On the controllability of a class ofnonlinear systems with time-varying

multiple delays in controlKrishnan Balachandran, B.Ed., M.Sc, M.Phil., Ph.D.

Indexing terms: Nonlinear systems, Control theory

Abstract: Sufficient conditions for global relative controllability of nonlinear systems with time-varying multipledelays in control and implicit derivatives are derived. The special case, when the delays are constant, is alsoconsidered and an example is provided. The results are obtained using the measure of noncompactness of a setand Darbo's fixed-point principle.

1 Introduction

Controllability of nonlinear systems with delays of varioustypes has been investigated by several authors [1-5], bymeans of Schauder's fixed-point theorem. With the help ofFan's fixed-point theorem in Reference 6 and the Leray-Schauder theorem in Reference 7 Chukwu discussed thecontrollability and null-controllability of nonlinear delaysystems with prescribed controls respectively. Dacka [8]introduced a new method of analysis to study the control-lability of nonlinear systems with implicit derivatives,based on the measure of noncompactness of a set andDarbo's fixed-point theorem. This method has beenextended to a larger class of nonlinear dynamic systems byBalachandran [9] and to nonlinear delay systems byDacka [10, 11]. Balachandran and Somasundaram[12, 13] have discussed the problem for nonlinear systemshaving implicit derivatives with different types of delays incontrol variables. The objective of the present paper is toshow that the method used in References 8 and 9 can beextended to the study of systems of nonlinear equationswith time-varying multiple delays in control variables. Theresults obtained are a generalisation of results given in Ref-erences 11 and 13.

2 Mathematical preliminaries

Let (X, || • ||) be a Banach space and E a bounded subset ofX. In this paper, the following definition of the measure ofnoncompactness of a set E is used [14]:

n(E) = inf{r > 0; E can be covered by a finite number

of balls whose radii are smaller than r}

The following version of Darbo's fixed-point theorembeing a generalisation of Schauder's fixed-point theoremshows the usefulness of the measure of noncompactness. 'If5 is a nonempty bounded closed convex subset of X andP: £—> S is a continuous mapping, such that for any set£ c 5 w e have

where k is a constant, 0 ^ k < 1, then P has a fixed point.'For the space of continuous functions Cn[t0, t{] the

measure of noncompactness of a set E is given by

H(E) = iwo(£) = i lim w(E, h)

Paper 4802D (C8), received 9th September 1985

The author is with the Department of Mathematics, Bharathiar University, Coim-batore 641 046, Tamil Nadu, India

IEE PROCEEDINGS, Vol. 133, Pt. D, No. 6, NOVEMBER 1986

where w(E, h) is the common modulus of continuity of thefunctions which belong to the set E, that is

w(E, h) = s u p [ s u p | x(t) — x(s) \:\t — s\^h]x e £

where as in the space of differentiable functions Cln[tQ, t{]

we have

where

DE = {x : x e E)

3 Basic assumptions and definitions

Consider the following nonlinear systems with time-varying multiple delays in control represented by the equa-tion

x(t) = A(t, x(t), x(t))x(t)

Bit, x(t),i = 0

+ f(t, x(t), x(t)) (1)

where the state x(t) is an n-vector and the control u(t) is ap-vector, A(t, x, x) is an n x n matrix, B((t, x, x) fori = 0, 1, . . . , M are n x p matrices and f(t, x, x) is an n-vector function.

Assume that the elements ajk of A(j, k = 1, 2, . . . , n) andbijk of Bjj = 1, 2, . . . , n, k = 1, 2, . . . , p) for i = 0, 1, . . . , Mare continuous functions and fulfil the following condi-tions :

IaJk(t,

Ibijk{t, { for each t e [t0, t j

and x, y e R" (2)

where N, L{(i = 0, 1, . . . , M) and K are positive constants.Further, for every y, y e R" and x e R", t e [t0, t1],

1/

I ajk(t, x, y) - ajk(t, x, y) | ^ -± \ y - y \

a-I bijk{t, x, y) - bijk(t, x, y) \ ^ — | y - y

\f(t,x,y)-J{t,x,y)\^k2\y-y\

(3)

where kx, k2 and at{i = 0, 1, ..., M) are positive constantsand 0 ^ k2 <3- Define the norm of a continuous n x p

297

matrix valued function D(t) by

||Z)(t)||=max £ \djk(t)\j k = i

where djk are elements of D.Assume that the functions ht : [t0, t j —» R, i = 0, 1, . . . ,

M are twice continuously differentiate and strictlyincreasing in [t0, t{\. Moreover,

hit) ^ t for t e [r0, t j i = 0, 1, . . . , M (4)

Let us introduce the time lead functions [2]:

r& : Mto), hfrj] -> [t0, tj, i = 0, 1, . . . , M

such that rih&t)) = t for t e [f0, *i]- Further assume thatfjo(t) = t and for t = tx, the functions Ait-(0 satisfy theinequalities

M ' i ) Ah)= U ' i ) ^ *«-i(t,) < • • • < M*i) < M'i) = ti (5)

The following definitions of complete state of the systemeqn. 1 at time t and global relative controllability areassumed [2, 5].

