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2URAL MATHEMATICAL JOURNAL, Vol. 8, No. 2, 2022, pp. 13-26
DOI: 10.15826/umj.2022.2.002
OUTPUT CONTROLLABILITY OF DELAYED CONTROL SYSTEMS IN A LONG TIME HORIZON1
Boris I. Ananyev
Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, 16 S. Kovalevskaya Str., Ekaterinburg, 620108, Russian Federation
Abstract: In this paper, we consider the output controllability of finite-dimensional control systems governed by a distributed delayed control. For systems with ordinary controls, this problem was investigated earlier. Nevertheless, in many practical and technical problems the control acts with some delay. We give the necessary and sufficient condition for the output controllability. The main goal of our control is to govern the output of the system to some position on a subspace in a given instant, and then keep this output fixed for the remaining times. This property is called the long-time output controllability. For this, sufficient conditions are given. The introduced notions are applied for the investigation of averaged controllability of systems with delayed controls. The general approach for that is to approximate the system by the ordinary one. Some examples are considered.
Keywords: Output and averaged controllability, Delayed control, Approximation.
1. Introduction
In this paper, we deal with the output controllability of finite-dimensional control systems governed by a distributed delayed control. For systems with ordinary controls, this problem is investigated in [2]. It is known that in many practical and technical problems the controlling actions take place with some delays. The main goal of our control is to govern the output of the system to some position on a subspace in a given time T > 0, and then keep this output fixed for the remaining times t > T.
Consider a linear autonomous system with delayed controls and observation:
x(t) = Ax(t) + [ dB(s)u(t + s), t > 0, x(t) € Rn, u(t) € Rm, (1.1)
J-h
y(t) = Cx(t), C € Rpxn, (1.2)
where elements of the matrix function B(s) belong to BV[—h, 0] (the space of functions of bounded variation) and they are left continuous on (—h,0], B(s) = 0 for Vs > 0, and B(s) = B(—h) for Vs < —h. Since the matrix B(s) generates a Borelian measure, any bounded Borelian m-vector function u(t) can be used as a control. The notion of output controllability is as follows.
Definition 1. We say that the system (1.1) is C-output controllable, if for every x0 € Rn and every y € im C = {y | y = Cx, x € Rn} there exist an instant T > 0 and a bounded Borelian control u on [—h, T] such that the solution x(t) of (1.1) with initial condition x(0) = x0 satisfies y(T ) = Cx(T )= y.
1This work was performed as part of research conducted in the Ural Mathematical Center with the financial support of the Ministry of Science and Higher Education of the Russian Federation (Agreement number 075-02-2022-874).
Let us recall that the ordinary (state) controllability definition of the system (1.1) follows from Definition 1 when C = In. The symbol In € Rnxn means the unity matrix.
In this paper, we are interested in the conditions for having long-time output controllability. This controllability notion means that the output of the system enters the subspace and then remains on it for later times. This is defined as follows.
Definition 2. Given y € im C, the system (1.1) is said to be C-long-time output controllable (briefly C-LTOC on y), if for every x0 there exist a time T > 0 and a control u such that the solution of (1.1), with initial condition x(0) = xo satisfies y(t) = y for every t € [T, to).
It is obvious that state controllability implies C-output controllability for any matrix C. But in order to save the property y(t) = y for every t > T we need extra conditions on a delayed control. In this paper we only assume the C-output controllability of the system and give a criterion for this. Our main attention is directed to conditions of C-LTOC (or simply LTOC) for systems of the form (1.1), (1.2) and their applications.
The notions of output controllability and C-LTOC can be applied to averaged controllability property of finite-dimensional, parameter dependent systems with delayed controls. The averaged controllability has been introduced in the paper [4]. More precisely, let us consider d realizations of control systems,
Xi(t) = Aix(t) + f dBi(s)u(t + s), t > 0, xi(t) € Rn, u(t) € Rm, i € 1 : d, (1.3)
J-h
and d parameters pi > 0, ^d=1 pi = 1. Here the matrices Bi(s) have the same properties as B(s) in (1.1).
Definition 3. We say that the flock of systems (1.3) is controllable in average for the weights pi > 0 if for all initial states x10,...,xd0 and every y € Rn there exist an instant T > 0 and a bounded Borelian control u on [—h, T] such that the solutions of (1.3) satisfy the equal-ityH d=i Pixi(T) = y.
