Değişken gecikmeli kısıtlı stokastik kontrol geçiş sistemleri için regülatör problemi

Yıl 2020, , 997 - 1005, 26.09.2020

Çerkez Ağayeva

https://doi.org/10.17798/bitlisfen.633210

Öz

Bu makalede,
doğrusal stokastik denklemler sınıfıyla  
ifade olunan geçiş sistemleri ele alınmıştır. Gecikmeli faz ve kontrol
parametreleri içeren diferansiyel denklemler için karesel amaç fonksiyonu olan
optimal kontrol problemi oluşturulmuş ve sağ uç noktasında  kısıta sahip olan durum için  optimizasyon problemi incelenmiştir. Literatürde
Doğrusal Karesel Regülatör  olarak
bilinen ve sabit katsayılı stokastik diferansiyel denklemlerle ifade olunan bu
problemin optimal lığı için yeter ve gerek koşul, maksimum prensibi şeklinde  ispatlanmıştır. Bunun yanı sıra geçiş
sistemleri için önemli olan geçiş noktalarının bulunması için karşıtlık
koşulları  bulunmuştur. Sonda ise  Doğrusal Karesel Regülatör  problemleri için önem taşıyan optmal
kontrolün geri dönüşüm  şekli
bulunmuştur. Çözümü,   Rikkati
denklemleriyle  ifade olunan geri
dönüşüm  problemi, bu çalışmada   değişken gecikmeli stokastik sistemler için
uygulanmıştır.

Anahtar Kelimeler

Optimal kontrol problemi, Gecikmeli stokastik diferansiyel denklemler, Geçiş sistemleri

Kaynakça

• Kolmanovsky V., Myshkis A. 1992. Applied Theory of Functional Differential Equations. Dordrecht: Kluwer Academic Publishers.
• Anderson B., Ilchmann A, Wirth F. 2013. Stabilizability of time-varying linear systems. Systems and Control Letters, 62: 747–755.
• Gikhman I., Skorokhod A. 1972. Stochastic Differential Equations. Germany, Berlin: Springer.
• Hoek J., Elliott R. 2012. American option prices in a Markov chain model, Applied Stochastic Models in Business and Industry, 28: 35-39.
• Mao X., 1997. Stochastic Differential Equations and Their Applications. Chichester: Horwood Publication House.
• Shen H., Xu Sh., Song X. , Luo J. 2009. Delay-dependent robust stabilization for uncertain stochastic switching system with distributed delays. Asian Journal of Control, 5: 527-535.
• Chojnowska-Michalik A. 1978. Representation theorem for general stochastic delay equations. Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys., 7: 635–642 .
• Elsanosi I., Øksendal B., Sulem A. 2000. Some solvable stochastic control problems with delay. Stochastics and Stochastics Reports, 1-2: 69–89 .
• Kohlmann M., Zhou X. 2000. Relationship between backward stochastic differential equations and stochastic controls: A linear-quadratic approach. SIAM, Journal on Control and Optimization, 38:1392–1407.
• Agayeva C., Allahverdiyeva J. 2007. On one stochastic optimal control problem with variable delays. Theory of stochastic processes, 13: 3–11.
• Balakrishnan A. 1975. A note on the structure of optimal stochastic control. Applied Mathematics and Optimization, 1: 87 -94.
• Chernousko F., Ananievski I., Reshmin S. 2008. Control of Nonlinear Dynamical Systems: Methods and Applications (Communication and Control Engineering). Germany,Berlin: Springer.
• Federico S., Golds B., Gozzi F. 2011. HJB equations for the optimal control of differential equations with delays and state constraints, II: optimal feedbacks and approximations. SIAM, Journal on Control and Optimization, 49:2378–2414 .
• Larssen B. 2002. Dynamic programming in stochastic control of systems with delay. Stochastics and Stochastics Reports, 3–4: 651–673 .
• Kalman R., 1960. Contributions to the theory of optimal control. Boletin De La Sociedad Matematica Mexicana, 5:102–119. Bellman R. 1955. Functional equations in the theory of dynamic programming, positivity and quasilinearity. Proceeding of National Academy of Science, USA, 41:743–746.
• Bensoussan A., Delfour M., Mitter S. 1976. The linear quadratic optimal control problem for infinite dimensional systems over an infinite horizon; survey and examples. In: IEEE Conference on Decision and Control; Clearwater, Fla, USA, December 1-3: pp.746-751.
• Delfour M.C. 1986. The linear quadratic optimal control problem with delays in state and control variables: a state space approach. SIAM J Control Optim. 24:835-883.
• Ichikawa A. 1982. Quadratic control of evolution equations with delays in control. SIAM, Journal on Control and Optimization, 20:645-668.
• Bismut J. M. 1976. Linear quadratic optimal stochastic control with random coefficients, SIAM , Journal on Control and Optimization, 14:419–444.
• Wonham W. 1968. On a matrix Riccati equation of stochastic control, SIAM ,Journal on Control and Optimization; 6: 312–326.
• Boukas E.-K. 2006. Stochastic Switching Systems. Analysis and Design. Boston, USA:Birkhauer.
• Kharatatishvili G., Tadumadze T. 1997. The problem of optimal control for nonlinear systems with variable structure, delays and piecewise continuous prehistory. Memoirs on Differential Equations and Mathematical Physics, 11: 67-88.
• Tadumadze T., Arsenashvili A. 2008. Optimization of a delay variable structure system with mixed intermediate condition. Bulletin of the Georgian National Academy of Sciences , 2(3):22–26.
• Aghayeva Ch. 2014. Necessary condition of optimality for stochastic switching systems with delay. In: International Conference on Mathematical Models and Methods in Applied Sciences; 23-25 September 2014; Saint Petersburg, Russia: MMAS’14. pp. 54-58.
• Abushov Q., Aghayeva Ch. 2014. Stochastic maximum principle for the nonlinear optimal control problem of switching systems, Journal of Computational and Applied Mathematics, 259: 371-376.
• Agayeva Ch. 2016. Linear Quadratic Control Problem of Stochastic Switching Systems with Delay, Anadolu University Journal of Science and Technology-B, Theoretical Sciences, 4(2), pp.52-58.
• Ağayeva Ç., Takan A.M. 2018. Restricted Optimal Control Problem for Stochastic Switching Systems with Variable Delay, Muş Alparslan Üniversitesi Fen Bilimleri Dergisi, 6(2), s. 565-569.
• Ekeland I. 1974. On the variational principle. Journal Mathematical Analysis and Applications , 47:324–353.
• Agayeva Ch., Abushov Q. 2005. Linear-square stochastic optimal control problem with variable delay on control and state. Transactions ANAS, math.- ph. series, informatics and control problems, Baku, 3: 204-208.
• Bellman R. 1955. lFunctional equations in the theory of dynamic programming, positivity and quasilinearity. Proceeding of National Academy of Science, USA, 41:743–746.