CC BY
10
УДК 517.926, 517.977.1
Differential Controllability of Linear Systems of Differential-algebraic Equations
Pavel S. Petrenko*
Matrosov Institute for System Dynamics and Control Theory SB RAS
Lermontov, 134, Irkutsk, 664033,
Russian
Received 05.08.2016, received in revised form 06.03.2017, accepted 10.04.2017 Linear controllable system of first order ordinary differential equations is considered. The system is unresolved with respect to the derivative of the unknown function and it is identically degenerate in the domain. An arbitrarily high unresolvability index is admitted. Differential controllability of the system is investigated under assumptions that ensure the existence of a global structural form that separates "algebraic" and "differential" subsystems.
Keywords: differential-algebraic equations, differential controllability, full controllability. DOI: 10.17516/1997-1397-2017-10-3-320-329.
1. Introduction and preliminaries
Let us consider the system of ordinary differential equations (ODEs)
A(t)x'(t) = B(t)x(t) + U(t)u(t), t G I = [0, +œ), (1.1)
where A(t), B(t) are known (n x n)-matrices, U(t) is known (n x /)-matrix, x(t) is unknown n-dimensional function of state, u(t) is /-dimensional function of control. It is assumed that det A(t) = 0.
Such systems are called differential-algebraic equations (DAEs). The measure of unresolvability for DAEs with respect to the derivative is an integer r : 0 ^ r ^ n, which is called index [1, 2].
There are a lot of various concepts of controllability in the theory of DAEs. The property of the full controllability (see., eg, [3, 4]) was first introduced and investigated for the DAEs with constant coefficients in [5]. The most important concepts of fl-controllability and impulse controllability were first introduced for the DAEs with constant coefficients and regular matrix pencil [6]. The resulting algebraic criteria are used in the analysis of the problem of minimizing a quadratic functional on solutions of the linear DAEs. Conditions of fl-controllability for the linear DAEs with infinitely differentiable coefficients were obtained [7]. Conditions of fl-controllability for the DAEs with variable coefficients and arbitrarily high unresolvability index were obtained [8]. Differential controllability of systems resolved with respect to the derivative (ODEs) was investigated [3].
In this paper we investigate differential controllability of linear DAEs systems with variable coefficients. The necessary and sufficient conditions of the differential controllability of such systems are obtained. The analysis is carried out under assumptions that ensure the existence of a global structural form that separates "algebraic" and "differential" subsystems. They have the same solutions as the original system [8-10].
* petrenko_p@mail.ru © Siberian Federal University. All rights reserved
2. Equivalent structural form
Let us introduce the following matrices for system (1.1)
Dr,z (t)
{ C\A(t)
C\A (t) + C2B(t)
O
C2A(t)
V C^A(r-1)(t) + C2rB(r-2)(t) C?A(r-2)(t) + C?B(r-3)(t)
O O
Cr A(t)
Dry (t) =
I C0A(t)
C0A'(t)+ C}B(t)
\ V C0A(r)(t) + C}.B(r-1)(t)
O
Dr
Drx(t) = ( B(t) Dry ) .
/
They have dimensions nr x nr, n(r +1) x n(r + 1) and n(r +1) x n(r + 2), respectively. From i!
this point on Ci =.,..'—— are binomial coefficients, O is the null matrix of appropriate size, j!(i - J)!
B(t) = colon (B(t),B '(t),..., B(r)(t)) t.
Let us suppose that condition iankDr z(t) = p = const yt G I holds for some r (0 ^ r ^ n) and there is non-special minor yt G I of the order n(r + 1) in the matrix Dr K(t). This minor includes p columns of the matrix Dr,z and n first columns of the matrix Dry. Such minor is called resolving.
Let us assume that we know exactly which columns of the matrix Dr,x(t) are included into the resolving minor. We delete n — d columns of the matrix B(t) which are not included in this minor, where d = nr — p. After the appropriate column permutation of Dr x(t) we obtain the matrix
O
Ar (t) = Dr,x (t) diag Q
U)
Q- 1 ,...,Q
i
(2.1)
where Ed is the identity matrix of order d, Q is (n x n)-permutation matrix§.