Let us use the Matlab notation for matrices and vectors. We can see that the averaged controllability notion is exactly the C-output controllability of (1.1)-(1.2) with matrices:
A = diag[Ai,...,Ad], B(s) = [Bi(s);...; Bd(s)], C = [piln,... ,pdIn],
where x — [xi;...; xd] € Rn . The flock of systems (1.3) is called simultaneously controllable if corresponding system (1.1) is state controllable. Of course, the simultaneous controllability of (1.3) implies the averaged controllability. We can also define the notion of long-time averaged controllability (briefly LTAC on y). We say that systems (1.3) are LTAC on y for the weights pi > 0 if for every initial states xi0,..., xd0 there exist an instant T > 0 and an admissible control u such that the corresponding mean value is the following
d
Epixi(t) = y
i=i
for every t € [T, to).
In this paper, we obtain conditions of C-output controllability and C-LTOC for general systems (1.1)-(1.2) and apply them for the LTAC property of (1.3). Besides, we get the algorithm for constructing of necessary control in special cases. Some examples are considered.
2. Output controllability of the system
First, note that the initial condition x0 does not play any role in Definition 1. System (1.1), (1.2) is C-output controllable iff for every y € im C there exist an instant T > 0 and an admissible control u on [—h, T] such that
¡■T ,0
c eA(T-d) dB (s)u(0 + s)d0 = y. 0-h
i- o t-e
/ dB(s)u(6» + s) = dB (a - 0)u(a), J-h Je-h
rT f0
eA(T-e) /
/0 J-h
Setting a = 0 + s we have
ro re
dB(s)u(0 + s) = /
-h e-h
and by Fubini's theorem we get the equivalent equality
ç T ç (a+h) AT
/ B(T,a)u(a)da = y, where B(T,a) = C eA(T-e)dB(a - 0). (2.1)
J-h JaV0
The n x m-matrix function B(T, a) is of bounded variation with respect to a and, therefore, belongs to the space Lpxm[—h,T] (the space of square integrable matrices or vectors). We can prove the following lemma.
Lemma 1. System (1.1), (1.2) is C-output controllable iff there is a segment [a, b], —h < a < b < T, such that
rank ^J B(T,a)B'(T,a)da^ = rankC. (2.2)
Proof. Let condition (2.2) be satisfied. Since
rb
im í B(T, a)B' (T, a) da C im C,
Ja
we obtain the equality of subspaces in this inclusion. For every y € im C there exists a vector v € Rp such that ^
/ B(T,a)B' (T,a)dav = y.
J a
Then u(a) = B'(T, a)v is a bounded Borelian control on [a,b]. We can take u(a) =0, a € [a,b], and satisfy (2.1) for any T > b. On the contrary, let condition (2.1) be valid, but there is a vector y € im C such that
rT
y £ im í B(T,a)B'(T,a)da. -h
'-h
Then we have a contradiction with (2.1) as
Lm[-h, T] = (u(a) : u(a) = B'(T, a)v, v € Rp} 0 |u(a) : J B(T, a)u(a)da = 0 j . Therefore, there are no functions u € L^—h,T] satisfying (2.1). □
Corollary 1. Condition (2.2) holds iff the equality
l'B(T,a) = 0 a.e. on [a,b] implies that l € kerC'. (2.3)
Proof. Condition (2.2) is equivalent to the equality
ker C' = ker f B(T, a)B'(T, a)da,
J a
or, in other words, the implication (2.3) is fulfilled. □
Corollary 2. The function B(T, a) from (2.1) can be expressed in the form
ЛraA0
(a-T )V(-h)
If T > h and a = 0, b = T - h, then
f 0 i-h
on [a, b]. Hence, the implication (2.3) is equivalent to the rank condition
r aA0
B(T,a) = CeA(T-a)b(T, a), where &(T,a)=/ eAsdB (s). (2.4)
(a-T)V(-h)
hen
o
b(T, a) = / eAsdB(s) = const -h
rank C
0 0 0
= rank C. (2.5)
-h -h -h
Proof. Setting a — 0 = s in (2.1) we get (2.4). As b(T,a) =const on segment [a,b], the relation l'B(T, a) = 0 can be differentiated with respect to a many times. So, we come to the equivalence of implication (2.3) and rank condition (2.5) by the theorem of Cayley-Hamilton [5, Theorem 7.2.4]. □
