Abstract
A general class of stochastic mechanical systems (including closed-loop control ones) developing under impacts generating both jumps and switchings is considered. We establish existence of a quadratic Lyapunov function for a system of this class: provided that the corresponding switching systems of ordinary differential equations have a common quadratic Lyapunov function and stochastic actions supplemented to them are moderate, this ordinary Lyapunov function is also a Lyapunov function for the system. Parameters of stochastic actions must satisfy the following constraints: disturbances in the force field are wiener processes with constant matrix coefficients; for jumps a certain combination of their distributions and of impacts arrival intensities is bounded relative to the ordinary common Lyapunov function. Systems of the class may have discontinuous coefficients and impacts arrival intensities; both are allowed to be time and state dependent.
Similar content being viewed by others
References
Campillo, F., Optimal Ergodic Control for a Class of Nonlinear Stochastic Systems—Application to Semi-Active Vehicle Suspensions, Proc. 28th Conf. Dec. Cont., 1989, pp. 1190–1195.
Campillo, F. and Pardoux, E., Numerical Methods In Ergodic Optimal Stochastic Control and Application, in Applied Stochastic Analysis, Karatzas, I. and Ocone, D, Eds., Berlin: Springer, 1992, pp. 59–73 (http://www.springerlink.com/index/y13p561655383ht7.pdf).
Anulova, S.V., Veretennikov, A.Yu., and Shcherbakov, P.S., Exponential Convergence of Multi-Dimensional Stochastic Mechanical Systems with Switching, Proc. 52nd IEEE Conf. Decision and Control, Florence, Italy, December 10–13, 2013, pp. 1217–1222.
Anulova, S.V. and Veretennikov, A.Yu., On Exponential Convergence of Stochastic Mechanical Systems with Switching, in Modern Stochastics and Applications, Springer Optimization and Its Applications 90, Switzerland: Springer Int. Publishing, 2014, pp. 159–174, DOI 10.1007/978-3-319-03512-310.
Anulova, S.V. and Veretennikov, A.Yu., Exponential Convergence of Multi-Dimensional Stochastic Mechanical Systems with Switching Impacts, Dokl. Math., 2015, vol. 91, no. 1, pp. 60–63.
Arapostathis, A., Borkar, V.S., and Ghosh, M.K., Ergodic Control of Diffusion Processes, Encyclopedia of Mathematics and Its Applications, vol. 143, Cambridge: Cambridge Univ. Press, 2012
Meyer, C.D., Matrix Analysis and Applied Linear Algebra (incl. CD-ROM and solutions manual), Philadelphia: Society for Industrial and Applied Mathematics (SIAM), 2000
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © S.V. Anulova, 2015, published in Avtomatika i Telemekhanika, 2015, No. 10, pp. 67–73.
Rights and permissions
About this article
Cite this article
Anulova, S.V. Quadratic Lyapunov function for stochastic mechanical systems with switching impacts. Autom Remote Control 76, 1765–1770 (2015). https://doi.org/10.1134/S0005117915100045
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0005117915100045