Matrix Q- 1 is constructed as follows. Let us denote the numbers of columns of B(t) by i 1, i2, ..., id and id+1, id+2, ■■in as the numbers of columns of B(t). They are included and not included in the resolving minor, respectively. Being left multiplied by matrix B(t), matrix Q-1 puts every (id+fc)-th column (k = 1,n — d) into k-th place and every (ij)-th column (J = 1, d) into the place with number n — d + J in the matrix B(t). Matrix Q-1 is invertible and it consists of zeros and n ones, wherein ones are the elements with indices (id+k,k) and (ij,n — d + J).
Lemma 1. Let us assume that
1) A(t),B(t),U(t),u(t) G Cr(I);
2) idnkDr z(t) = p = const yt G I;
3) there is the resolving minor in matrix Drxx(t). Then there exists linear differential operator
R = Ro(t) + Ri(t)dt +... + Rr(t) ( d:
(-)
\dt J
(2.2)
tcolon (xi, X2, ..., xn) =
fx 1 \
X2
\ Xn J
* The notation diag(Ax, A2,..., As) denotes quasi-diagonal matrix with the blocks listed in parenthesis on the main diagonal. Other elements are zero.
§ See [11] about row and column permutation matrix.
with continuous coefficients flj (t) (j = 0,r) that converts system (1.1) into the form
xl(t) = J1(t)x1(t) + H(t)u(t), (2.3)
x2(t) = J2(t)xl(t) + G(t)u(t), (2.4)
where colon (x}(t), x2(t)) = Qx(t), u(t) = colon(u(t), u'(t),..., u(r)(t)),
(m) (
Htt) St) ... GHrr{t))=(fl°(t) fll (t) ... flr (t)) Pr U M,
( C0U (t) O ... O \
C0U '(t) C} U (t) ... O
Pr [U(t)] =
(2.5)
\C0U(r)(t) C}U(r-1)(t) ... crU(t) J
JJ(l O )=(R°(t) fli (t) ... flr (t)) B(t)Q-1
( J2(t) Ed N
V Ji(t) O J
Matrices Rj (t) are uniquely determined by the resolving minor as
(Ro(t) Ri(t) ... Rr(t)) = (En O ... O) Aj(t) (Ar(t)Aj(t))-1. (2.6)
Definition 1. With a given control function u(t) n-dimensional vector function x(t) £ C1(I) is the solution of (1.1) if it identically satisfies this system on I.
Theorem 1. Let us assume that
1) A(t), B(t), U(t),u(t) £ C2r+1(I);
2) rankDrz(t) = p = const yt £ I;
3) there is the resolving minor in matrix Drx(t);
4) rankDr+1jy (t) = rankDrjy (t) + n yt £ I.
Then every solution of (1.1) is the solution of (2.3), (2.4) and vice versa. Definition 2. System (2.3), (2.4) is called the equivalent form of DAE (1.1). Let us define the initial conditions
x(to) = xo, (2.7)
where t0 £ I, x0 £ Rn is a given vector.
Theorem 1 provides a criterion for the existence and uniqueness of solution to problem (1.1), (2.7).
Corollary 1. Let us suppose that all the assumptions of Theorem 1 are satisfied. Then problem (1.1), (2.7) is solvable if and only if
x2,o = J2 (to)x1,o + G(to)u(to), (2.8)
where ( xi'° ) = Qxo. Moreover, if a solution to problem (1.1), (2.7) exists then it is unique.
x2,o
Definition 3. Initial condition (2.7) which satisfies (2.8) is called consistent with system (1.1) in the point to.
There are proofs of all results of this section [10].
3. Differential controllability
Definition 4. System (1.1) is called completely controllable on interval T = [io,ii] C I if for any x0,x1 € R" there exists control function u(t) such that the appropriate solution x(t) of DAE (1.1) satisfies conditions x(t0) = x0, x(t1) = x1.
Definition 5. System (1.1) is called differentially controllable on the interval T if it is completely controllable on any set [t0, t{] C T.
3.1. Non-stationary systems
In this section we prove necessary and sufficient conditions for the differential controllability of DAE (1.1).
Theorem 2. Let us suppose that all the assumptions of Theorem 1 are satisfied. System (1.1) is differentially controllable on T if and only if the following conditions are satisfied:
1) rank G(t) = d Wt € T;
2) Wh € R"-d : h = 0, hTX-1(t)H(t) ± 0 for almost all t, i.e. for all points t € T except some sets with zero measure^. Herein X(t) is (n — d) x (n — d) fundamental matrix of the homogeneous system x[(t) = J1(t)x1(t).