Let us discuss the Lemma 1 and its Corollaries. If condition (2.2) does not hold for some segment [a, b], it can be hold for grater ones. The rank condition for C-output controllability is possible if the matrix function B(s) is piecewise-constant as in the case of lumped delays. The simplest case of lumped delays is given by
B (s) = — BoX(_œ,o](s) — BiX(-TO>-i](s), (2.6)
where the indicator function X(a,b](s) = 1 if s € (a,b], and X(a,b](s) = 0, elsewhere. Let T > 1. We can divide the segment [—1, T] = (T — 1, T] U [0, T — 1] U [—1,0) into three parts. On the first semi-interval the implication
l'CeA(T-a)Bo = 0 ^ l € ker C'
is equivalent to the condition
rank C [B0,AB0,...,An-1B0] = rank C (2.7)
by the theorem of Cayley-Hamilton. On [0, T — 1] we have
b(T, a) = B0 + e-A Bi,
and we are in the conditions of Corollary 2. On the remaining semi-interval the implication
l'CeA(T-a-1)Bi = 0 ^ l € ker C'
is equivalent to condition
rank CeA(T -1)[Bi,ABi,...,An-1Bi] = rank C. (2.8)
If 0 < T < 1, we have only two segments [0, T] and [—1,T — 1]. Therefore, we get two conditions in (2.7) and (2.8), where the matrix of eA(T-1) is absent. Conditions (2.5), (2.7) do not depend on T, but condition in (2.8) nevertheless depends. It distinguishes case of delayed controls from the ordinary one.
Remark 1. Of course, we can also consider partly different Definitions 1-3 with null initial controls, i.e. when u(t) = 0 for t < 0. Then the integral in (2.1) is considered on [0,T], the parameter a > 0 in Lemma 1, and condition (2.8) is not necessary. In addition, we may demand that u(t) = 0 when t € [T — h, T], T > h. Then we have 0 < a < b < T — h in Lemma 1.
3. The property of LTOC
Suppose further that system (1.1)-(1.2) is C-output controllable at some instant T > h and u(t) = 0 if t € [0, T — h] as in the Remark 1. Then it is easily seen that the C-output controllability on [0, T] is equivalent to C-output controllability on [a, T + a] for all a > 0 when u(t) = 0 if t € [a, T + a — h]. Therefore, in order to get y(t) = y, Vt > T, we need to obtain the conditions for the property
Cxo = Cx(t) Vt > 0. (3.1)
By derivation with respect to t in (3.1), we have:
ro i-h
Introduce the subspace
CAx(t) + O / dB(s)u(t + s) = 0 Vi > 0. (3.2)
-h
U = |x € Rn : 3 an admissible function -u(-) s.t. x = J dB(s)u(s)| . (3.3)
To satisfy (3.2), one needs to have the inclusion CAx(t) € CU for Vt > 0. Since Rp = CU®CUX, we can take an orthonormal basis {h1,..., hq} in CUwhere q = p — dim (CU), and the corresponding matrix H = [h1,...,hq] € Rpxq. As a result, we get a projector P0 = HH' € Rpxp on the subspace CU
Consequently, condition (3.2) is L0x(t) =0 Vt > 0, where L0 = P0CA. Therefore, if we introduce the matrix C1 = [C; L0] € R2pxn, then C1X0 = C1x(t) Vt > 0, as in (3.1).
We can iterate the process similar to ordinary case with no delays as in [2] to define C2 = [C, L1], L1 = P1C1A, and so on. After k steps we get
Cfc+1 = [C; Lk], Lk = PkCkA € R(k+1)pxn, (3.4)
where Pk € R(k+1)px(k+1)p is the orthogonal projector on CkUThe process stops when ker Ck+1 = ker Ck. The condition (3.1) can be fulfilled iff Lkx0 = 0. To be more exact, the following assertion holds.
Lemma 2. We have ker Ck+1 C ker Ck C Rn and ker Lk+1 c ker Lk c Rn for every k € NU{0}. There exists a number K € 0 : n such that ker CK+1 = ker CK. Here C0 = C. For every i € N we have ker CK + = ker CK.
