We consider a problem of optimal control for parabolic-hyperbolic equations with nonlocal boundary conditions and semidefinite quality criterion. The optimality conditions are constructed by reducing the problem to a sequence of one-dimensional problems, the optimal control is obtained in a closed form, and its convergence is proved. The form of the quality criterion is substantiated.
Similar content being viewed by others
References
I. M. Gel’fand, “Some problems of analysis and differential equations,” Usp. Mat. Nauk, 14, No. 3, 3–19 (1959).
S. A. Beilin, “Existence of solutions for one-dimensional wave equations with nonlocal conditions,” Electron. J. Different. Equat., 56, 1–8 (2001).
M. Renardy, W. Hrusa, and J. A. Nohel, Mathematical Problems in Viscoelasticity, Longman, London (1987).
A. Ashyralyev and A. Yurtsever, “On a nonlocal boundary value problem for semilinear hyperbolic-parabolic equations,” Nonlin. Anal., 47, 3585–3592 (2001).
K. B. Sabitov, “Boundary-value problem for parabolic-hyperbolic equations with nonlocal boundary conditions,” Differents. Uravn., 46, No. 10, 1468–1478 (2010).
V. E. Kapustyan and I. A. Pyshnograev, “Conditions for the existence and uniqueness of the solution of a hyperbolic-parabolic equation with nonlocal boundary conditions,” Nauch. Vestn. Nats. Tekh. Univ. Ukr. “KPI,” 4, 72–86 (2012).
N. I. Ionkin, “Solution of one boundary-value problem of heat conduction with nonclassical boundary condition,” Differents. Uravn., 13, No. 2, 294–304 (1977).
J.-L. Lions, Quelques Mèthodes de Rèsolution des Problémes aux Limites Non Linéaires, Dunod, Paris (1969).
Author information
Authors and Affiliations
Additional information
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 67, No. 8, pp. 1068–1081, August, 2016.
Rights and permissions
About this article
Cite this article
Kapustyan, V.E., Pyshnograev, I.A. Problem of Optimal Control for Parabolic-Hyperbolic Equations with Nonlocal Point Boundary Conditions and Semidefinite Quality Criterion. Ukr Math J 67, 1204–1218 (2016). https://doi.org/10.1007/s11253-016-1146-0
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-016-1146-0