Proof. Necessity. Let system (1.1) be differentially controllable on T. This means that it is completely controllable on any set [t0, t1] c T. It follows from Theorem 1 that system (2.3), (2.4) has the same property. Then, any solution of (2.3), (2.4) should satisfy the following relations
x2(T0) — J2(T0)x1(T0) = G (T0)u(T0), (3.1)
x2(t1) — j2(t1)x1(t1) = G (T1)u(T1), (3.2)
X-1 (T1)x1(T1) — x1(T0) = [ 1 X-1(t)H(t)u(t)dt. (3.3)
J To
If relations (3.1), (3.2) are fulfilled for all left-hand sides then we have completeness of the ranks of matrices G(t0) and Grespectively. Since points t0,t1 (t1 > t0) are arbitrarily selected, it implies completeness of the rank of the matrix G(t) Wt € T, and leads to condition 1) of the Theorem.
Let us assume that condition 2) is not satisfied, i.e. there exists a nonzero vector h € R"-d such that hTX-1(t)H(t) = 0 on some interval [tq,^*] C T. At the same time it follows from
equation (3.3) that matrix X-1 (t)H(t)u(t)dt = 0 Wt € [t0,t1 ] for arbitrary t0,t1 € T (t1 >
To
t0) including t0 = t*, t1 = t*. We have a contradiction. In this way,
hTX-1(t)H(t)^ 0 (3.4)
for almost all t.
Sufficiency. Let us suppose that conditions 1), 2) of the Theorem are satisfied. One needs to show that in this case there exists a vector-function of control u(t) € C2r+1(T) such that equalities (3.1)-(3.3) holds for all left-hand sides.
Let us assume that control function u(t) is a polynomial
r
u(t) = £ (aj (t — To)r+1+j + ^(t — T1)r+1+j + Yj (t — To)r+1(t — T1)r+1 (t — c)s+j) , (3.5)
j=o
1A set located on interval [a, b] is called a set of zero measure if for every e > 0 it can be covered by a finite or countable system of intervals. The sum of the lengths of intervals does not exceed e.
where a.j,3j,Yj G R" d are unknown coefficients, c G T is constant, and s > 2r +1 is some integer. Then
u(t) = Ar+i(t — to) colon(a0,..., ar) + Ar+}(t — t1) colon(3o,..., ¡r)+ +$(t — To,t — Ti)As(t — c) colon(Yo, ...,Yr ),
(3.6)
where
Ak(t)
k! k:
'^tk Ei
k!
(k — 1)!
k!
\ (k — r)!
tk-1E,
tk-r Ei
(k +1)! k+1-r
(k + r)! tk+r E
(k + r)!t El (k + r)! tk+r-1E
(k + r — 1)!
(k + 1 — r)!
tk+1-r El
(k + r)! tk E
T"t El
k = r + 1,..., s; *(t,T ) =
(
?o( è
\i=0
! ((r +1)!)2 (tT)T+1
((r + 1)!)2 i ((r + 1)!)2tT+iTT
-Ei
(r + i)!(r +1 -
i)! J
El
^ ((r + 1)!)2(tT)T
El
\
f^ T ((r + 1)!)2t1+itr+1-i\ JT— T
IS CV+WTTT-iTj El c1(g CT-1
((r + 1)!)2
((r + 1)\)2t2+iT T+
- )
Ei
O O
((r + 1)!)2(tT )T+1 E
T ((r + 1)!)2 7
Let us introduce the following designations
91 = x2(TO) — .h(To)x1(To), g2 = x2(n) — J2(n)x1(T1), g3 = X 1(T1)X1(TÎ) — X1(TO);
a = colon(ao,..., ar), 3 = co\on(po,..., ¡r), y = colon(Yo,..., Yr).
Substituting (3.6) into (3.1) and (3.2), we obtain the system of equations with respect to coefficients a and ¡3:
G(To)Ar+1(To — T1)3 = 91, (3.7)
G (t 1 )Ar+1(T1 — To)a = 92.
(3.8)
It is obvious that for to = t1 matrix Ak(t) is invertible for all t G [to,t\\. Therefore, taking into account condition 1) of the Theorem, we can uniquely determine coefficients a and 3 :
3 = A-11(To — T1)G(To)T (G(to)G(to)t) 1 91,
a = A-+1(T1 — To)G(T1)T (G(t1)G(t1)t) 1 92.