Proof. We will argue by induction. As ker C1 = ker C0 n ker L0, we trivially obtain ker Ci C ker C0. Suppose that ker Ck C ker Ck-1 for some k € N. Then we notice that
Lkx = 0 ^ 3 u € U s.t. Ck(Ax - u) = 0 ^ Ck-1(Ax - u) = 0 ^ Lk-1x = 0.
This means that ker Lk C ker L^-1. Therefore, ker Ck+1 C ker Ck. It is obvious that there exists a number K € 0 : n such that ker CK+1 = ker CK C Rn. It follows that ker LK+1 = ker LK C Rn. Indeed,
LKx = 0 ^ 3 u € U s.t. CK(Ax - u) = 0 ^ CK+1(Ax - u) = 0 ^ LK+1x = 0.
Hence, by induction, we obtain the final assertion. □
Note also that im Lk C im Ck for every k € N U {0}. This is equivalent to the inclusion ker A'CkPk D ker Ck. Indeed, if C'kz = 0, then z ± CkU ^ z € CkUx ^ Pkz = z.
The problem of control with delays to ensure equality (3.1) is more difficult than for ordinary controls. Let us prove the lemma.
Lemma 3. Let Ck, k € N U {0}, be the sequence defined by (3.4), and let K € 0 : n such that ker CK+1 = ker CK. Then there exists a function v(t) € U such that (3.1) holds where
x(t) = Ax(t) + v(t), x(0) = xo, (3.5)
if and only if LKx0 = 0 with LK defined by (3.4).
Proof. It follows from (3.1) that CKx0 = CKx(t) and LKx0 = 0. On the contrary, assume that LKxo = 0. We need
Ckx0 = Ckx(t) Vt > 0.
After derivation we get
Ckx(t) = Ck(Ax(t) + v(t)), v(t) € U. (3.6)
If we find v(t) with CKxc(t) = 0, then the lemma is proved. We can write
Ckx(t) = Lkx(t) + (/(k+1)p - Pk)CkAx(t) + Ckv(t).
Here 1(K+1)p - PK is a projector on CKU. Hence, there exists a continuous closed-loop control v(x) such that
(1(k+1)p - Pk)CkAx + Ckv(x) = 0. Relation (3.6) under such a control reduces to
CK x(t) = LK x(t).
Let us write the orthogonal expansion for x(t):
x(t) - x0 = x0(t) + x1(t), (3.7)
where x0(t) € ker CK and x1(t) € im CK. Then
Ck x(t) = Ck (x0 + x1 (t))
and
LK x(t) = LK x1(t)
as
ker CK = ker CK+1 = ker C0 n ker LK.
Thus, we get that
Ck x 1(t) = Lk X1(t).
The matrix CK is invertible on the subspace im CK and x1(0) = 0 from (3.7). Therefore, x1(t) = 0 Vt > 0. The lemma is proved. □
It follows from Lemma 3 that conditions
Lkxo = 0 and (/(K+1)p — Pk) CkAx + Ckv(x) = 0
are necessary and sufficient for the solution x(t) of equation (3.5) to satisfy (3.1). They define the function v(t) = v(x(t)), but for our purposes we need a function u(t) such that
f dB(s)u(t + s) = u(t), t > 0. (3.8)
J (—h) V(-1)
This is an integral equation. It can have no solutions. Therefore, in next sections we consider the approximation scheme to exclude equations like (3.8). Now, we formulate the general result.
Theorem 1. Let be given y € im C. For every x0 € Rn there exists an admissible control for the system (1.1) such that the solution satisfies Cx(t) = y Vt > T if and only if
[y;0] € im Ck+1
and system (1.1), (1.2) is CK+1-output controllable in the sense of Remark 1, i.e. the condition like (2.2) holds for some 0 < a<b < T — h with
B(T,a) = Ck+1eA(T-a)b(T,a)
and rank CK+1. Here Ck is the sequence defined by (3.4) and the number K is defined by Lemma 2. Besides, equation (3.8) has to be resolved for the function y(t) defined in Lemma 3.
