OPTIMAL CONTROL FOR SYSTEMS DESCRIBED BY HYPERBOLIC EQUATION WITH STRONG NONLINEARITY

Simon Serovajsky

doi:10.11948/2013014

2013 Volume 3 Issue 2

Article Contents

Article navigation > Journal of Applied Analysis & Computation > 2013 > 3(2): 183-195

Next Article Previous Article

Simon Serovajsky. OPTIMAL CONTROL FOR SYSTEMS DESCRIBED BY HYPERBOLIC EQUATION WITH STRONG NONLINEARITY[J]. Journal of Applied Analysis & Computation, 2013, 3(2): 183-195. doi: 10.11948/2013014

Citation:

Simon Serovajsky. OPTIMAL CONTROL FOR SYSTEMS DESCRIBED BY HYPERBOLIC EQUATION WITH STRONG NONLINEARITY[J]. Journal of Applied Analysis & Computation, 2013, 3(2): 183-195. doi: 10.11948/2013014

OPTIMAL CONTROL FOR SYSTEMS DESCRIBED BY HYPERBOLIC EQUATION WITH STRONG NONLINEARITY

Simon Serovajsky

Differential Equations and Control Theory Department, al-Farabi Kazakh National University, 71 al Farabi av., 050078 Almaty, Kazakhstan

Abstract

Abstract

The optimization control problem for a hyperbolic equation is considered. The system is nonlinear with respect to the state derivative. The regularization technique for the state equation is applied. The necessary conditions of optimality for the regularized control problem are proved. It uses the extended differentiability of the control-state mapping for the regularized equation. The convergence of the regularization method is proved. Therefore the optimal control for the regularized problem with small enough regularization parameter can be chosen as an approximate solution of the initial optimization problem.
- Optimization /
- nonlinear hyperbolic equation /
- regularization /
- extended derivative /
- variational inequality
MSC: 49K20

FullText(HTML)

References

[1]	P. Burgmeier and H. Jasinski, Uber die Auwendung des Algorithmus von Krylov und Chernous'ko and Problems der Optimalen Steurung von Systemen mit verteilten Parametern, Wiss. Z. TH Leuna-Mereseburg, 21(1981), 470-476. Google Scholar
[2]	F.L. Chernous'ko and V.V. Kolmanovsky, Computing and approximate optimization methods, Math. Anal. Itogi nauki i texniki, 14(1977), 101-166. Google Scholar
[3]	I. Ekeland and R. Temam, Convex Analysis and Variational Problems, NorthHolland Publish. Comp.; New York, American Elsevier Publish. Comp., Inc., Amsterdam-Oxford, 1976. Google Scholar
[4]	A.V. Fursikov, Optimal control of distributed systems. Theory and applications, Amer. Math. Soc., Providence, 1999. Google Scholar
[5]	V.S. Gavrilov and M.I. Sumin, Parametric suboptimal control problem of a system, described by Goursat-Darboux system with pointwise constraints, Izv. Vuzov. Math., 6(2005), 40-52. Google Scholar
[6]	J. Ha and S. Nakagiri, Optimal control problem for nonlinear hyperbolic distributed parameter systems with damping terms, Funct. Equat., 47(2004), 1-23. Google Scholar
[7]	L.V. Kantorovich and G.P. Akilov, Functional analysis, Nauka, Moscow, 1977. Google Scholar
[8]	S.G. Krein, etc., Functional analysis, Nauka, Moscow, 1972. Google Scholar
[9]	J.L. Lions, Controle Optimal de Systemes Gouvernes par des equations aux Derivees Partielles, Dunod, Gauthier-Villars, Paris, 1968. Google Scholar
[10]	J.L. Lions, Controle de systemes distribues singuliers, Gauthier-Villars, Paris, 1983. Google Scholar
[11]	J.L. Lions, Quelques methods de resolution des problemes aux limites non lineaires, Dunod, Gauthier-Villars, Paris, 1969. Google Scholar
[12]	S. Matveev and V. Yakubovich, Abstract optimization control theory. SP Univ., SP, 1994. Google Scholar
[13]	N.I. Pogodaev, On solutions of the goursat-Darboux system with boundary controls and distributed controls, Differential Equations, 43(2007), 1142-1152. Google Scholar
[14]	M.B. Suryanarayana, Necessary conditions for optimal problems with hyperbolic partial differential equations, SIAM J. Control, 11(1973), 130-147. Google Scholar
[15]	S.Ya. Serovajsky, Optimization for nonlinear hyperbolic equation without uniqueness theorem for the solution of the boundary problem, Izv. Vuzov. Math., 1(2009), 76-83. Google Scholar
[16]	S.Ya. Serovajsky, Regularization method for the optimization control problem for a nonlinear hyperbolic equation, Differential Equations, 28(1992), 2187-2190. Google Scholar
[17]	S.Ya. Serovajsky, Calculation of functional gradients and extended differentiation of operators, Journal of Inverse and Ill-Posed Problems. 13(2005), 383-396. Google Scholar
[18]	D. Tiba, Optimal control for second order semilinear hyperbolic equations, Contr. Theory Adv. Techn., 3(1987), 33-46. Google Scholar
[19]	D. Tiba, Optimal control of nonsmooth distributed parameter systems, Lecture Notes in Mathematics, WJ Springer-Verlag, Berlin, 1459, 1990. Google Scholar