ON APPROXIMATELY CONTROLLABLE SYSTEMS 1

Appl. Comput. Math., V.15, N.3, 2016, pp.247-264

ON APPROXIMATELY CONTROLLABLE SYSTEMS SURVEY NAZIM I. MAHMUDOV1 , MARK A. MCKIBBEN2 Abstract. The aim of this article is to provide a thorough analysis of the approximate controllability of abstract systems, beginning with the inception of the concept in the 1960s. The study begins with a discussion of abstract infinite-dimensional linear systems where the notion of approximate controllability originated. The extension of this foundational theory to nonlinear systems is reviewed, followed by a modern treatment of fractional approximate controllability which has significant applications in the mathematical modeling of phenomena across disciplines. Keywords: Approximate Controllability, Fractional Evolution Equation, Resolvent Condition, Controllability Operator. AMS Subject Classification: 34A08, 34K30, 34K37, 93B05.

1. Introduction The notion of controllability represents a major concept of modern control theory. In [15, 16], concept of (exact) controllability for finite-dimensional dynamical systems is introduced. Intuitively, a dynamical system is exactly controllable if it can be steered from an arbitrary initial state to an arbitrary final state in finite time using a set of admissible controls. This work set the stage for the theory of controllability developed in the more than half a century that followed, and which was first expanded upon for abstract linear systems in the works [7–9]. In the decade that followed, researchers investigated this notion of controllability for infinitedimensional abstract systems and encountered an interesting finding: for abstract linear systems taking values in an infinite-dimensional, separable Banach space, under conditions comparable to those established in the finite-dimensional theory, the system could not be exactly controllable on any finite interval. This observation is available in [40–42]. This prompted the formulation of the weaker notion of approximate controllability. Intuitively, a dynamical system is approximately controllable if an arbitrary initial state can be steered toward an arbitrarily small neighborhood of any given final state. For infinite-dimensional systems in which one can formulate an exact controllability result, typically the hypotheses one must impose to establish the result are too restrictive or are extremely difficult to verify in practical situations. As such, the notion has limited applicability. However, approximate controllability results are much more prevalent and from the viewpoint of applications, approximate controllability is quite adequate. As such, studying the weaker notion of approximate controllability for nonlinear systems is not only important, it is necessary. The remainder of the article is structured as follows. We begin in Section 2 with a brief review of foundational controllability results for abstract linear control systems. We primarily focus on the discussion provided in the seminal papers [7–9], and in the mid 1970s , [40–42], and more recently [4]. The discussion in Section 3 then focuses on some initial pinnacle contributions to the theory of approximate controllability of abstract systems formulated by equipping linear 1

Department of Mathematics, Eastern Mediterranean University, Gazimagusa, Mersin 10, Turkey e-mail: [email protected] 2 Department of Mathematics, West Chester University West Chester, Pennsylvania 19383 USA e-mail: [email protected] Manuscript received 23 September 2016. 247

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APPL. COMPUT. MATH., V.15, N.3, 2016

systems with a nonlinear forcing term. Among the papers reviewed [6, 26, 33, 45, 49, 50]. The conditions that ensure the approximate controllability of abstract nonlinear systems are typically established using tools of semigroup theory, operator theory and fixed-point theory under the assumption that the linear part of the associated nonlinear system is, itself, approximately controllable. The discussion and theory developed in the papers reviewed in Sections 2 and 3 paved the way for the study of approximate controllability of a diverse collection of abstract systems, including stochastic evolution equations of various types [28,29], equations of Volterra type [17], equations of Sobolev type [18, 25, 27], neutral equations [21, 24, 46], delay systems [19, 20, 22, 37, 39, 46], nonlinear differential inclusions [34, 43], and equations equipped with nonlocal initial conditions [31], just to name a few. The ever-expanding abstract theory developed over the past half-century is important from the viewpoint of applications, and has been applied to a continually increasing collection of concrete PDEs and nonlinear ODEs arising in the mathematical modeling of phenomena occurring in various disciplines, including physics, engineering, population ecology, etc. thereby broadening the scientific community’s understanding these phenomena. The article finally concludes with a survey of very recent efforts in developing a theory of approximate controllability of abstract fractional differential equations in Section 4.