Definition 1: The set y(t) = {x(t), P(r, s)} whereP(t, s) = u(s) for s e [min, /i,(t), 0 is said to be the completestate of the system eqn. 1 at time t.

Definition 2: System eqn. 1 is said to be globally rela-tively controllable on fj0 , t{\ if, for every complete statey(t0) and every vector xt e R", there exist a control u(t)defined on [t0, tx] such that the corresponding trajectoryof the system eqn. 1 satisfies x(tx) = x : .

For each fixed z e Cl[t0, t{], consider the followinglinear system:

M

x(t) = A(t, z, z)x(t) + X Bit, z, zMhft)) +f(t, z, z) (6)i = O

The solution of the system (6) with x(t0) = x0 can beexpressed in the form

x(t) = F(t, t0; z, z)x0 + F(t, s;z,z)Jto

M

x Z Bi(s> z, Z)«(/J,-(S)) dsi = 0

t

+ F(t, s; z, z)f(s, z, z) ds (7)

where F(t, t0; z, z) is the transition matrix of the linearsystem

x(t) = A(t, z(t), z(t))x(t)

with F(t0, t0; z, z) = I, the identity matrix.Using the time lead function rt{t), eqn. 7 can be written

as

x(t) = F(t, t0; z, z)

[ M Chiit)xo + Z F(*o> ri(s);z, z)

i = 0 Jhi(to)x Biir^s), z, z)ri{s)u(s) ds

F(to,s;z, z)f(s, z, z)ds\ (8)J

the following form:

x(ti) = F(tl, t0; z, z)

[ m froxo + Z ^(fo> r .( s) ' z, z)

i = O Jhi(to)

xBi(ri(s),z,z)ri(s)fi(to,s)dsM

+ Z /U0,r,<s);z, fc)i = m + 1 Jh,(f o)

x Bi(rl{s),z,z)ri{s)P{t0,s)ds

+ Z r^Wo.^jz.z); = o Jto

x UjCXs), z, z)rt{s)u(s) ds

+ lF(t0,s;z,z]f{s,z,z)ds\ (9)

For brevity, let us introduce the following notations:

Gtt0,t;z,z)= Z ̂ o . rfi); z, i)Bfrf& z, z)r}{t) (10)j=o

m fto

H(t, z, z) = Z F(lo»'"Xs); z> z)

+ Z ^0,r,<s);z, z)i = m + 1 J/ii(f o)

(11)

Jto

By eqn. 5, the equality eqn. 8 for t = tx can be expressed in

298

x Bi(ri(s),z,z)ri(sMt0,s)ds

t0), Xi; z, z) = F(t0, tx; z, z)xx - x0

f'1- H(tltz, z)- F(t0, s; z, z)

Jto

xf(s,z,z)ds (12)

Define t h e c o n t r o l l a b i l i t y m a t r i x W(t0, tx; z , z) b ym - l ChAt\)

W{t0, ti; z, z) = Z G.(fo. s ; z> ̂ )

x G-(t0, s; z, z) ds (13)

where the prime indicates the matrix transpose.

4 Controllability results

Theorem: Given the system eqn. 1 with conditions givenby eqns. 2 to 5, and

inf d e t W{t0 ,tl;z,z)>0

then the system eqn. 1 is globally relatively controllable on

Proof: Define the nonlinear operator T on the Banachspace C;|rj0, t{] by the formula

T(z)(t) = F(t, t0; z, z)Oo + H{t, z, z)t

F(t0, s; z, z)/(s, z, z)ds

(14)

Jtom - l p*,-(t)

+ Z F(to,ri(s);z, z)« = 0 J/i;+i(t)

x fi^Xs), z, )̂̂ ,<s)«(s) ds

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where the control u(t) is given by

u(t) = G[{t0, t; z, z)W~\t0, tx; z, z)q{y{t0), xx; z, z)

for r e [ W i X Wi)) (15)

where y(t0) and xt are chosen arbitrarily. Substituting eqn.15 into eqn. 14, we obtain

C2= sup

T(z)(t) == F{t, t0; z, z)\ x0 + H{t, z, z)

F(t0, s; z, z)f(s, z, z) ds

F(t0, riis);z,z)

JfO

m - l

1 = 0 Jh,+ i(

x Bfrfa), z, '£tOi t; z, z)

x Wl{t0, tx; z, z)qiy{t0), xx\ z, z) ds (16)