Proof. According to the Lemma 3 the control exists iff the system is transferred to the state x(T) such that CK+1x(T) = [y;0]. This is possible for every y € im C and every x0 € Rn iff the system is CK+1-output controllable. After that we solve the problem as in Lemma 3 which does not depend of initial instant T. □
We do not give any sufficient conditions for the existence of a solution of integral equation (3.8). This is considered in some special cases. For example, in simplest case (2.6) we have the difference equation
Bou(t) + B1u(t — 1) = u(t), u(t) =0 if t < 0 which can be resolved step-by-step on segments [i — 1,i]:
BoUi(t) = v(t) — B1Ui-1(t — 1), t € [i — 1,i], i € N, (3.9)
where u0(t) = 0.
Example 1. Consider the flock of two systems of the form:
X1 = —x} + ij=i biijUi(t - j + 1), Xf = first system, X1 = + + ij=1 b2ijui(t — j + 1), Xf = ; second system,
where we have the case with = p2 = 1/2 and C = [/2,/2]/2. Condition (3.1) reduces to the requirement: Cx0 = 0 implies Cx(t) = 0 if t > 0. Here x(t) € R4 is the composed vector. Below we study the example in detail for various coefficients b1ij.
4. The system in the infinite-dimensional space
Let us now rewrite the system (1.1)-(1.2) in the infinite-dimensional space following [1]. We can write
f dB(s)u(s) = Bou(0)+ Bu, where Bo = — B(0), Bu = / dB(s)u(s). (4.1)
J-h J [—h,0)
Formula (4.1) is true for continuous vector-functions u, but we want to use functions {u € H = Lm[-h, 0]}. In this case we consider the operator B as unbounded with dense domain b(B) = W = W^-h, 0] (the Sobolev space). If u € W, the function 0(t, s) = u(t + s) satisfies the equation in partial derivatives:
0(t, s) = D0(t, s), 0(0, s) = u0(s), 0(t, 0) = u(t), (4.2)
with the operator D = d/ds. Equation (4.2) is considered in H with unbounded D. The left-shift C0-semigroup St on H is defined by
(Stu)(s) = J u(t + S), s € [-h -t] if t < h, and (Stu)(s) = 0 if t > h. [0, s € (-t, 0]
The infinitesimal generator for St is D with dense domain
b(D) = W0 = {u € W : u(0) = 0} C H.
As shown in [1, Lemma 1.1], the solution
|u0(t + s), s € [-h, -1] „ , , . .
0(t,s) = { 0V ' 1 J if t < h, and 0(t,s)= u(t + s) if t > h,
I u(t + s), s € (-1, 0]
of equation (4.2), 0(t, ■) € H, can be represented by
0(t) = St00 + / St-rAu(r)dr, (4.3)
0
where the operator A € L(Rm, W*) (the space of linear operators) is given by the relation (Au,w) = u'w(0) for all w € W. So, in spite of the fact that equality (4.3) is considered in W* D H D W and the integration is also fulfilled in W*, we have 0(t, ■) € H for every
u € Doc [0
Introducing the operators A = [A,B;0, D], B = [B0; A], and C0-semigroup T by
Tt-rz =
eA(t-r) x + /1 eA(t-a) BS«-r 0da; Sa_r 0
r
where z = [x; 0] € Z = Rn xH, we can write the mild solution
z(t) = Ttzo + / Tt-rBu(r)dr, for the equation (4.4)
Jo
z(t) = Az(t) + Bu(r), z(0) = zo. (4.5)
Here the operator A is unbounded on Z with domains b(A) = Rn x W0. For the operator B we have B € L(Rm, Rn x W*).
Equation (4.5) has no delays in control, but a recurrent procedure like in (3.4) is, unfortunately, impossible for infinite-dimensional system (4.5) to find a C-LTOC control. Therefore, we pass to finite-dimensional approximations of the obtained system.
5. Finite-dimensional approximation
We use the averaging approximation of the delayed system following [3]. For every positive integer N, we define the finite-dimensional linear subspace HN of H by
N
/N
HN = { u eH : u = ViXi, V € R
Vi
¿=1
where denote the characteristic function of [ij ,ti-1) for i € 1 : N and ij = — ih/N, i € 0 : N. The subspace is isometrically isomorphic to RmN by means of the embedding yn : RmN ^ such that (YNg)(s) = Vj, s € [ii,ii-1), i € 1 : N, where g = [v1;...; ]. On RmN, we define the induced inner product
"In
</,g)N = /'QNg, /,g e RmN,
where
Qn = diag [Im,..., Im]h/N € RmNxmN.
