A Jurdjevic-Quinn theorem for stochastic differential systems under weak conditions

• Autorzy: Florchinger Patrick

Identyfikatory

DOI

10.2478/candc-2022-0002

• Języki publikacji: EN

Abstrakty

EN

The purpose of this paper is to provide sufficient conditions for the stabilizability of weak solutions of stochastic dif- ferential systems when both the drift and diffusion are affine in the control. This result extends the well–known theorem of Jurdjevic–Quinn (Jurdjevic and Quinn, 1978) to stochastic differential systems under weaker conditions on the system coefficients than those assumed in Florchinger (2002).

Słowa kluczowe

EN

stochastic differential system; weak solution of stochastic differential equation; asymptotic stability in probability; Barbashin–Krasovskii theorem; stabilizing state feedback law

• Wydawca: Systems Research Institute, Polish Academy of Sciences
• Czasopismo: Control and Cybernetics
• Rocznik: 2022
• Tom: Vol. 51, No. 1
• Strony: 21--29
• Opis fizyczny: Bibliogr. 21 poz.

Twórcy

autor

Florchinger Patrick

• 23 Allée des Oeillets, F 57160 Moulins Les Metz, France

Bibliografia

• Cherny, A.S. (2002) On the uniqueness in law and the pathwise uniqueness for stochastic differential equations. SIAM Journal on Theory of Probability and its Applications 46 (3) 406–419.
• Cherny, A.S. and Engelbert, H.J. (2005) Singular Stochastic Differential Equations. Lecture Notes in Mathematics 1858 Springer-Verlag, Berlin. DOI:10.1007/b104187.
• Deng, H., Krstić, M. and Williams, R.J. (2001) Stabilization of stochastic nonlinear systems driven by noise of unknown covariance. IEEE Transactions on Automatic Control 46 1237–1253. DOI:10.1109/9.940927.
• Faubourg, L. and Pomet, J.-B. (1999) Design of control Lyapunov functions for “Jurdjevic-Quinn” systems. In: D. Aeyels et al., eds., Stability and Stabilization of Nonlinear Systems. Lecture Notes in Control and Information Sciences 246. Springer Verlag, Berlin, Heidelberg, New York 137–150.
• Florchinger, P. (1994) A stochastic version of Jurdjevic–Quinn theorem. Stochastic Analysis and Applications 12 (4) 473-480.
• Florchinger, P. (2001) A stochastic Jurdjevic–Quinn theorem for the stabilization of nonlinear stochastic differential systems. Stochastic Analysis and Applications 19 (3) 473-480.
• Florchinger, P. (2002) A stochastic Jurdjevic-Quinn theorem. SIAM Journal on Control and Optimization 41 (1) 83–88.
• Hofmanová, M. and Seidler, J. (2012) On weak solutions of stochastic differential equations. Stochastic Analysis and Applications 30 (1) 100–121. DOI: 10.1080/07362994.2012.628916
• Ikeda, N. and Watanabe, S. (1989) Stochastic Differential Equations and Diffusion Processes. North–Holland Publishing, Amsterdam.
• Jurdjevic, V. and Quinn, J.P. (1978) Controllability and stability. Journal of Differential Equations 28 381-389.
• Karatzas, I. and Shreve, S.E. (1991) Brownian Motion and Stochastic Calculus. Springer-Verlag, New York.
• Khalil, H. K. (1996) Nonlinear Systems. 2nd ed. Prentice-Hall, Upper Saddle River.
• Khasminskii, R.Z. (1980) Stochastic Stability of Differential Equations. Sijthoff & Noordhoff, Alphen aan den Rijn.
• Kushner, H.J. (1972) Stochastic stability. In: R. Curtain, ed., Stability of Stochastic Dynamical Systems. Lecture Notes in Mathematics 294 Springer Verlag, Berlin, Heidelberg, New York, 97-124.
• Lee, K.K. and Arapostathis, A. (1988) Remarks on smooth feedback stabilization of nonlinear systems. Systems and Control Letters 10 41–44.
• Li, F. and Liu, Y. (2014) Global stability and stabilization of more general stochastic nonlinear systems. Journal of Mathematical Analysis and Applications 413 841–855. DOI:10.1016/j.jmaa.2013.12.021
• Morin, P. (1996) Robust stabilization of the angular velocity of a rigid body with two actuators. European Journal of Control 2 (1) 51–56.
• Ondreját, M. and Seidler, J. (2018) A note on weak solutions to stochastic differential equations. Kybernetika 54 888–907. DOI:10.14736/kyb-2018-5-0888
• Outbib, R. and Sallet, G. (1992) Stabilizability of the angular velocity of a rigid body revisited. Systems and Control Letters 18 93–98.
• Tsinias, J. (1989) Sufficient Lyapunov–like conditions for stabilization. Mathematics of Control Signals and Systems 2 343–357.
• Yang, H., Kloeden, P.E. and Wu, F. (2018) Weak solution of stochastic differential equations with fractional diffusion coefficient. Stochastic Analysis and Applications 36 (4) 613–621. DOI:10.1080/07362994.2018.1434005