To prove the continuity of the operator T, let us assumethat the sequence of functions zn e C\[t0, t{] tends to z innorm, that is zn tends uniformly to z and zn tends uni-formly to z. It is known from the theory of linear differen-tial equations that the fundamental matrix F(t, t0; zn, zn)tends uniformly to F(t, t0; z, z) and the uniform con-vergence of their derivatives follows by the equality

— F(t, to;zn, zn) = A(t, zn, zn)F(t, t0; zn, zn)

and the local uniform continuity of the matrix A. Similarlythe uniform convergence of the sequence B£t, zn(t), zn(t))for i = 0, 1, 2, . . . . , M and f(t, zn(t), zn{t)) to Bt(t, z(t), z(t))and f(t, z, z), respectively, follow from the continuity of thematrices Bt and the function / If we denote un the controlcorresponding to the function zn by eqn. 15, we easily seethat un tends uniformly to the control u corresponding tothe function z as above. As the integrals involved in eqn.14 improve the uniform convergence upon uniform con-vergence with the first derivative, and hence the operator Tis continuous. Further, it maps the space C*[f0, t{] intoitself.

Let us consider the closed convex subset of C^[t0, t jdefined by

H = [_z : z e C j [ t 0 , t{\, \\z\\^NY, || Dz \\ ^ N 2 ]

where the positive real constants N^ and N2 are defined by

[ m - l

Ci+ I (Mi)-Wii = 0

x exp (nN(tl - r&„

with

a,. = |r,(s)|, b = \fl(to,s)\

m

C1=\x0\+ X(«o - Hi = O

x exp {nN(ri(t0) - t0))M

i = m + 1

x exp (nN(t1 - t0)) + K{t^ - t0) exp (nN(tt - t0))

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x 2. exp(ni;=o

x npLiai[_C1 +\xx\ e x p (nN(t1 - t o

and

N2 = n<M

npL{ C2i = 0

0 ^ k, N, < i 0 ^ (a0 aM)C2 < \ (17)

The operator T maps H onto itself. All the functionsT(z)(t) are equicontinuous, because they have uniformlybounded derivatives. Now we shall find an estimate of themodulus of continuity of the functions DT(z)(t) for t, s e[*o.*i]

\DT(z)(t)-DT(z)(s)\

^\A(t,z(t),z(t))T(z)(t)

- A(t, z(t), z(s))T(z)(t)\

+ \A(t, z(t), z(s))T(z)(t)

- A(s, z(s), z(s))T(z)(s)\

; = o

- Bfr, z(t), zi

i = 0

- B^s, z(s), zisMhm

+ \f(t,z(t),z(t)-f(t,z(t),z(s))\

+ \f(t,z(t),z(s))-f(s,z(s),z(s))\ (18)We can take the upper estimates of the second, fourth andsixth terms of the right-hand side of inequality 18 as /?0(| t— s\), Pi(\t — s\) and /?2(11 — 51), respectively, where /?,-

are the non-negative functions, such that lim,,_,0+ /?,(/i) = 0.The first, third and fifth terms of inequality 18 can be madeas

*iAU*W-*(s)|, f.*tC2\z(t)-z(s)\> = o

and k2 \ z(t) — z(s) |, respectively.Setting k = klNl +(a0+<*! + •

p = p0 + fit + f}2, we finally obtain

| DT(z)(t) - DT(z)(s) | ^ k \z{t) - z(s) \ + f3

hence, we infer that

w(DT(z), h) ^ kw(Dz, h) + P{h)

Thus, we have for any bounded set E a H,

kfi(E)

+ aM)C2 + k2 and

t - s \

Consequently, by the Darbo fixed-point theorem, the oper-ator T has at least one fixed point; therefore, there exists afunction z* e Cj,\_t0, t{\ such that

z*(t) = T(z*(t)) (19)

Differentiating with respect to t, we easily verify that x(t)given by eqn. 19 is a solution of the system eqn. 1 for thecontrol uit) given by eqn. 15. The control steers the systemeqn. 1 from the initial complete state y(t0) to x(tl) = xt eR" on the interval [f0, t x ] , and, because y(t0) and xx have

299

been chosen arbitrarily, then, by definition 2, the systemeqn. 1 is globally relatively controllable on \_t0, fx].

Remark: If we assume that the nonlinear function in eqn.1 also satisfies the Lipschitz condition, with respect to thestate variable, then we can obtain the unique responsedetermined by any control.

