Asymptotics of the Solution to a Singularly Perturbed Time-Optimal Control Problem with Two Small Parameters
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ymptotics of the Solution to a Singularly Perturbed Time-Optimal Control Problem with Two Small Parameters A. R. Danilin1,2,∗ and O. O. Kovrizhnykh1,2,∗∗ Received January 10, 2019; revised February 4, 2019; accepted February 11, 2019
Abstract—The paper continues the authors’ previous studies. We consider a time-optimal control problem for a singularly perturbed linear autonomous system with two independent small parameters and smooth geometric constraints on the control in the form of a ball ⎧ x˙ = y, x, y ∈ R2 , u ∈ R2 , ⎪ ⎪ ⎪ ⎪ ⎨ ε3 y˙ = Jy + u, u ≤ 1, 0 < ε, μ 1, 3 ∗ ⎪ ⎪ x(0) = x0 (ε, μ) = (x0,1 , ε μξ) , y(0) = y0 , ⎪ ⎪ ⎩ x(Tε,μ ) = 0, y(Tε,μ ) = 0, Tε,μ → min,
where J=
0 1 0 0
.
The main difference of this case from the systems with fast and slow variables studied earlier is that here the matrix J at the fast variables is the second-order Jordan block with zero eigenvalue and, thus, does not satisfy the standard asymptotic stability condition. Continuing the research, we consider initial conditions depending on the second small parameter μ. We derive and justify a complete asymptotic expansion in the sense of Erdelyi of the optimal time and optimal control with respect to the asymptotic sequence εγ (εk + μk ), 0 < γ < 1. Keywords: optimal control, time-optimal control problem, asymptotic expansion, singularly perturbed problem, small parameter.
DOI: 10.1134/S0081543820040033 1. PROBLEM STATEMENT We consider a time-optimal control problem [1] for a linear autonomous system with fast and slow variables (see review [2]) in the class of piecewise continuous controls with smooth geometric constraints ⎧ ⎪ x˙ = y, x, y ∈ R2 , u ∈ R2 , ⎪ ⎪ ⎪ ⎪ ⎨ ε3 y˙ = Jy + u, u ≤ 1, 0 < ε, μ 1, (1.1) 3 μξ)∗ , ⎪ (ε, μ) = (x , ε y(0) = y , x(0) = x ⎪ 0 0,1 0 ⎪ ⎪ ⎪ ⎩ x(Tε,μ ) = 0, y(Tε,μ ) = 0, Tε,μ → min, 1
Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia 2 Ural Federal University, Yekaterinburg, 620000 Russia e-mail: ∗ [email protected], ∗∗ [email protected]
S10
ASYMPTOTICS OF SOLUTION
S11
where ξ ∈ R is a known constant, ξ = 0, * stands for transposition, and J=
0 1 0 0
.
(1.2)
Note that the small parameter ε enters the equations of system (1.1) in the third power for convenience, in order to avoid in what follows fractional rational powers of the parameter in asymptotic expansions. Basic relations for a system of general form with a polygon as a bounding set were obtained in [3]. The behavior of reachable sets as the small parameter tends to zero was studied in [4, 5]. The papers [6–8] are devoted to deriving a complete asymptotic expansion of the solution in control problems for linear systems with fast and slow variables and a bounding set in the form of a ball in Euclidean space. A distinctive feature of the problem under consideration is that the eigenvalues of the matrix at the fast variables are zero and thereby the standard condition for the asymptotic stability of this matrix (see, e.g., [5, Sect. 3.2, Assumption A1]) is
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