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On Exact Controllability of First-Order Impulsive Differential Equations
Advances in Difference Equations volume 2010, Article number: 136504 (2010)
Abstract
Many dynamical systems have an impulsive dynamical behavior due to abrupt changes at certain instants during the evolution process. The mathematical description of these phenomena leads to impulsive differential equations. In this work, we present some new results concerning the exact controllability of a nonlinear ordinary differential equation with impulses.
1. Introduction
Many evolution processes in nature are characterized by the fact that at certain moments in time they experience an abrupt change of state. Such behavior is seen in a range of problems from: mechanics; chemotherapy; population dynamics; optimal control; ecology; industrial robotics; biotechnology; spread of disease; harvesting; physics; medical models. The reader is referred to [1–8] and references therein for some models and applications to the above areas.
The branch of modern, applied analysis known as "impulsive" differential equations furnishes a natural framework to mathematically describe the aforementioned jumping processes. Consequently, the area of impulsive differential equations has been developing at a rapid rate, with the wide applications significantly motivating a deeper theoretical study of the subject [9–11].
Impulsive control systems have been studied by several authors [12–18]. In [15] the problem of controlling a physical object through impacts is studied, called impulsive manipulation, which arises in a number of robotic applications. In [16] the authors investigated the optimal harvesting policy for an ecosystem with impulsive harvest. For some recent references on different control strategies, including impulsive control, we refer the reader to [13, 19–26] and the references therein.
Now, let and
and
, and
is an
real matrix.
Consider the following impulsive control differential equation:


where is an operator defined on a set of admissible controls
and
As usual, and
Our purpose is to the system (1.1)-(1.2) from the initial state to a desired final state
in the finite time
We say that system (1.1)-(1.2) is exactly controllable in the time if for any
there exists a control
such that a solution
of (1.1)-(1.2) satisfies
Of course, we specify below the space of solutions and controls.
The main idea of our approach is to transform the controllability problem to the existence of a fixed point of an appropriate nonlinear operator generated by the original problem. This approach is not new and has been used by some authors such as Bhat [27], Chang et al. [28, 29], Sakthivel et al. [30], and Tonkov [31].
2. Some General Results on Exact Controllability
Consider the following finite-dimensional linear system:

where is an
matrix, and
This linear system is completely controllable if for any there exists a
and a control function
defined for
such that the solution to (2.1) with initial condition
satisfies
. It is well known [32] that the linear system (2.1) is completely controllable if and only if

If the system is infinite dimensional, that is,

where is the infinitesimal generator of a strongly continuous semigroup in a Hilbert space
, and
a linear bounded operator from a Hilbert space
into
then if the semigroup is compact the linear system (2.3) is not exactly controllable [33–35].
In this paper we study the finite-dimensional nonlinear impulsive control problem (1.1)-(1.2).
3. Exact Controllability without Impulses
Consider (1.1) without impulses, that is,

with the initial condition

Here, continuous and
In what follows, and hence if
is a solution of the initial problem (3.1)-(3.2), then

We can define the following operator defined by the right-hand side of (3.3):

In what follows is the identity operator so that the control space is
Note that
depends on the initial condition
and for any control

Now, for suppose that we are able to find a control
such that

This means that for the control the system transfers the initial state to the desired final state if

has a fixed point. Consequently, if the operator , which of course depends on the initial state
, has a fixed point for any initial state, then the system is exactly controllable.
To clarify the ideas exposed above, suppose that and let us introduce the control

Define by the right-hand side in(3.8)

Thus, for this control,

We have thus the following result.
Theorem 3.1.
If for any initial condition and final condition
the operator

has a fixed point, then system (3.1) with is exactly controllable.
4. Exact Controllability with Impulses
As usual, see any of the references on impulsive differential equations, we consider the Banach space

with the norm

Let and
the restriction of
to that subinterval
The space

with the norm

is a Banach space.
Now consider the impulsive control differential equation


where is continuous and there exist the limits

and
are continuous
By a solution of (4.5)-(4.6), and for continuous controls we mean a function
satisfying (4.5) for every
and the impulses (4.6). In the case that the control
is, for example, in the space
locally, the solution must satisfy (4.5) for almost every
for each
and the impulses indicated in (4.6).
Lemma 4.1.
If is a solution of (4.5)-(4.6), then
satisfies

Reciprocally, if satisfies (4.8), then
is a solution of (4.5)-(4.6).
See [2] for the proof.
Now, define the operators by

and

As in the nonimpulsive case, this control steers the system for the initial state
to the final state
in the finite time
Consequently we have the following result.
Theorem 4.2.
If has a fixed point for any initial condition
and final condition
, then the impulsive system (4.5)-(4.6) is exactly controllable.
5. Main Results
Schauder fixed point theorem states that any continuous mapping of a nonempty convex subset of a normed space into a compact set of that normed space has a fixed point [36, Theorem 4.1.1]. One of the most useful consequences is Schaefer's theorem [36, Theorem 4.3.2].
Theorem 5.1.
Let be a normed space with
a compact mapping. If the set

is bounded then has at least one fixed-point.
The operators and
are continuous and compact [2, 37]. Consequently,
is also continuous and compact and we can apply Schaefer's theorem.
Theorem 5.2.
Suppose that has a sublinear growth, that is, there exist constants
and
such that for every

Assume that the impulses have sublinear growth. For every there exist
and
such that for every
one has

then the operator has a fixed point for any
and the impulsive system (4.5)-(4.6) is exactly controllable.
Proof.
Let Using (5.2) and (5.3), it is evident that for any

where are constants.
Also, there exist constants such that for any

Combining these two last inequalities we get

for some constants Consequently, for any
we have

If is a solution of the equation
then

For define
Noting that
we see that
Taking
we deduce that the set
is a bounded set.
Hence all the possible solutions of the equation are bounded "a priori." By Schaeffer's theorem,
has a fixed point, which is equivalent to the exact controllability of the impulsive system (4.5)-(4.6).
As a consequence we have the following.
Theorem 5.3.
Assume that is bounded and the impulses
are also bounded. then the operator
has a fixed point for any
and the impulsive system (4.5)-(4.6) is exactly controllable.
When the nonlinearity is bounded, (4.5) is not exactly controllable in general. Even in the linear case
the equation is not exactly controllable in general; see condition (2.2). However, by adding adequate impulses we can control the equation and hence the system becomes exactly controllable.
Example 5.4.
Let be an
real matrix. Consider the system

such that does not satisfy (2.2). Then, (5.9) is not completely controllable. However, by adding the impulse

for some the impulsive system (5.9)-(5.10) is completely controllable.
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Acknowledgments
The research of C. C. Tisdell was supported by funding from The Australian Research Council's Discovery Projects (DP0450752). The research of J. J. Nieto was partially supported by Ministerio de Educación y Ciencia and FEDER, Project MTM2007 { 61724, and by Xunta de Galicia and FEDER, Project PGIDIT06PXIB207023PR.
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Nieto, J., Tisdell, C. On Exact Controllability of First-Order Impulsive Differential Equations. Adv Differ Equ 2010, 136504 (2010). https://doi.org/10.1155/2010/136504
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DOI: https://doi.org/10.1155/2010/136504