Constrained Controllability of the h-Difference Fractional Control Systems with Caputo Type Operator

The problem of controllability to a given convex target set of linear fractional systems with h-difference fractional operator of Caputo type is studied. Necessary and sufficient conditions of controllability with constrained controllers for such systems are given. Problem of approximation of a continuous-time system with Caputo fractional differential by a discrete-time system with h-difference fractional operator of Caputo type is discussed.


Introduction
In the control theory there are three fundamental concepts: controllability, observability, and stability of the given control system.Controllability means that there is a possibility to transfer the considered system from a given initial state to a final state using controls from some set.Observability means the possibility of the reconstruction of an initial state on the base of controls (inputs) and output sequences.It seems that these concepts in the classical linear control systems (for both continuous-time and discrete-time cases) are quite well investigated.In last years some of theirs extensions on nonuniform time domain have been done; see, for example, [1,2].The other extension of the classical theory of control systems is introducing instead of the classical time derivative a fractional one (see, e.g., [3,4]) or instead of the classical difference operator a fractional difference (see, e.g., [5][6][7]).In the recent years the fractional calculus is viewed as a power tool in descriptions of real system' behaviours; see, for example, [8][9][10].
The controllability problem of fractional linear systems was studied, for example, in [3,7,[11][12][13][14].In [15] controllability of linear systems with Caputo type difference operators with two different fractional orders was investigated.Later on these results were extended to ℎ-difference linear control systems with  different fractional orders in [16].
In many cases it is assumed that the set of control values coincides directly with the whole control space, but in practise this set should be bounded; see, for example, [17].A restriction on controls possesses some difficulties for controllability conditions.Necessary and sufficient conditions for controllability of classical linear systems with control constrains were proven in [18][19][20].In [21] some of results of [20] were extended to any time model.Conditions for constrained controllability for a special class of linear systems with two fractional orders and with Caputo type difference operators were tackled in [22].Generally, in the field of systems with fractional order, this topic is worthy of investigation.To the best of our knowledge, there exist only few works in continuous-time case; see [23].
The goal of the present work is to give conditions for possibility of steering of linear ℎ-difference control system with fractional difference Caputo type operator to a given convex target set  from a specified initial state.To this aim in Section 2 there are presented the needed definitions and facts from fractional ℎ-difference calculus.Next, in Section 3, using the separation hyperplane theorem, necessary and sufficient conditions for constrained controllability of the considered system are discussed.In [22] the first step in the investigation of the problem of constrained controllability for discrete-time fractional two-order linear systems, using the specific transition and gramian controllability matrices, was discussed.Now, in Section 3, using the Z-transform method and Mittag-Leffler function introduced in [24] and similar reasoning (that idea in fact comes from [20]), necessary and sufficient conditions for constrained controllability of the 2 Discrete Dynamics in Nature and Society considered system are presented.These results can be quite easily extended to multistep and/or multiorder fractional systems, but notation would be more complicated.
As real continuous-time models in some cases should be approximated by discrete-time models, the problem of approximation of a continuous fractional coast function by a discrete-time one is discussed in Section 4. Section 5 discussed a particular case of constrained controllability, namely, the case when the target set consists only of one point.This specific situation obtains when a given object is transferred to a predetermined position.
For our goal let us recall the separation theorem.Suppose that  is a real normed space and ,  ⊂ .
Theorem 1 (see [25]).Suppose that  and  are convex and disjoint sets.Let the interior of  be nonempty.Then sets  and  are separable.

Preliminaries
Let us start from introduction of the basic notations and facts needed in the subsequent sections.

