# 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. [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. [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. [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. [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. [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. [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. [8] M. Yu. Romanovsky and Yu. M. Romanovsky, Introduction to Econophysics. Statistical and Dynamic Models, Moscow, IKI, 2012. [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. [10] F. P. Vasil'ev, Numerical Methods of Solving Extremal Problems, Nauka, Moscow, 1980. [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.

show all references

##### References:
 [1] A. I. Egorov, Optimal Control for Linear Systems, Kyiv, Naukova dumka, 1988. [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. [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. [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. [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. [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. [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. [8] M. Yu. Romanovsky and Yu. M. Romanovsky, Introduction to Econophysics. Statistical and Dynamic Models, Moscow, IKI, 2012. [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. [10] F. P. Vasil'ev, Numerical Methods of Solving Extremal Problems, Nauka, Moscow, 1980. [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.
 [1] Rafael Abreu, Cristian Morales-Rodrigo, Antonio Suárez. Some eigenvalue problems with non-local boundary conditions and applications. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2465-2474. doi: 10.3934/cpaa.2014.13.2465 [2] Nikolai Dokuchaev. On forward and backward SPDEs with non-local boundary conditions. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5335-5351. doi: 10.3934/dcds.2015.35.5335 [3] Lu Yang, Meihua Yang, Peter E. Kloeden. Pullback attractors for non-autonomous quasi-linear parabolic equations with dynamical boundary conditions. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2635-2651. doi: 10.3934/dcdsb.2012.17.2635 [4] Zhaoquan Xu, Jiying Ma. Monotonicity, asymptotics and uniqueness of travelling wave solution of a non-local delayed lattice dynamical system. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 5107-5131. doi: 10.3934/dcds.2015.35.5107 [5] Alessio Fiscella, Enzo Vitillaro. Local Hadamard well--posedness and blow--up for reaction--diffusion equations with non--linear dynamical boundary conditions. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5015-5047. doi: 10.3934/dcds.2013.33.5015 [6] Lucas Bonifacius, Ira Neitzel. Second order optimality conditions for optimal control of quasilinear parabolic equations. Mathematical Control & Related Fields, 2018, 8 (1) : 1-34. doi: 10.3934/mcrf.2018001 [7] Kareem T. Elgindy. Optimal control of a parabolic distributed parameter system using a fully exponentially convergent barycentric shifted gegenbauer integral pseudospectral method. Journal of Industrial & Management Optimization, 2018, 14 (2) : 473-496. doi: 10.3934/jimo.2017056 [8] Walter Allegretto, Yanping Lin, Shuqing Ma. On the box method for a non-local parabolic variational inequality. Discrete & Continuous Dynamical Systems - B, 2001, 1 (1) : 71-88. doi: 10.3934/dcdsb.2001.1.71 [9] Shu Luan. On the existence of optimal control for semilinear elliptic equations with nonlinear neumann boundary conditions. Mathematical Control & Related Fields, 2017, 7 (3) : 493-506. doi: 10.3934/mcrf.2017018 [10] Getachew K. Befekadu, Eduardo L. Pasiliao. On the hierarchical optimal control of a chain of distributed systems. Journal of Dynamics & Games, 2015, 2 (2) : 187-199. doi: 10.3934/jdg.2015.2.187 [11] Stig-Olof Londen, Hana Petzeltová. Convergence of solutions of a non-local phase-field system. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 653-670. doi: 10.3934/dcdss.2011.4.653 [12] Hongjie Dong, Doyoon Kim. Schauder estimates for a class of non-local elliptic equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (6) : 2319-2347. doi: 10.3934/dcds.2013.33.2319 [13] A. V. Bobylev, Vladimir Dorodnitsyn. Symmetries of evolution equations with non-local operators and applications to the Boltzmann equation. Discrete & Continuous Dynamical Systems - A, 2009, 24 (1) : 35-57. doi: 10.3934/dcds.2009.24.35 [14] Yuanhong Wei, Xifeng Su. On a class of non-local elliptic equations with asymptotically linear term. Discrete & Continuous Dynamical Systems - A, 2018, 38 (12) : 6287-6304. doi: 10.3934/dcds.2018154 [15] William G. Litvinov. Optimal control of electrorheological clutch described by nonlinear parabolic equation with nonlocal boundary conditions. Journal of Industrial & Management Optimization, 2011, 7 (2) : 291-315. doi: 10.3934/jimo.2011.7.291 [16] Qinqin Zhang. Homoclinic orbits for discrete Hamiltonian systems with local super-quadratic conditions. Communications on Pure & Applied Analysis, 2019, 18 (1) : 425-434. doi: 10.3934/cpaa.2019021 [17] Ciro D'Apice, Olha P. Kupenko, Rosanna Manzo. On boundary optimal control problem for an arterial system: First-order optimality conditions. Networks & Heterogeneous Media, 2018, 13 (4) : 585-607. doi: 10.3934/nhm.2018027 [18] Elena Goncharova, Maxim Staritsyn. Optimal control of dynamical systems with polynomial impulses. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4367-4384. doi: 10.3934/dcds.2015.35.4367 [19] Lijuan Wang, Qishu Yan. Optimal control problem for exact synchronization of parabolic system. Mathematical Control & Related Fields, 2019, 9 (3) : 411-424. doi: 10.3934/mcrf.2019019 [20] Igor Chueshov, Björn Schmalfuss. Qualitative behavior of a class of stochastic parabolic PDEs with dynamical boundary conditions. Discrete & Continuous Dynamical Systems - A, 2007, 18 (2&3) : 315-338. doi: 10.3934/dcds.2007.18.315

2018 Impact Factor: 1.008