Numerical approach to the controllability of fractional order impulsive differential equations

1 Introduction

Sometimes, integer-order differential equation becomes inadequate to model some physical phenomena such as in anomalous diffusion. Such phenomena give rise to fractional order differential equations. The main advantage of studying fractional order systems is that they allow greater degrees of freedom in the model. Differential equations of fractional order appear more often in diverse areas of science and engineering, such as image processing, signal processing, bio-engineering, viscoelasticity, fluid flow and control theory [1,2,3,4,5,6]. For the fundamental understanding of fractional calculus and related numerical methods, one can refer to [7,8,9,10,11].

On the other hand, some phenomena are characterized by rapid changes. The first kind of changes takes place over a relatively short time compared to the overall duration of the entire process. Mathematical models in these cases are developed using impulsive differential equations. In the second kind, the changes are not negligibly short in duration and these changes begin impulsively at some points and remain active over certain time intervals. The mathematical model of these situations gives rise to a differential equation with non-instantaneous impulses.

The study of non-instantaneous impulsive differential equations has significant applications in different areas, for example, in hemodynamical equilibrium and the theory of rocket combustion. An excellent application of non-instantaneous impulse is the introduction of insulin into the bloodstream. It produces an abrupt change in the bloodstream. The consequent absorption is a gradual process that remains active over a finite time span.

Recently, many researchers have shown their interest in existence, uniqueness of solutions, stability and controllability of impulsive problems with non-instantaneous impulses [12,13,14,15,16]. Hernández and Regan [17] studied mild and classical solutions for the impulsive differential equation with non-instantaneous impulses. Wang and Fečkan [18] have shown existence, uniqueness and stability of solutions of such a general class of first-order impulsive differential equations. Later, Muslim et al. [12] investigated existence, uniqueness of solutions and stability of second-order differential equations with non-instantaneous impulses. However, the controllability of the non-instantaneous impulsive control system is the less treated topic as compared to the existence and uniqueness of solutions.

In the setting of controllability, the control system is an interconnection of components forming a system configuration that will result in a desired system response. Controllability is one of the structural properties of dynamical systems. It provides the ability to move a system around entire configuration space using only certain feasible manipulations. It deals with whether or not the state of a state-space dynamic system can be controlled from the input. Many authors dealt with controllability problems that can be found in many recently published papers [19,20,21,22].

Recently, Wang et al. [19] discussed controllability of fractional non-instantaneous impulsive differential inclusions. However, Wang et al. achieved exact controllability by only applying control in the last subinterval of time. But, Wang et al. did not propose any computational scheme for the steering control. In this manuscript, the control is applied for each subinterval of time, due to which the concept of total controllability arises. Moreover, none of the research papers have so far discussed the numerical approach for the controllability of the non-instantaneous impulsive differential equation of order α∈(1,2]. Therefore, this manuscript is devoted to the study of numerical controllability for the following fractional order nonlinear differential equation with non-instantaneous impulses in a space ℝn:

where α∈(1,2] and u(t) is a state function with time interval 0=s0=t0<t1<s1<t2,…,tm<sm<tm+1=T<∞. Let A be a coefficient matrix of system (1.1). The control function w(⋅)∈L2(J1≔⋃i=0m[si,ti+1],ℝm). Let B be a bounded linear operator from ℝm to ℝn. Consider the state function u∈C((ti,ti+1],ℝn),i=0,1,…,m, and there exist u(ti−) and u(ti+),i=1,2,…,m with u(ti−)=u(ti). The functions Ψi1(t,u(ti−)) and Ψi2(t,u(ti−)) represent non-instantaneous impulses during the intervals (ti,si],i=1,2,…,m. Ψi1,Ψi2 and g are suitable functions. These functions will be explained briefly in the subsequent sections.

The manuscript proceeds as follows. In Sections 1–3, the introduction, notations, results and required assumptions are given, which will be required for the later sections. In Section 4, controllability of problem (1.1) is investigated by the iterative scheme. Later, controllability of the integro-differential equation is briefly mentioned in Section 5. In Section 6, a few numerical examples are given to show the application of the obtained results.

2 Preliminaries and assumptions

In this section, some useful definitions related to fractional calculus are briefly reviewed. Also, some necessary properties concerned with the Mittag-Leffler function are discussed.

Let L1(ℝ+,ℝn) and C1(ℝ+,ℝn),ℝ+=(0,∞) be the space of all integrable functions and continuously differentiable functions, respectively. Furthermore, ℝn be the n-dimensional Euclidean space, where n∈ℕ.

The Laplace transform of the Riemann-Liouville fractional derivatives is defined by

The Laplace transform of the Caputo fractional derivatives is defined by

In particular, if α∈(1,2], then

3 Controllability for the linear system

We consider the linear impulsive differential equation of order α∈(1,2] with control function u(t) as follows:

where u∈ℝn and control function w(⋅)∈L2(J1≔⋃i=0m[si,ti+1],ℝm). Let B be a bounded linear operator from ℝm to ℝn. The solution of the linear impulsive system (3.1) is given by the following integral equation:

In order to prove the controllability for the nonlinear system (1.1), the following assumptions are taken:

(A1) g:J1×ℝn→ℝn, J1=⋃i=0m[si,ti+1] is a continuous function and there exist positive constants λ1 and λ2 such that

and ∥g(t,u)∥≤λ2, for every u,v∈ℝn,t∈J1.

(A2) There exist positive constants CΨi1andCΨi2,i=1,2,…,m such that CΨi1=maxt∈Ii∥Ψi1(t,⋅)∥andCΨi2=maxt∈Ii∥Ψi2(t,⋅)∥, where Ii≔[ti,si].

(A3) Ψik∈C(Ii×ℝn,ℝn) and there are positive constants LΨik,i=1,2,…,m,k=1,2, such that ∥Ψik(t,u)−Ψik(t,v)∥≤LΨik∥u−v∥, ∀t∈Iiandu,v∈ℝn.

Let ℋ=PC([0,T],ℝn) be the space of piecewise continuous functions.

PC([0,T],ℝn)≔{u:[0,T]→ℝn:u∈C((tk,tk+1],ℝn),k=0,1,…,m, and there exist u(tk−) and u(tk+),k=1,2,…,m}. It can be easily proved that PC([0,T],ℝn), for all t∈[0,T], is a Banach space endowed with the supremum norm. For the sake of notational convenience, let us define

4 Controllability for the nonlinear system

Steering of a dynamical control system from an arbitrary initial state to an arbitrary final state on each sub-interval [tk,tk+1] using the set of admissible controls is called a totally controllable system. In this section, total controllability of system (1.1) is investigated through the iterative scheme.