Controllability of Completely Integrable Linear Nonstationary Pfaffian Systems
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Controllability of Completely Integrable Linear Nonstationary Pfaffian Systems O. V. Khramtsov1∗ and S. A. Prokhozhii1∗∗ 1
Vitebsk State University, Vitebsk, 210038 Belarus e-mail: ∗ [email protected], ∗∗ [email protected]
Received January 3, 2020; revised May 13, 2020; accepted May 14, 2020
Abstract—We consider linear completely integrable Pfaffian systems of differential equations with matrices whose entries are analytic functions. The property of complete controllability of the system in a neighborhood of a regular point is studied. A sufficient condition for a Pfaffian system to have this property is derived. Whether the system has the complete controllability property is determined by a rank equality for some matrix constructed based on the known matrices of the original Pfaffian system. DOI: 10.1134/S0012266120080133
1. STATEMENT OF THE PROBLEM Consider a process described by a completely integrable linear nonstationary Pfaffian system dx = (A1 (t1 , t2 )x + B1 (t1 , t2 )u(t1 , t2 )) dt1 + (A2 (t1 , t2 )x + B2 (t1 , t2 )u(t1 , t2 )) dt2 ,
(1.1)
where t = (t1 , t2 ) ∈ D ⊂ R2 is a vector argument, D is a connected and simply connected convex domain, x ∈ Rn is the output (the system state), u ∈ Rr (r ≤ 2n) is the input (a control), which is a continuously differentiable vector function, and A1 (t), A2 (t), B1 (t), and B2 (t) are real matrices of appropriate sizes with entries analytic in t ∈ D. If the control u(t) ≡ 0, then the complete integrability condition for the corresponding homogeneous system has the form [1, p. 44] ∂A1 (t) ∂A2 (t) + A1 (t)A2 (t) ≡ + A2 (t)A1 (t), ∂t2 ∂t1
t ∈ D.
(1.2)
The complete controllability conditions for the inhomogeneous system (1.1) consist of requirement (1.2) and a differential constraint for the control u(t1 , t2 ) of the form B1 (t)
∂u ∂u − B2 (t) ≡ P (t)u, ∂t2 ∂t1
where P (t) ≡ A2 (t)B1 (t) − A1 (t)B2 (t) +
t ∈ D,
(1.3)
∂B2 (t) ∂B1 (t) − . ∂t1 ∂t2
Conditions (1.2) and (1.3) must be satisfied identically; therefore, the solutions of system (1.3) are taken for the admissible controls u. For system (1.3) to be solvable, it is necessary and sufficient that the following rank condition be satisfied [2, p. 91; 3]: there exists a constant real vector α = (α1 , α2 ) and a number m ≤ n such that one has the matrix rank equality rank [α1 B1 (t) + α2 B2 (t)] = rank [B1 (t), B2 (t)] = rank [B1 (t), B2 (t), P (t)] = m,
t ∈ D.
(1.4)
Note that condition (1.4) does not hold for all systems of the form (1.1). If this condition is not satisfied, then the question on the existence of a solution of system (1.1) and, if it exists, on the domain where the solution is defined remains open. The answers can be given, for example, by the 1108
CONTROLLABILITY OF PFAFFIAN SYSTEMS
1109
method of extension into involution [2, pp. 337–342; 4, pp. 267–268]. Throughout the remaining part of the paper, we only consider systems (1.1) that satisfy condition (1.4). We say that a point t0 = (t01 , t02 ) ∈ D is regular if it satisfies condition (1.4); otherwise this point is said to
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