(3.9) (3.10)
Substituting (3.6), (3.9) and (3.10) into equation (3.3) we obtain the system of equations with respect to 7:
i X-1(t)H(t) (Ar+1(t — To)a + Ar+1(t — T1)3) dt =
To
= i 1 X-1(t)U(t)^(t — To, t — T1)As(t — c)dt Y.
T0
(3.11)
C
O
0
Matrix &(t — T0,t — t1 ) is invertible on (t0, t\) and &(t — T0,t — t\) = O in points t = t0 and t = t1. Then, by virtue of condition 2) of the Theorem for any nonzero vector h € R"-d the relation
hTX-1(t)H(t)$(t — To, t — T10.
is fulfilled. As discussed above, matrix As(t — c) is invertible for all t = c. In this case, it is easy to see that for sufficiently large s the relation
/■ ti
hT X-1(t)H(t)$(t — T0,t — T1)As(t — c)dt = hTN = 0 Wt € [t0,t1],
To
is fulfilled for all nonzero h € R"-d. This means that matrix N has full row rank. Thus, the solvability of the equation
Y = NT (NNT)-1 g3
follows from (3.11), where
g3 = 93 — i X-1(t)H(t) (Ar+1(t — To)a + Ar+1(t — n)p) dt. To
Consequently, we have found control function (3.5) such that equalities (3.1)-(3.3) hold for all left-hand sides. It means complete controllability of (2.3), (2.4) on [t0,t1]. Since the values of t0,t1 C T are arbitrary, so one can conclude that system (2.3), (2.4) is completely controllable on any set [t0, t1] c T. Therefore, it is differentially controllable on T by Definition 5. □
Differential controllability condition can be formulated in terms of the controllability matrix
5(t) = (So(t) S\(t) ... S"-d-1(t)), (3.12)
where S0(t) = H(t), Si(t) = J1 (t)Si-1 (t) — S'i-1(t), i = 1,n — d — 1.
Theorem 3. Let us assume that
1) A(t), B(t) € c2r+"-d-1 (T),U(t),u(t) € c2r+"-d(T);
2) Assumptions 2) - 4) of Theorem 1 are satisfied.
System (1.1) is differentially controllable on T if and only if the following conditions are satisfied:
i) rank G(t) = d Wt € T;
ii) rankS(t) = n — d for almost all t, i.e. for all points t € T except some sets with measure zero.
Proof. Sufficiency. Let us assume the opposite. Suppose that conditions i) and ii) are satisfied but system (1.1) is not differentially controllable on T. According to Theorem 2, it means that hTX-1(t)H(t) = 0 on some interval [t0,t1] c T. Denote F(t) = X-1(t)H(t). Taking into account that (X-1(t)) = J1(t)X-1(t) and sequentially differentiating matrix F(t) n — d — 1 times, we obtain
F (t)j = ( — 1)j X-1(t)Sj (t), _
where Sj are matrices from (3.12) and F(t)(j) =0 Wt € [t0, t{], j = 0,n — d — 1.
Then hTS(t) =0 Wt € [t0,t1] for any nonzero h € R"-d so rank S(t) < n — d. We have the contradiction.
Necessity. Let system (1.1) be differentially controllable on T. According to Theorem 2:
a) condition i) of the Theorem is satisfied;
b) for any nonzero h € R"-d relation (3.4) is fulfilled for almost all t € T.
Let us assume that in this case the rank of matrix S(t) < n — d on some interval [t0, t1] c T. Then, for any nonzero vector p € R"-d we have pTS(t) =0 Wt € [t0, t1] or
pTSo(t)=0, pTS1 (t) = 0, ..., pTS"-d-1(t)=0. (3.13)
Taken into account that S0(t) = H(t), it follows from (3.13) that pTH(t) = 0. Assuming that pT = hTX-1(t), we obtain the contradiction with (3.4). □
3.2. Stationary systems
In this case the original system has matrices with constant coefficients. Differential controllability conditions have a simple structure. Let us consider a stationary system of DAEs
Ax'(t) = Bx(t) + Uu(t), t G I,
(3.14)
where A, B and U are constant matrices with sizes n x n, n x n and n x l, respectively, det A = 0.