The corresponding vector and matrix norms will be denoted by || ■ ||n. The dual mapping yn* : HN ^ RmN has the natural extension nN : H ^ RmN defined by
rti-1
nN u = [vi;...; vN ], Vj = / u(s)dsN/h, i € 1: N.
Jti
We have that PN = 7NnN is an self-adjoint orthogonal projector onto HN and nN7N = /mN. Introduce the the following matrices:
BN = lim (B(s + h/N) - B(s)) = B(ti-i) - B(tj), i € 1 : N.
sfti
Note that the matrix B(s) is left-continuous. For 0 € H, let nN0 = g = [vi;...; vN] € RmN. Then we can approximate the infinite-dimensional operators as follows:
NN
B0 w BPN 0 = ^ BN Vi; D0 w VPN 0 = ^ N (v,-i - v,)xi/h, Vo = 0; i=i i=i Bu w [B0u; N%iu/h].
Denote by ZN the space RnxHN. Introduce the approximating operators AN = [A, BPN; 0, VPN] : ZN ^ ZN and BN = [Bo; Nxi/h] : Rm ^ ZN. Let TtN denote the Co-semigroup generated by AN on ZN and let nN = [/„,0;0,nN], 7N = [/„,0;0,yn] be the operators on Z and on Rn+mN, respectively. The following theorem is true.
Theorem 2 [3, Theorem 3.1]. Let the matrix B(s) have the form
f 0
B(s) = -n=0BiX(-^,-hi](s) - B0i(r)dr, 0 = h0 < ••• <hq = h, (5.1)
Js
where B01(-) € Lnxm[-h, 0]. Then there exist constants M and w independent of N such that
||enNan,n< Mewt. It follows from definitions of operators that
An7N = [A,B7N;0,nNV7N] € R(n+mN)x(n+mN). Therefore, the finite-dimensional approximation for (4.4), (4.5) is written as
X(t) = Ax(t) + By n g(t) + B0u(t),
(5.2)
gg(t) = nNV7Ng(t) + N[u(t); 0;...; 0]/h.
Since g(t) = [v1(t);...; vN(t)] we can write the matrices of system (5.2), where the state vector is [x(t);g(t)], in the following form
An = [A, BN,..., BN; 0mNxn, (qn)-1 V] ,
V = [ -Im, 0, ... , 0, 0;
^m, ^mo . . . , 0, 0; ,
. . . . . (5.3)
0, 0, ... ,1m, -Im],
BN = [B0; ImN/h;...; 0; 0] .
By Trotter-Kato theorem and Theorem 2 the following estimates are true [3, Theorems 4.4 and 4.10].
(i) If z € b(A), then
|| [In; PN]TtZ - TN [In; PN]z|| < aieat(h/N)||z||Rn xw, VN € N, t> 4h.
(ii) For t > 5h and VN > N0,
||[In; PN]Tt - TtN[In; PN] || < ^(h/N).
(iii) There exists a positive constant a3, dependent on t but independent on N, such that for
every u(-) € Lm[0,t] and all N € N, we have
f TN(t - r)BNu(r)dr 0
< a3 |u|Lm[0,t].
ZN 2
From (iii) it follows that
lim [ TN(t - r)BNu(r)dr = [ T(t - r)Bu(r)dr.
N ./0
It is unknown whether estimates in Theorem 2 and in (i)-(iii) without an assumption (5.1) are true.
6. Application to averaged controllability and examples
For the flock (1.3), Lemma 1 can be reformulated in the following way.
Lemma 4. The flock of systems (1.3) is controllable in average for the weights pi > 0 iff there is a segment [a, b], — h < a < b < T, such that
rank ^ J B(T,a)B'(T,a)da^ = n, (6.1)
where
d r (a+h)AT
B(T, a) = V pi / eAi (T-0) dBi (a — 0).
i=1 aVO
Of course, the condition (6.1) holds iff the equality
l'B(T, a) = 0 a.e. on [a, b] implies that l = 0. (6.2)
Corollary 2 has the form.
Corollary 3. The function B(T, a) from (6.1) can be expressed in the form
d z-aAO
B(T,a) = V pieAi(T-a)bi(T,a), where bi(T,a)=/ eAiS dBi (s).
i=1 J (a-T) V(-h)
If T > h and a = 0, b = T — h, then
f o
bi (T, a) = eAiSdBi(s) = const
J-h
on [a, b]. Hence, the implication (6.2) is equivalent to the rank condition
d . o d „0
rank Vpi / eAi(s+h)dBi(s), VpiA* / eAi(s+h)dBi(s), L i=i i=i
d f o 1
...,^PiAf-1 / eAi(s+h)dBi (s) = n.
