Journal of Applied Sciences and Environmental Management World Bank assisted National Agricultural Research Project (NARP) - University of Port Harcourt ISSN: 1119-8362 Vol. 9, Num. 3, 2005, pp. 71-76

Chukwu (1980) showed that if the linear delay system

is uniformly asymptotically stable and

is proper, then

is Euclidean null controllable, provided f satisfies certain growth and continuity condition.

Sinha (1985) studied the non-linear infinite delay system

and showed that (1) is Euclidean null controllable if the linear base system

                          (2)

is proper and the free system

                                                                       (3)

is uniformly asymptotically stable, provided that f  satisfies some growth conditions.

Balachandran and Dauer (1996) studied the null controllability of the non-linear system

                         (4)

Hale (1974) provided sufficient conditions for the stability of systems of the form

                                                   (5)

The aim of this paper is to study the null controllability of systems of the form

     (6)

whereL(t, Ø) is continuous in t, linear in Ø, and is given by

                                                                                             (7)

 BASIC ASSUMPTIONS AND PRELIMINARIES.

Let Eⁿbe an n-dimensional linear space with norm |.| . In equation (6) A_i is a continuous n x n  matrix function for 0 ≤ h_i ≤ h and H (t, θ. x(t)) is n x n matrix valued function which is measurable in (t, θ. x(t)) , and H (t, θ. x(t)) is of bounded variation in ( θ, x(t)) in ( - ∞, ∞) The matrix function B_i(t) i = 2, 1, 2, ........n  are n x p , continuous in t and h + t₀ - mini h_i(t_o) where h₁ (t)are defined below. Here x ε Eⁿ and u ε E^p. Let 0 ≤ h ≤ y  be given real numbers (y may be + ∞). The function η : [- y, 0] → (0, ∞)is lebesque integrable on [- y, 0], positive and non-decreasing. Let B([- y, 0] Eⁿ) be the Banach space of functions which are continuous. Let B([- y, 0] Eⁿ be the Banach space of functions which are continuous and bounded on [-y , 0] such that

for any t ε R  and any function  be defined by

                Assume that the function h_i[t0, t1] → R       i = 0, 1, 2, .........n are twice continuously differentiable and strictly increasing in [t₀, t₁], further

                     .

Let us introduce as in Kantorovich (1992) the following time lead function r₁ with

such that  . Further assume that

h_i(t)=tand for t=t₁ the function h_i(t) satisfies the inequalities

     (8)

                Let the fundamental matrix X satisfy the equation

with initial state  is the solution of  . Using the time lead function and the inequalities (8) we have,

  For brevity introduce the following notations

Consider the homogeneous systems

The controllability matrix of system (14) at time t is given by

                                                                  (16)

where T denotes matrix transpose.

Chukwu (1992), established null controllability of systems of the form (14)

DEFINITION: The system (6) is said t o be null controllable if for each Ø ε B([- y, 0], Eⁿ) there is a t₁≥ t0, u ε L₂ ([t₀, t₁])u is a compact convex subset of E^p such that the solution x(t; t₀, Ø , u) of (6) satisfies x_to(t; t₀, Ø , u)= θ and x(t; t₀, Ø , u)= 0

MAIN RESULT

THEOREM: Suppose that the constraint set u is an arbitrary compact subset of E^p and that

i.                     Assume that system (15) is uniformly asymptotically stable, so that the solution    

ii.                    The linear control system (14) is controllable

iii.                  The continuous function f satisfies

Where

then (6) is Euclidean null controllable.

Proof: Since (15) is controllable, hence it is proper in Eⁿ exists for each w^-1(t₀, t₁) . Suppose the pair of functions x, u  form a solution pair to the following equations

                                                                                  (17)

for some suitable chosen  and

then u  is square integrable on [t₀ - h, t₀] and x is a solution of (6) corresponding to the control u  with initial state Y(t_o) = x(t_o; Ø , η) = 0. Now it is shown that u : [(t₀, t₁) ] → u  is a compact constraint subset of Eⁿ that is |u| ≤ a  for some constant a > 0  Since (15) is uniformly asymptotically stable and B_i are continuous in t, it follows that

and therefore

since β - α ≥ 0 and s ≥ t_o ≥ 0. From (18) t₁ can be chosen so large that which proves that u  is an admissible control for this choice of t₁

It remains to prove the existence of a solution pair of the integral equation (16) and (17). Let B be the Banach space of all functions  where  with the norm defined by

  Define the operation  T : B → by T(x, u) = (y, v) where

Because of the various assumptions on our system and the estimate from (17) to (19) it is clear that  

It follows that T : Q(r → Q(r)) since Q(r) is closed, bounded and convex, by Rieze’s theorem (Kantorovich and akilov, 1992) it is relatively compact under T. The Schauders theorem implies that T has a fixed point (x, u) ε Q(r), this fixed point (x, u) of T is a solution pair of the set of integral equations (20) and (21). Hence the system (6) is Euclidean null controllable.

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