal control for parabolic-hyperbolic equations with nonlocal boundary conditions and general quadratic quality criterion, Advances in Dynamical Systems and Control. Springer International Publishing, 69 (2016), 387-401. Google Scholar

[5]

V. O. KapustyanO. A. Kapustian and O. K. Mazur, Problem of optimal control for the Poisson equation with nonlocal boundary conditions, Journal of Mathematical Sciences, 201 (2014), 325-334. doi: 10.1007/s10958-014-1992-y. Google Scholar

[6]

V. O. KapustyanO. V. KapustyanO. A. Kapustian and O. K. Mazur, The optimal control problem for parabolic equation with nonlocal boundary conditions in circular sector, Continuous and Distributed Systems II. Springer International Publishing, 30 (2015), 297-314. doi: 10.1007/978-3-319-19075-4_18. Google Scholar

[7]

V. S. Mel'nik and M. Z. Zgurovsky, Nonlinear Analysis and Control of Physical Processes and Fields, Berlin, Springer, 2004. doi: 10.1007/978-3-642-18770-4. Google Scholar

[8]

M. Yu. Romanovsky and Yu. M. Romanovsky, Introduction to Econophysics. Statistical and Dynamic Models, Moscow, IKI, 2012.Google Scholar

[9]

P. N. Vabishchevich and A. A. Samarskii, Solving the problems of the dynamics of an incompressible fluid with alternating viscosity, Zhurnal Vychislitel'noi Matematiki i Matematicheskoi Fiziki, 12 (2000), 1813-1822. Google Scholar

[10]

F. P. Vasil'ev, Numerical Methods of Solving Extremal Problems, Nauka, Moscow, 1980. Google Scholar

[11]

L. A. Zolina, On a boundary-value problem for a model equation of hyperbolas-parabolic type, Zhurnal Vychislitel'noi Matematiki i Matematicheskoi Fiziki, 6 (1966), 991-1001. Google Scholar

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