2. Approximate controllability of linear systems We begin our discussion with the seminal works [40–42]. The theory developed in these papers is concerned with abstract control systems of the form y ′ (t) = Ay (t) + Bu (t) , t ≥ t0 ,

(1)

where y (t) belongs to a complex, separable infinite-dimensional Banach space X (called the state space) with norm ∥·∥X ; u belongs to a complex, separable Banach space U (called the control space); B : U → X is a bounded linear operator; and the operator satisfies the following assumption: Assumption 1. A : dom (A) ⊂ X → X is a closed, linear densely-defined operator that is the infinitesimal generator of a strongly continuous semigroup of bounded operators {S (t) : t ≥ t0 } on X. If the control space U is finite-dimensional with dimension m, say with basis {ei : i = 1, ..., m}, (1) can be written as m ∑ y ′ (t) = Ay (t) + bi ui (t) , t ≥ t0 , (2) i=1

where bi = Bei (i = 1, ..., m) and ui are the components of u. The following concept of approximate controllability of (1) was introduced: ˙ System (1) is approximately controllable on if for every ε > 0 and Definition 1. (i): arbitrarily chosen initial starting position y0 and final ending position y1 in X, there is an admissible control u on [t0 , T ] for which ∥y (T, t0 , x0 , u) − y1 ∥X < ε,

(3)

where y (T, t0 , x0 , u) is the solution of (1) corresponding to the initial point y0 and control u, evaluated at time t = T . (Equivalently, for each initial starting position y0 , the set of all points to which y0 can be steered by admissible controls on [t0 , T ] is dense in X.) (ii): If ε = 0 in (i), we say that (1) is exactly controllable on [t0 , T ]. An admissible control u on [t0 , T ] is a U -valued Bochner integrable function with bounded norm ∥u (t)∥U on [t0 , T ]. For each such admissible control u and initial condition y0 , it is

N.I. MAHMUDOV, M.A. MCKIBBEN: ON APPROXIMATELY CONTROLLABLE SYSTEMS ...

249

known that under assumption 1, (1) has a unique mild solution on expressed by the variation of parameters formula ∫t y (t, t0 , y0 , u) = S (t − t0 ) y0 +

S (t − τ ) Bu (τ ) dτ.

(4)

t0

For brevity, we shall henceforth write y (t) in place of y (t, t0 , y0 , u) . Remark 1. Definition 1 can be suitably modified for the more specific control system (2). For the finite-dimensional case (when X = Rn and U = Rm ), parts (i) and (ii) of Definition 1 are equivalent. A well-known result in the finite-dimensional case (when X = Rn and U = Rm and so, A and B are matrices)[ is that (1) is exactly controllable (and hence, approximately ] controllable) if and only if rank B, AB, ..., An−1 B = n. The following characterization is the extension of this result to the infinite-dimensional case Theorem 1. System (1) is approximately controllable on [t0 , T ] if and only if cl (span {An BU : n ≥ 0}) = X.

(5)

Regarding system (2), condition (5) can be written as cl (span {An bi : n ≥ 0, i = 1, ..., m}) = X.

(6)

This result provides an algebraic test for the approximate controllability of (1) in contrast to the results established in [8, 9], which employ a much more technical apparatus involving the use of ordered representation theory of a Hilbert space for self-adjoint operators. Triggiani discusses several examples of Volterra integral equations and specific PDEs. As part of this discussion it is shown that while some of these systems are indeed approximately controllable under appropriate conditions, they are not exactly controllable on any interval. As such, when X is infinite-dimensional there are systems of the form (1) that are approximately controllable on [t0 , T ], but not exactly controllable on [t0 , T ]. More precisely, the following theorem is proven in [40]: Theorem 2. Let X be an infinite-dimensional Banach space. The system (2) defined on X can never be exactly controllable on any (fixed) finite interval . The same is true for system (1) defined on X under the assumptions that is a compact operator and that X has a Schauder basis. Consequently, the introduction of the more general concept of approximate controllability is not only nontrival, but it is essential. Even though the concept is weaker, it proves more widely applicable than exact controllability (especially considering the highly restrictive assumptions that must be imposed in order to ensure such a system in exactly controllable) and it has been shown that in practical applications, the notion of approximate controllability is more than adequate. Remark 2. The more general non-autonomous case in which the operators A and B depend on t is also studied in [40], but for simplicity and interest of uniformity of discussion throughout the manuscript, we focus only on the autonomous case. The above investigation is continued in the second seminal paper [42]. A more specialized framework for (1) and (2) (in the sense of enhanced assumptions being imposed on the operators A and B and restricting the underlying state space X) in an effort to establish readily-verifiable criteria for the approximate controllability of these systems involving the eigenvalues of A is explored. Specifically, in addition to Assumption 1, the following additional restrictions are imposed: Assumption 2. The state space X is a Hilbert space. Assumption 3. The operator A : dom (A) ⊂ X → X is normal and there exists µ0 for which the resolvent operator R (µ0 , A) is a compact operator on X. In such case, the following result is proved:

250


Theorem 3. Suppose that assumptions 1 – 3 are satisfied. Then, (1) is approximately controllable on if and only if Pj (rngB) = Xj ,

j = 1, 2, ...,

(7)

where Xj is the rj -dimensional eigenspace associated with the eigenvalue λj and Pj is the orthogonal projection of X onto Xj . Remark 3. The condition in Theorem 3 holds, for instance, when the range of the operator B is all of X. A result concerning the approximate controllability of (2) is established under the following additional analyticity assumption, which holds, for instance, if A satisfies assumption 1 and is self-adjoint. { } Assumption 4. The eigenvalues of A are contained in a sector Σ = λ : |arg (λ − a)| ≤ π2 + β , where a is a real number and 0 < β < π2 . Let {ejk : k = 1, ..., rj , j = 1, 2, ...} be a complete orthonormal set of eigenvectors of A. (Note ∑ ∑rj that every x in X can be expressed uniquely as x = ∞ j=1 k=1 ⟨x, ejk ⟩ ejk .) For each j = 1, 2, . . . and every m-tuple of vectors (bi ) in X, the rj × m matrix Bj is defined as follows:   ⟨b1 , ej1 ⟩ . . . ⟨bm , ej1 ⟩  ⟨b1 , ej2 ⟩ . . . ⟨bm , ej2 ⟩    Bj =  .. (8)  ..  .  . . . .⟨ ⟩ ⟨b1 , ejr1 ⟩ . . . b1 , ejrj The result concerning the approximate controllability of (2) can now be stated as follows: Theorem 4. Suppose that assumptions 1 – 4 are satisfied. Then, (2) is approximately controllable on [0, T ] if and only if for all vectors bi , i = 1, ..., m in X, rankBj = rj ,

for all j = 1, 2, ...

(9)

Much work on the approximate controllability of abstract linear systems was published in the years following the appearance of these two papers. Of them, we next focus on the work in [4]. New necessary and sufficient conditions based on the convergence of a certain sequence of operators involving the resolvent of the negative of the so-called controllability operator are established in this paper for the approximate controllability of (1). Remark 4. The results we mention below are actually special cases of the ones proved in that paper. Indeed, a nonlinear version of (1) is considered, as well as a related stochastic variant. We present a simpler case, however, to highlight the nature of the conditions for approximate controllability being formulated. The nature of these conditions is novel in and of itself, and are useful for the study of controllability issues for many different types of problems. Assume that the operator A satisfies assumption 1 and recall the variation of parameters formula (4). For every 0 ≤ t < T , the operator ΓTt defined by ∫T ΓTt

=

S (T − s) BB ∗ S ∗ (T − s) ds

(10)

t

is called a controllability operator. The following theorem provides necessary and sufficient conditions for the approximate controllability of (1) in terms of (10) and the associated resolvent operators. Theorem 5. The following statements are equivalent: i) The control system (1) is approximately controllable on [0, T ]. ii) If B ∗ S ∗ (t) x = 0, for all t ∈ [0, T ] then x = 0. ( )−1 iii) λ λI + ΓT0 converges to the zero operator as λ → 0+ in the strong operator topology. ( ) −1 iv) λ λI + ΓT0 converges to the zero operator as λ → 0+ in the weak operator topology. This work was continued and expanded upon in [26].


251

3. Approximate controllability of nonlinear systems We now investigate the controllability of abstract systems obtained by equipping the linear control systems reviewed in Section 2 with a nonlinear forcing term. Controllability results for the nonlinear infinite dimensional case primarily concern semilinear control systems consisting of a linear part and a nonlinear forcing term. Various sufficient conditions for approximate controllability have been established in the past three decades. We begin with the works [49,50]. This work is an outgrowth of the study [13]. Unlike Henry’s results, these results apply to a nonlinear abstract control system in a Hilbert space, as well as finite-dimensional differential equations. The semilinear abstract evolution equation { ′ y (t) = Ay (t) + F (y (t)) + (Bv) (t) , 0 < t < T, (11) y (0) = y0 is studied. Here, y : [0, T ] → H is the state trajectory and H is a Hilbert space; the control v belongs to V = L2 (0, T ; U ) (the space of square-integrable V -valued functions on (0, T )), where U is a real Hilbert space; A : dom (A) ⊂ H → H is a linear operator on H; F : H → H is a nonlinear operator; B : V → L2 (0, T ; H) is a bounded linear operator; and y0 ∈ H. Generally, the approximate controllability for semilinear systems is linked to the operators B and F and their relationship to one another. This fact becomes apparent when reviewing the nature of the following assumptions imposed. Assumption 5. A : dom (A) ⊂ H → H is a linear operator that generates a differentiable semigroup {S (t) : 0 ≤ t ≤ T } on H. Assumption 6. B : V → L2 (0, T ; H) is a bounded linear operator. Assumption 7. F : H → H is a nonlinear operator for which there exists a positive constant MF > 0 such that ∥F (y1 ) − F (y2 )∥H ≤ MF ∥y1 − y2 ∥H ,