It is easy to see that the concepts of complete and differential controllability coincide for stationary systems. So, in this case, we deal with just controllability.
It is known [11] that in the case of a regular matrix pencil AA — B there exist invertible (n x n)-matrices P and S such that
PAS =
{eZ, PBS =( G E )
(3.15)
where N is upper triangular matrix with p square zero blocks on the diagonal so that Np = O, G is some square matrix of size n — a.
It follows directly from [12] that the presence of the resolving minor in matrix Drx is a necessary and sufficient condition of existence of operator (2.2) in the case of constant coefficients. We show that there is such minor for r = p.
By left multiplying matrix Dpx by the matrix diag{P,..., P} and right multiplying by matrix diag{S,... ,S} and taking into account (3.15), we obtain
/ O
G
E, O
\
o
E
N O
O G
E, O
O
En
N O
O
E
N
o
O G
E, O
\
o
En
N O
(3.16)
Obviously, that the rank of the matrix to the right of the double line in (3.16) is equal to the rank of matrix Dp,z. Using the block transform of matrices, it is easy to show that iankDpz = np — a, since Np = O. There is the resolving minor in matrix (3.16). It includes all columns of the blocks that contain the identity matrixes and d = a. Thus, in the case of a regular matrix pencil AA — B there is the resolving minor in matrix Dr,x with r = p.
It is also easy to see that the relation
rank (diag{P,..., P}Dr+i,y diag{S,..., S}) = rank (diag{P,..., P}Dry diag{S,..., S}) + n is satisfied with r = p.
Thus, in the case of a regular matrix pencil AA — B all the assumptions of Theorem 1 are fulfilled for DAE (3.14) with r = p. The equivalent form in this case is given by
where
(t) = Jx xi(t) + Hu(t), X2(t) = J2xi(t) + Gu(t), t G I,
(H) = (
colon (x\(t), x2(t))
Go ... Gr \
Ho ... Hr I
Qx(t), xi(t) G Rn-d, x2(t) G Rd;
( J2 Ed )
V J1 O )
= (R0U ... RrU),
(3.17)
= RoBQ
Taking into account Theorems 2 and 3, we come to the following statements.
Theorem 4. Let a matrix pencil AA — B be regular (i.e. det(AA — B) ^ 0). System (3.14) is controllable if and only if the following conditions are satisfied:
1) rank G = d;
2) Vh € Rn-d : h = 0, hTX-1(t)H^ 0.
Theorem 5. Let a matrix pencil AA — B be regular (i.e. det(AA — B) ^ 0). System (3.14) is controllable if and only if the following conditions are satisfied:
1) rank G = d;
2) rank Q = n — d.
Here Q = (H J{H ... J[i-d-1H) is the controllability matrix of system (3.17).
Conditions of Theorems 4 and 5 can be formulated in terms of input data for system (3.14), using the following substitutions
J
( O E- ) R
( En-d \ O,
H = (ro,2u ... Rr,2U)
( Ri1 ) ' V Ri, 2 J
R, i = 0,r.
Coefficients Ri (i = 0,r) are determined in terms of coefficients of DAE (3.14) and their derivatives defined in (2.1), (2.6).
4. Example
Let us consider the linear system of DAE
+ cos t 1 — cos t 1 — sin t cos t 1 — sin t — sin t I x' (t) + 0 0 0
0 0 + ( 10 01
j x'(t) + ^ ( Ü )
— cos t — sin t 1 + sin t + cos t 0
1 + sin2 t
cos t
i(t) = 0,
— cos t 1
x(t) +
(4.1)
where t € I = [0, +œ), x(t) : I ^ R3 is unknown function.
We investigate system (4.1) on differential controllability on I. To do this, we verify all the assumptions of Theorem 3.
