Sufficient Conditions for Terminal Invariance of Stochastic Jump Diffusion Systems
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PICAL ISSUE
Sufficient Conditions for Terminal Invariance of Stochastic Jump Diffusion Systems M. M. Khrustalev∗,a and K. A. Tsarkov∗,b ∗
Trapeznikov Institute of Control Sciences, Russian Academy of Sciences, Moscow, Russia e-mail: a [email protected], b [email protected] Received March 2, 2020 Revised May 18, 2020 Accepted July 9, 2020
Abstract—Sufficient conditions for the terminal invariance of nonlinear dynamic stochastic controlled systems (jump diffusions) are formulated and proved. The jump component has the form of an integral over a random Poisson measure. The parameters of this measure (the intensity and distribution of the values of jumps) are assumed to change over time. The conditions of invariance with respect to perturbations for a given initial state and also the conditions of absolute invariance (which ensure the constancy of a terminal criterion for any initial state) are proposed. The results are applied to a number of model examples, which include the numerical simulation and analytical study of the designed terminally invariant dynamic systems. Keywords: sufficient conditions for terminal invariance, nonlinear stochastic systems, jump diffusion processes, systems with impulsive actions DOI: 10.1134/S0005117920110089
1. INTRODUCTION Theoretical research in the field of terminal invariance of dynamic controlled systems began in 1963; see [1]. Thereafter, in 1968, global necessary and sufficient conditions for invariance with respect to perturbations (or weak invariance, in the terminology of [1]), and the conditions of absolute invariance with respect to both perturbations and initial state, were obtained for the class of deterministic controlled systems in [2]. Later on, in 1987, these conditions were combined and structured in the monograph [3], which also described in detail an extensive base of applications of the theory, remaining relevant to the present day. More examples of solving particular applied problems of flight dynamics by the methods of invariance theory can be found in [4] and the references presented in [3]. The studies continued in 2017 with the appearance of [5, 6], where the terminal invariance conditions were generalized to the case of stochastic controlled systems of the diffusion type. Finally, in 2018, the paper [7] was published, in which the theoretical results were given a more rigorous and precise mathematical formulation, and their possible use for solving particular applied problems was shown on several nontrivial examples. Terminal invariance conditions for various significant special cases of controlled diffusion processes are investigated at the present time as well; see [8]. In this paper, the authors intend to start a new spiral in the development of the theory of terminal invariance and to consider a generalization of controlled systems of the diffusion type in the form of stochastic systems containing not only a continuous Gaussian part, but also a discontinuous Poisson component [9]. Such systems (and the processes corresponding to them) in various sources 2062
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