(6.3)
Let us pass to the property of LTAC. For the sake of example, we restrict the analysis to the case of the null control, i.e. the goal is to steer and keep the average equal to zero. We also consider the case with d = 2 components, and we chose p1 = p2 = 1/2, and B1(s) = B2(s) = B(s). We do the remark.
Remark 2. All the statements in Section 3 are still valid, if at step k + 1 in (3.4) we consider any matrix Ck+1 = [RC; Lk], with R e Rpxp, det R = 0, and Lk is a matrix of n columns such that ker C n ker Lk = ker C n ker Lk, where Lk is defined by (3.4). With this modification, y has to be modified in Ry.
Let U be the subspace defined by (3.3). In what follows, P denotes the orthogonal projector of Rn on Uand we set E = (A — A)/2, F = (A1 + A2)/2.
Instead of the sequence Ck introduced in (3.4), we use the sequence sk defined by
-fc =
[
PE, PEE,
^n;
—PE; —PEE ;
(6.4)
k-1
—PEE
k-1
G R(k+1)nx2n
PEE'
We can note the following.
• For k = 0 So = 2C = [/„,/„].
• For k = 1, let P0 be the orthogonal projector of Rn on S0[U;U]x = Ux. We see P0 = P. Then we set L1 = [S0;PS0A] = [/n,/n;PA^PA2]. Since kerL1 = kerSi, matrix Si is suitable, according to Remark 2.
• Assume that at step k the matrix Sk given by (6.4) is suitable. We define Pk, the orthogonal projector of R(fc+1)n on Sk[U;U]x = diag [P, /„,..., In]. Then we set
[
Lk+i = [-0; Pk SkA] =
PAi, PEAi
PA2; -PEA
2;
PEE k-1A1, —PEEk-1A2 ].
It is obvious that ker Lk+1 = kerSk+1. So, Sk+1 is suitable.
As in Lemma 2, we have kerS^+1 c kerS^ c kerS0 c R2n. Since dim (ker S0) = n there exists K G 0 : n such that kerSK+1 = kerSK, and we have kerSK = kerSn (see Lemma 2).
As a consequence of Theorem 1 and the above considerations, we obtain the following result.
Corollary 4. Let d = 2 and let A1, A2 G Rnxn, and B1(s) = B2(s). Then for every x10, x20 G Rn the flock of systems (1.3) is LTAC to 0 for p1 = p2 = 1/2 if and only if the condition like (2.2) holds for some 0 < a < b < T — h with
B(T, a) = Sn diag
r-(a+h)AT
'aV0
r (a+h)AT
eAi(T-0)dB(a — 0), / eA2(T-0)dB (a — 0)
J a
'aV0
and rankSn, where the matrix Sn is given by (6.4) for k = n.
Remark 3. The Corollary 4 ensures that the solutions x1(t) and x2(t) of (1.3) (with d = 2 and B1(s) = B2(s) = B(s)) can be steered to some [x1(T); x2(T)] G kerSn. This condition can be equivalently rewritten as
X1 (T )+ X2(T ) = 0, X1(T) — X2(T) G {g G RN : EE kg = 0 Vk G 0: n — 1} .
Let g = (x1 — x2)/2 and f = (x1 + x2)/2. Then for every control v(t) G U we have
x 1 = A1x1 + v(t),
/ = Ef + Eg + v(t), g = Ef + Eg,
x 2 = A2X2 + v(t).
Now it becomes obvious that f (t) = 0 for t > T if and only if v(t) = —Eg(t) and g(t)=eF(t T)g(T) such that Eg(t) = 0 for t > T. Note that v(t) e U.
Of course, we need a control u(t), t > T, such that
f dB(s)u(t + s) = v(t), t > T, (6.5)
./(-h)V(T-t)
similarly to (3.8).
