Estimation of Reachable Sets from Above with Respect to Inclusion for Some Nonlinear Control Systems
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timation of Reachable Sets from Above with Respect to Inclusion for Some Nonlinear Control Systems M. S. Nikol’skii1 Received April 4, 2019; revised April 16, 2019; accepted April 29, 2019
Abstract—The study of reachable sets of controlled objects is an important research area in optimal control theory. Such sets describe in a rough form the dynamical possibilities of the objects, which is important for theory and applications. Many optimization problems for controlled objects use the reachable set D(T ) in their statements. In the study of properties of controlled objects, it is useful to have some constructive estimates of D(T ) from above with respect to inclusion. In particular, such estimates are helpful for the approximate calculation of D(T ) by the pixel method. In this paper, we consider two nonlinear models of direct regulation known in the theory of absolute stability with a control term added to the right-hand side of the corresponding system of differential equations. To obtain the required upper estimates with respect to inclusion, we use Lyapunov functions from the theory of absolute stability. Note that the upper estimates for D(T ) are obtained in the form of balls in the phase space centered at the origin. Keywords: reachable set, Lyapunov function, absolute stability, direct regulation.
DOI: 10.1134/S0081543820040124 INTRODUCTION The problem of estimating the reachable sets D(T ) of controlled objects from above with respect to inclusion is of certain interest for mathematical control theory and its applications. Such estimates are useful in the analysis of dynamic possibilities of controlled objects and in the approximate calculation of D(T ) by the pixel method. In this paper we consider two nonlinear control systems of general form connected with classical models of the theory of absolute stability of direct regulation (see [1, 2]). The first system (case 1 below) contains one nonlinearity, and the second system (case 2) contains m nonlinearities, m ≥ 2. We estimate from above with respect to inclusion the reachable set D(T ) (see, for example, [3,4]) using the techniques of Lyapunov functions, which first appeared in motion stability theory (see, for example, [2, 5, 6] and many other papers). Note that the techniques of Lyapunov functions were used in earlier papers (see, for example, [6]) not only for traditional problems of motion stability theory but also for other qualitative problems of the theory of differential equations. 1. Consider the nonlinear control system x˙ = Ax + bϕ(σ(x)) + M u, 1
Steklov Mathematical Institute of the Russian Academy of Sciences, Moscow, 119991 Russia e-mail: [email protected]
S98
(1)
ESTIMATION OF REACHABLE SETS
S99
where x ∈ Rn (n ≥ 1), b ∈ Rn , A is an n × n matrix, M is an n × r matrix (r ≥ 1), ϕ(σ) is a continuously differentiable scalar function of a variable σ ∈ R1 , σ(x) = c, x
(2)
for c ∈ Rn , and u is a control vector from a compact set U ⊂ Rr . We agree to denote by Rk (k ≥ 1) the arithmetic Euclidean space whose elements are ordered columns of k
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