Condition 1) is obviously satisfied. To verify condition 2) we construct the matrices
D
l,x
/ -Pi - P2 l + P2 + P2 0 1 + Pi 1 - Pi l 0 0 0 \
i + P2 -P1 0 -P1P2 l - P2 -P2 0 0 0
P1 l l 0 0 0 0 0 0
P2 - Pi P1 - sin(2i) 0 -2 2 2P2 + p2 + 1 0 1 + Pi 1 - Pi 1
sin(2i) P2 0 3P2 -2P1 -P1 -P1P2 l - P2 -P2
V -P2 0 0 P1 l l 0 0 0 /
D
/ 1 + Pi Pi P2 1 - Pi 1 | 0 0 0| 0 0 0 0
1 - P2 -P2 1 0 0 0| 0 0
0 0 0| 0 0 0| 0 0 0
-Pi - 2P2 2P2 + Pi2 + 1 0| 1 + Pi 1 - Pi 1| 0 0 0
3P2 -2Pi -Pi 1 Pi P2 1 - P2 -P2 | 0 0 0
Pi 1 1| 0 0 0| 0 0 0
2P2 - 3Pi 3Pi - 2sin(2t) 0| -Pi - 2P2 2P2 + p2 + 1 0| 1+P1 1-P1 1
4sin(2t) 3P2 P2 | 3P22 -2P1 -P1 | Pi P2 1 - P2 -P2
\ -2P2 0 0| Pi 1 1| 0 0 0
2,VZ
where Pi = cos t and P2 = sin t.
It is easy to see that i&n\kDiz = p = 2 Vt € I. The columns that are included into the resolving minor are framed in matrix Di x. It includes p = 2 columns of matrix Di z, n = 3
l
first columns of matrix Diy and the third column of matrix It is easy to verify that
rank D2y = rank Diy + n = 8.
Thus, the conditions are fulfilled wherein system (4.1) has an equivalent form. Let us find the coefficients of operator (2.2)
1 cos t 0 \ ¡00 cos t sin t — 1 R = | 0 1 0 I + I 0 0 sin t
0 0 1 \ 0 0 0
d dtt'
(4.2)
that converts DAE (4.1) into the system
i(t) +
0 1 + sin t \ ,, s ( cos t 0
j xi(t) +
1 cos t
10
) u(t) + (
0 cos t sin t — 1 0 sin t
x2(t) + (cos t 1 )x1(t)+(0 1 )u(t)=0.
Finally we obtain
H(t)
cos t 1
cos t sin t — 1 sin t
Then rank G (t) = d =1, i.e. condition i) of Theorem 3 is satisfied. We construct the controllability matrix of system (4.3)
5(t) = (So(t) Si(t)),
where S0(t) Then
H(t), Si(t) = Ji(t)H(t) — H'(t). S (t) = (
1 + 2sin t 0
0 sin t + 3 sin21 — 1 0 1 cos t
.
'(t) = 0, (4.3)
(4.4)
G(t) = 0 1 0 0
We have rank S(t) = 1 when 1 + 2 sin t = 0 and —1 — cos t = 0, i.e. in the points t = —n/6 + 2nn, t = 7n/6 + 2nn, t = n + 2nn, n G N. Thus, DAE (4.1) is not differentially controllable on any interval T C I, containing the points t = —n/6 + 2nn, t = 7n/6 + 2nn, t = n + 2nn, n G N. 1 0 0 1
Alternatively, S(0) =
0 0 0 2
so rank S(0) = 2 = n — d yt G I. According to Lemma
4 [10], system (4.1) is completely controllable on any interval T c I that contain point t = 0 and also points t = —n/6 + 2nn, t = 7n/6 + 2nn, t = n + 2nn, n G N.
This work was partially supported by the Russian Foundation for Basic Research (project no. 16-31-00101), by the Complex Program of Fundamental Scientific Research of Siberian Branch of RAS (no. II.2) and by the Council for Grants of the President of Russian Federation for state support of the leading scientific schools (project NSh-8081.2016.9).
x
u
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Дифференциальная управляемость линейных систем дифференциально-алгебраических уравнений
Павел С. Петренко
Институт динамики систем и теории управления имени В. М. Матросова СО РАН
Лермонтова, 134, Иркутск, 664033
Россия
Рассматривается линейная система обыкновенных дифференциальных уравнений с переменными коэффициентами, не разрешенная относительно производной искомой вектор-функции и тождественно вырожденная в области определения. Допускается произвольно высокий индекс неразре-шенности системы. Исследуется дифференциальная управляемость такой системы в предположениях, обеспечивающих существование эквивалентной в смысле решений структурной формы с разделенными "дифференциальной" и "алгебраической" подсистемами.
Ключевые слова: дифференциально-алгебраические уравнения, дифференциальная управляемость, полная управляемость.
