# American Institute of Mathematical Sciences

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. Kapustyan, O. 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. Kapustyan, O. V. Kapustyan, O. 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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##### 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. Kapustyan, O. 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. Kapustyan, O. V. Kapustyan, O. 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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