Example 2. Let us return to the flock in the Example 1. We have A1 = [0, —1;1,0], A2 = [1,2; 0,1]. The system has 8 parameters. Let b111 = b112 = b211 = b212 = 1. Other parameters are equal to zero. It corresponds to one control with one delay in the form [u(t) + u(t — 1); 0]. The flock is controllable in average for every T > 1 in the sense of the Remark 1, as condition (6.3) is fulfilled. Here we have the projector P = [0,0; 0,1]. It was shown in [2] that the systems with one scalar ordinary control:
X 1 = A1X1 + [v(t); 0], X2 = A2X2 + [v(t); 0],
are controllable in average, but not simultaneously controllable. Moreover, this system has the long-time averaged controllability property. Hence, there is a control v(t), t > T, owing to the Remark 3. We can find a control u(t), t e [0,T — 1], such that x1(T) + x2(T) = 0 due to controllability. Equation (6.5) is u(t) + u(t — 1) = v(t), t > T. As in (3.9), it can be resolved step-by-step on segments [T + i — 1, T + i]:
ui(t) = v(t) — Ui-1 (t — 1), t e [T + i — 1,T + i], i e N,
where u0(t) = 0.
We can also analyze the property of LTAC for the case B1(s) = B2(s) when d = 2. Then we use the general considerations of Section 3. Note that equation (3.9) can be easily resolved only if the matrix B0 is square and det B0 = 0. For our examples, it corresponds to the condition det [b111 ,b121; b211,b221] = 0. This determinant equals zero in Example 2, but, nevertheless, we found the u(t).
Example 3. Let b112 = b211 = 1 and others parameters equal zero. It corresponds to one control with one delay in the forms [u(t — 1);0] for the first system and [u(t);0] for the second one. The average controllability for every T > 1 is easily verified due to condition (6.3). Introduce B1 = [0,1;0, 0], B2 = [1, 0;0, 0], and v(t) = [v1(t); v2(t)]. The corresponding systems with ordinary controls have the form:
x 1 = A1X1 + B1v(t), x 2 = A2X2 + B2v(t).
This system has the LTAC property with v1(t) = v2(t). We cannot solve the equation u(t — 1) = v2(t), u(t) = v1(t). It may be solved only if v1(t — 1) = v2(t). Let us pass to the approximation from Section 5. Let b=[1;0], then our flock of systems is written as
X 1(t) = A1x1(t) + bu(t — 1), X 2(t) = A2x2(t) + bu(t), y(t) = (X1(t)+ X2(t)) /2.
We need to approximate only the first system. Here m = 1 and ti = —i/N, i e 0 : N. As B(s) = —b%(-TO -1](s), the matrix B^ = b and BN = 0, i e 1 : N — 1. Therefore, matrices (5.3)
have the form
AN = [Ai, 02X(N-i), b; Onx2, (QN)-iV] , V = [ -1, 0, ... , 0, 0;
1, -1, . . . , 0, 0;
0, 0, ... , 1, -1], bN = [02xi; N;...; 0; 0] ,
where Af € r(2+n)x(2+n), V € RNxN, and bN € R2+N. We compose the matrices A = diag [[Af] , B = [[bN] ; b], and C = [/2, 02xf ,/2]/2. For the obtained system x = Ax + Bu, we verify the property of C-LTOC. It does not hold for any N. The flock has not the property of LTAC for the case bii2 = b2ii = 1 and others equal zero.
It can be verified that the approximating system in Example 2 has the LTAC property.
7. Conclusion and open problems
In this paper, we considered the notion of output controllability for ordinary systems with retarded controls and gave the necessary and sufficient condition for that. For the notion of longtime output controllability, we obtained only sufficient conditions. This notions were applied for the investigation of averaged controllability of mentioned systems. The general approach for that is to approximate the systems by the ordinary ones. In connection with the results obtained, a number of interesting open questions arise.
• Assume that there exists a number N0 € N such that for every N > N0 the approximating system has the C-LTOC property. Is it sufficient for C-LTOC property of the original system? And vice versa, if the original system has C-LTOC property, whether it is sufficient for C-LTOC of the approximating system?
• How to obtain any rank conditions for output controllability of systems with delays in the state and control? The same question about the C-LTOC property of such a systems.
We considered the LTAC property for flocks with finite number of members. Can the results be extended for flocks with infinite members?
• Does output controllability imply output feedback stabilisation? Suppose that the system is output controllable, does it exist a feedback control u(t) = Ky(t) such that y(t) goes to zero as t goes to to?
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