On Exact Controllability of First-Order Impulsive Differential Equations

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:

(1.1)

(1.2)

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:

(2.1)

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

(2.2)

If the system is infinite dimensional, that is,

(2.3)

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,

(3.1)

with the initial condition

(3.2)

Here, continuous and

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

(3.3)

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

(3.4)

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

(3.5)

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

(3.6)

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

(3.7)

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

(3.8)

Define by the right-hand side in(3.8)

(3.9)

Thus, for this control,

(3.10)

We have thus the following result.

Theorem 3.1.

If for any initial condition and final condition the operator

(3.11)

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

(4.1)

with the norm

(4.2)

Let and the restriction of to that subinterval

The space

(4.3)

with the norm

(4.4)

is a Banach space.

Now consider the impulsive control differential equation

(4.5)

(4.6)

where is continuous and there exist the limits

(4.7)

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

(4.8)

Reciprocally, if satisfies (4.8), then is a solution of (4.5)-(4.6).

See [2] for the proof.

Now, define the operators by

(4.9)

and

(4.10)

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

(5.1)

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

(5.2)

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

(5.3)

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

(5.4)

where are constants.

Also, there exist constants such that for any

(5.5)

Combining these two last inequalities we get

(5.6)

for some constants Consequently, for any we have

(5.7)

If is a solution of the equation then

(5.8)

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

(5.9)

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

(5.10)

for some the impulsive system (5.9)-(5.10) is completely controllable.