NECESSARY AND SUFFICIENT CONDITIONS OF OPTIMALITY FOR HYPERBOLIC EQUATION WITH NON-LOCAL BOUNDARY CONDITIONS
DOI:
https://doi.org/10.5281/3kyrgd05Abstract
The theory of boundary value problems for hyperbolic equations
is one of the most important sections of the theory of partial differential equations, which is explained both by the theoretical significance of the results and
by the presence of their practical applications in gas dynamics, in the theory
of infinitesimal bendings of surfaces, in the momentless theory of shells, in
magnetic hydrodynamics, mathematical biology and other areas.
In the XX century in connection with applied technical and economic needs,
it became necessary to solve control and optimization problems. The most famous works in this area are J.I. S. Pontryagin and his students V. G. Boltyanskiy, R. V. Gamkrelidze, E. F. Mishchenko, studied issues of control of processes described by systems of ordinary differential equations [18,19].
Further development of applied research leads to the need to control objects, the behavior of which is described using methods with partial differential
equations. The corresponding control problems were considered in the works
of A.G Butkovsky and co-authors [3,4], A.I Egorov [6-9], J.-L. Lyons [12], K.
A. Lurie [13], T. K. Sirazetdinov [20], as well as V. A. Ilyin and E. I. Moiseeva
[21,22], [14,15], A.T.Ramazanova [26], Kuliyev Hamlet [27].
Optimal control problems for the processes described by hyperbolic differential equations are encountered in various applications [1,2]. Such problems
have been most fully investigated in cases where the boundary conditions for
the equations of state are classical. However, there are numerous problems
in physics, technology, biology, etc., in which the processes are described by
hyperbolic equations, where the boundary conditions are nonclassical or nonlocal [23,24]. Among the nonlocal boundary value problems for the hyperbolic
equations , a special place is occupied by boundary value problems with integral boundary conditions [25]. Optimal control problems for hyperbolic-type
equations with nonlocal boundary conditions, including those with integral
boundary conditions, have been little studied.
References
Egorov A.I., Optimal processes in systems with distributed parameters and some problems
of the theory of invariance / AI Egorov // Izv. Academy of Sciences of the USSR, series of
mat. - (1965)1205-1260.
[8] Egorov A.I., Optimal control of thermal and diffusion processes. / A. I. Egorov - Moscow:
Nauka, (1978) 464 p.
[9] Egorov A.I ., Foundations of control theory. / A. I. Egorov - M .: FIZMATLIT, (2004) 504
p.
[10] Kowalewski A ., Time-optimal control of infinite order hyperbolic systems with time delays
/ A. Kowalewski // Int. J. Appl. Math. Comput. Sci.19(4), (2009) 597-608.
[11] Kozlova E.A ., The boundary control problem for a system of equations of hyperbolic type
/ EA Kozlova // Izv. Sarat. un-that. New ser. Ser. Maths. Mechanics. Computer science.
1(4.2) (2013) 51-56.
[12] Lyon J.-L ., Optimal control of systems described by partial differential equations. / J.-L.
Lyon - M .: MIR, (1972) 416 p.
[13] Lur’e K. A ., Optimal control in problems of mathematical physics. K. A. Lurie - Moscow:
Nauka, (1975) 480 p.
[14] Moiseev E.I ., Optimal boundary control of the displacement in the U / p string with a free
end / EI Moiseev // Differential equations.44(5), (2011) 709-711
[15] Moiseev E.I ., Optimal boundary control of displacement of vibrations of a string with a
nonlocal parity condition of the second kind / EI Moiseev, AA Holomeeva // Differential
Equations. 47(1)(2011) 127-134.
[16] Micu S ., An introduction to the controllability of partial differential equations. / S. Micu E.
Zuazua ., // ”Quelques questions de the’orie du contro’le”. Sari, T., ed., Collection Travaux
en Cours Hermann. - (2004) 69-157.
[17] Pawlow I., Boundary control of degenerate two-phase Stefan problems / I. Pawlow // Math.
Inst. Univ. Augsburg. - (1984) Prepr. 44 - pp. ten.
[18] Pontryagin L.S., Mathematical theory of optimal processes. / L. S. Pontryagin, V. G. Boltyansky, R. V. Gamkrelidze, E. F. Mishchenko - Moscow: Nauka, (1983)392 p.
[19] Pontryagin L.S ., Mathematical theory of optimal processes and differential games / LS
Pontryagin // Tr. MI AN. - (1985) 119-158.
[20] Sirazetdinov T. K., Optimization of systems with distributed parameters. / T.K.Sirazetdinov
- Moscow: Nauka, (1977) 480 p.
[21] Ilyin V.A., On boundary control at one end of the process described by the telegraph equation
/ V.A Ilyin, E.I Moiseev // Dokl. RAS. - (2002) - T. 387, No. 5.600-603.
[22] Ilyin V.A., Boundary control of the process of vibrations of a string at one end of it with
a fixed second end and under the condition of the existence of finite energy / V.A Ilyin //
Dokl. RAS. 378(6), (2019)743-747
[23] Yusubov Sh.Sh., On an optimality of the singular with respect to components controls in the
Goursat-Darboux systems. Probl. Upr., 5, (2014) 2-6.
[24] Yusubov Sh.Sh., Nonlocal Problem with integral conditions for a high-order hyperbolic equation. Ukrainian Mathematical Journal, 69 (1), (2017) 148-160.
[25] Ramazanova Aysel T., Necessary and Sufficient optimality conditions in an optimal control
problem with nonlocal conditions,19th French-German-Swiss conference on Optimization 17-
20 Sep 2019 Nice (France),https://fgs-2019.sciencesconf.org/program (2019).
[26] Ramazanova Aysel ., On Determining Initial Conditions of Equations Flexural-Torsional
Vibrations of a Bar, European Journal of Pure and Applied Mathematics, 12(1),(2019)
25–38.
[27] Ramazanova Aysel.T , Hamlet F Kuliyev, Arnd Roesch., An inverse problem for determining
right hand side of equations for hyperbolic equation of fourth order,Advances in Differential
Equations and Control Processes,20(2) ,(2019)143-16.
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