Terminal Invariance of Quasi-Linear Stochastic Diffusion Systems That Are Nonlinear in Control Variable
- PDF / 654,640 Bytes
- 16 Pages / 612 x 792 pts (letter) Page_size
- 42 Downloads / 194 Views
OCHASTIC SYSTEMS
Terminal Invariance of Quasi-Linear Stochastic Diffusion Systems That Are Nonlinear in Control Variable M. M. Khrustalev Trapeznikov Institute of Control Sciences, Russian Academy of Sciences, Moscow, Russia e-mail: [email protected] Received February 29, 2020 Revised May 11, 2020 Accepted May 25, 2020
Abstract—The terminal invariance conditions for stochastic diffusion systems established earlier are concretized for the class of quasi-linear systems that are nonlinear in the control variable. Some recommendations on the design of control laws ensuring terminal invariance are given. Keywords: stochastic system, control, terminal invariance DOI: 10.1134/S0005117920100045
1. INTRODUCTION In the theory of dynamic systems control, an important problem is to design a control law ensuring the constancy of a terminal criterion (in the general case, a vector-valued one) under any a priori unknown deterministic time-varying perturbations affecting the system. L. Rozonoer called it the problem of weak invariance and obtained local necessary conditions for it; see [1]. However, the term “terminal invariance” seems more natural for this type of invariance, which corresponds to the term “terminal control” in the literature devoted to control problems for the final state of dynamic systems. For this problem, necessary and simultaneously sufficient conditions were established by the author in [2]. In addition, its generalization (the problem of absolute invariance) was considered and sufficient conditions were derived therein. In [3, 4], a new problem of invariance theory, the terminal invariance of controlled stochastic dynamic systems of diffusion type, was formulated and general sufficient conditions for such invariance were derived. In [5], these conditions were examined for the simple case of linear stochastic systems. This paper is a direct continuation of the earlier research [3, 4]: the conditions established in [3, 4] are concretized for quasi-linear stochastic diffusion systems that are nonlinear in the control variable. For this class of systems, the resulting conditions are reduced to rather constructive control design algorithms ensuring invariance. Since only terminal invariance are under consideration, the word “terminal” will be omitted below. 2. TERMINAL INVARIANCE: PROBLEM STATEMENT Assume that a controlled dynamic system is described by a vector Itˆo differential equation of the form dx(t) = f (t, x(t), u(t))dt + g(t, x(t), u(t))dw(t)
(1)
with an initial condition x(t0 ) = x0 and the following notations: t ∈ [t0 , t1 ] as time (tmin ≤ t0 < t1 ); x := (x1 , . . . , xn )T ∈ Rn as the state of this system; w(t) := (w1 (t), . . . , wq (t))T ∈ Rq , w(t) = 1840
TERMINAL INVARIANCE OF QUASI-LINEAR STOCHASTIC DIFFUSION SYSTEMS
1841
w(t ˜ − t0 ), where w(·) ˜ is the standard q-dimensional Wiener process; (·)T as the transpose operation. The control variable u(t) ∈ Rm is the restriction to the time interval [t0 , t1 ] of a fixed function t→u ˜(t) : [tmin , t1 ] → Rm that is continuous on the half-open time i
Data Loading...