Corollary: Let h£t) = t — ht, ht ̂ 0. Then the system eqn.1 is globally relatively controllable on [r0, t{\, if

inf det Wl{to,ty\z,z)>0 (20)

where

m-l Ct\-hi

Gito,s;z,z)G'i{to,s;z,z)ds

with_

Gi(t0, tx; z, z) =j = 0

,t + hj; z, z)BJ(t + hj3 z, z)

Proof: If ht{t) = t - h{:, then

r,{t) = t + ht, r,(0 = 1 for i = 0, 1, ..., M (21)

Substituting eqn. 21 into W and using the above theorem,it is easily verified that the system eqn. 1 is globally rela-tively controllable on [t0, t{\ if the inequality eqn. 20holds.

5 Example

Consider the system

x (t)*i(0 = x2(t) + x2(t) sin - ^

lo

xx2(t) = Xl(t) + xx(0 sin -f

l

sin x2(t))

lo

+ tu2(t) + tu2(t - h)

In the matrix form, we have

A(t, x, x) =0

1 + s i n * "

0

B0(t, x , x) = Bx{t, * , * ) = * °t

It is easily seen that A, Bo, Bl and / are bounded andcontinuous. Moreover, the functions satisfy the Lipschitzconstant with respect to the variable x, with the constantk1 = 0.25, a, = 0, k2 = 0.25. For any fixed z e C2[t0, tx],

the state transition matrix F(t, t0; z, z) has the followingform:

., l[a + b a-blF(t, t0; z, z) = -

2\_a — b a + bj

1 + sin -7 ) ds16.

where

a = a(t) = exp I ( 1 + sin 77 } dsho

b = b(t) = exp - (1Jro V

and the controllability matrix Wx(t0, tl;z, z) is

wt _ j . f ( 1 "T a\ + b\ a1a2-^bib2~\4 Jfo \_aia2 + b

lb2 al + bl J

1 f'1 Ra + b)2 + s2(b - a)2 (b2 - a2)(l + s2) 1+ 4 J,, _ J_ (b2 - a2){\ + s2) (b - a)2 + s\a + b)2\

where a1=a + a + b + b with a = a(s + h), b = b(s + h)

a2 = b + b — a — a

b1 = s(b — a) + (s + h)(b — a)

b2 = s(a + b) + {s + h)(a + 6)

We observe that the infimum of det tVi(t0, tx; z, z) isgreater than zero. Further, choose Nt in such a way that itsatisfies the condition of eqn. 17 in the theorem, as 0 <JVt < 4/3. Hence, by the corollary, the system is globallyrelatively controllable.

6 References

1 MIRZA, K.B., and WOMACK, B.F.: 'On the controllability of non-linear time delay systems', IEEE Trans., 1972, AC-17, pp. 812-814

2 KLAMKA, J.: 'Relative controllability of nonlinear systems withdelays in control', Automatica, 1976,12, pp. 633-634

3 DAUER, J.P., and GAHL, R.D.: 'Controllability of nonlinear delaysystems', J. Optimiz. Theory & Appl., 1977, 21, pp. 59-70

4 KLAMKA, J.: 'Controllability of nonlinear systems with distributeddelays in control', Int. J. Control, 1980, 31, pp. 811-819

5 BALACHANDRAN, K., and SOMASUNDARAM, D.: 'Control-lability of nonlinear systems consisting of a bilinear mode with timevarying delays in control', Automatica, 1984, 20, pp. 257-258

6 CHUKWU, E.N.: 'Controllability of delay systems with restrainedcontrols', J. Optimiz. Theory & Appl., 1979, 29, pp. 301-320

7 CHUKWU, E.N.: 'Null controllability of function space of nonlinearretarded systems with limited controls', J. Math. Anal. & Appl., 1984,103, pp. 198-210

8 DACKA, C.: 'On the controllability of a class of nonlinear systems',IEEE Trans., 1980, AC-25, pp. 263-266

9 BALACHANDRAN, K.: 'Global and local controllability of nonlin-ear systems', IEE Proc. D, Control Theory & Appl., 1985, 132, (1), pp.14-17

10 DACKA, C : 'On the controllability of nonlinear systems with timevariable delays', IEEE Trans., 1981, AC-26, pp. 956-959

11 DACKA, C : 'Relative controllability of perturbed nonlinear systemswith delay in control', ibid., 1982, AC-27, pp. 268-270

12 BALACHANDRAN, K, and SOMASUNDARAM, D.: 'Control-lability of a class of nonlinear systems with distributed delays incontrol', Kybernetika, 1983,19, pp. 475-482

13 SOMASUNDARAM, D., and BALACHANDRAN, K.: 'Relativecontrollability of a class of nonlinear systems with delay in control',Indian J. Pure & Appl. Math., 1983, 14, pp. 1327-1334

14 SADOVSKII, B.J.: 'Limit compact and condensing operators', Russ.Math. Surveys, 1972, 27, pp. 95-156

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