Linear Control Systems with Fractional Order
Let us consider the following linear control system of the form with an initial condition () =  0 and  ∈ N 0 .Therin  : ℎ, and the values (ℎ) of control  are elements of an arbitrary set Ω ⊆ R  .Denote the state forward trajectory of system (10), that is, a solution which is uniquely defined by initial state  0 and control  ∈ Ω by (⋅,  0 , ).
Proof.The proof follows from the properties of Z-transform of the fractional Caputo type difference operator and it mimics (with respect to ℎ) the proof of the similar result with ℎ = 1 given in [24].
The set of all states that can be reached by system (10) from the initial state () =  0 in a finite number of steps  is called a reachable set and denoted by R  0 (); see, for example, [27]; that is, fl { ∈ R  :  ( 1 ) =  (,  0 , ) , for some  ∈ Ω} (12) and R  0 (0) fl {0}.Similarly as in [28] one can show that if Ω ⊂ R  is compact and convex set, then R  0 () is a compact and convex subset of R  for any  0 .
Let  ⊂ R  denote a target set and Ω ⊂ R  .
Definition 4. System (10) is Ω-controllable to  in a finite number of steps  from the initial state () =  0 if there exists a control  ∈ Ω which transfers this system to the set  at a prescribed number of steps .
Let  ∈ N 0 and () fl ℎ  (ℎ).For a vector  ∈ R  let us define the cost function in the following way: where   denotes the transposition of the vector  ∈ R  .From definition of the convolution it follows that (13) can be rewritten as Note that if  = 1 and ℎ = 1, then Theorem 5. System (10) is Ω-controllable to an open convex target set  in  steps from the given initial state () =  0 if and only if there exists a vector  ∈ R  such that ( 0 , , ) > 0.
Proof.The proof is similar to the proof of the respective results in [21].
"⇐" First let us note that if system (10) is not Ωcontrollable to an open and convex target set  in  steps from the given initial state () =  0 , then R  0 ( 1 ) ∩  = 0. So, Theorem 1 implies that sup (, 0 ,)∈R  0 () On the other hand, if there exists a vector  ∈ R  such that ( 0 , , ) > 0, then taking into account compactness of the set Ω, the following holds: So, we have contradiction with (20) and in the consequence with the assumption that system (10) is not Ω-controllable to an open convex target set  in the finite number of steps  from the given initial state.

Corollary 6.
If the target set  is closed and convex, then system (10) is Ω-controllable to  from initial state (0) =  0 in a finite number of steps  if and only if for any vector  ∈ R  such that ‖‖ = 1 it holds that min ‖‖=1 ( 0 , , ) ≥ 0.

Approximation of Continuous-Time Control Systems of Fractional Order
Since during the study of real phenomena sometimes continuous-time models should be approximated by discretetime ones, in this section we consider the problem of approximation a coast function associated with a continuoustime linear fractional control system with Caputo differential by the respective coast function associated with the discretetime linear control system with ℎ-difference fractional operator of Caputo type.Before stating the main result of this section let us recall some facts and results about approximation of a continuoustime linear control system with the Caputo fractional differential by an ℎ-difference linear control system with fractional Caputo type operator.
(33) By the same motivation as in [28] one can note that if Ω ⊂ R  is compact and convex set, then R  0 () is also a compact and convex subset of R  for any  0 .
Let  ⊂ R  denote a target set and Ω ⊂ R  .
Definition 10.System ( 29) is Ω-controllable to  in a time  ∈ (0, ], -positive arbitrary but fixed, from initial state (0) =  0 if there exists a control  ∈ Ω which transfers this system to the target set  in time .
Then E () (  ) can be approximated by  () (ℎ  ,  ℎ ).Proof.Result is a consequence of definitions of two-parameter Mittag-Leffler functions in continuous-and discretecases, respectively, and of Propositions 11 and 7.
Proof.If system (37) is the one that we can obtaine under the uniform sampling with sampling step ℎ > 0 from continuoustime linear system (29), then by Corollary 12 the trajectory of (29) can be approximated by the respective trajectory of (37) in values via the limit lim ℎ → 0 ( ℎ ) = ().Hence thesis is a simple consequence of definition of the coast function (34), Corollary 12, and Proposition 11.

Some Remarks
Let us, using the classical reasoning (see, e.g., in [30,31]), shortly remark a special case of constrained controllability of system (10) to the target set  that consists only of one point.
Let us consider system (10) and let  = {0}.By Corollary 3, the final state   that can be obtained in a finite number of steps  from the initial state () =  0 using controls  with values in Ω ⊂ R  is given by with () = ℎ  (ℎ).Note that in fact Denoting by it can be easy to see that the target set  = {0} to which system (10) should be steered from the given initial state () =  0 using controls   = ((0), (ℎ), . . ., (( − 1)ℎ)), (ℎ) ∈ Ω, is given by Corollary 14.Let 0 ∈ int Ω. Assume that system (10) is controllable in a finite number steps .Then there exists a neighborhood  0 of state () =  0 ∈ R  such that all elements of  0 can be steered to set  = {0} using the control   with controls values in Ω.
Corollaries 12 and 14 imply the following.

Conclusions
In the paper we consider the problem of steering in a finite number of steps of a linear ℎ-difference control system with Caputo type fractional operator to a given target set when the control is subject to specified magnitude constraints.Necessary and sufficient conditions for the constrained controllability for this class of systems to the target set were proposed.Since in some cases continuous-time models should be approximated by discrete-time ones, conditions for approximation of the coast function associated with a continuous-time linear fractional control system with Caputo differential by coast function associated with the respective (i.e., obtained by uniform sampling) discrete-time linear control system with ℎ-difference fractional operator of Caputo type were presented.