for all y1 , y2 ∈ H.

Zhou considers an intercept system related to (11) on the interval [t0 , T ] with a given initial value y0 ∈ H prescribed at the time t0 by { ′ y (t) = Ay (t) + F (y (t)) + B(t0 ,T ) u (t) , 0 < t < T, (12) y (0) = y0 where the state trajectory y : [t0 , T ] → H belongs to L2 (t0 , T ; H), the control u belongs to V , and B(t0 ,T ) : V → L2 (t0 , T ; H) is a bounded linear operator for which B(t0 ,T ) u (t) = B(0,T ) u (t) , for all t0 ≤ t ≤ T. Assumptions 5 – 7, taken together, imply that the Cauchy problem (12) has a unique mild solution y ∈ L2 (t0 , T ; H) satisfying the variation of parameters formula ∫t y (t) = S (t − t0 ) y0 +

[ ] S (t − s) F (y (s)) + B(t0 ,T ) u (s) ds, t0 ≤ t ≤ T.

(13)

t0

Motivated by Definition 1, we can use (13) to formulate the definition of approximate controllability as follows. Definition 2. System (12) is approximately controllable on [t0 , T ] if the reachable set given by R (t0 , z0 ) = {ξT : ∃u ∈ V such that ξT = S (T − t0 ) y0 +  ∫T [ ]  + S (T − s) F (y (s)) + B(t0 ,T ) u (s) ds , (14)  t0

where y is the solution of (12) corresponding to u, is dense in H, for every initial condition y0 ∈ H.

252


Equivalently, for any ε > 0 and ξT ∈ H there exists a control uε ∈ V such that

 

∫T

[ ]

ξT − S (T − t0 ) y0 + S (T − s) F (yε (s)) + B(t ,T ) uε (s) ds < ε, 0

t0

(15)

H

where yε is the solution of (12) corresponding to uε ∈ V . The following is the main sufficient condition imposed on (11) in order to guarantee the approximate controllability of (11) on [0, T ]: Assumption 8. For every ε > 0 and p ∈ L2 (t0 , T ; H), there exists u ∈ V such that

∫T [ ]

(i): S (T − s) p (s) − B(t0 ,T ) u (s) ds < ε;

t0

H (ii): there exists a positive constant MB such that

B(t ,T ) u 2 ≤ MB ∥p∥ 2 , 0

L (t0 ,T ;H)

L (t0 ,T ;H)

for all p ∈ L2 (t0 , T ; H); and (iii): The constant MB satisfies eMF MS T MS MB MF (T − t0 ) < 1 is sufficiently small. The following theorem is established in [49]. Theorem 7. System (11) is approximately controllable on [0, T ] (in the sense of Definition 2, suitably modified) if assumptions 5 – 7 are satisfied and there exists some t0 ∈ [0, T ) for which assumption 8 holds. Remark 5. Assumption 8 implies that the intercept system (12) corresponding to t0 is approximately controllable. Also, assumption 8 is satisfied if, for instance, the range of the operator B is dense in L2 (0, T ; H). Naito [33] studied (11) (with y0 = 0, for simplicity) under arguably simpler assumptions that avoid imposing inequality conditions involving the system components. Precisely, assumption 5 was weakened to Assumption 9. A : dom (A) ⊂ H → H is a linear operator that generates a strongly continuous semigroup {S (t) : 0 ≤ t ≤ T } on H. This assumption, together with assumptions 6 and 7, ensure the uniqueness of mild solutions to (11), for every u ∈ L2 (0, T ; V ). The following three conditions were imposed in place of assumption 8: Assumption 10. The solution mapping defined by W : L2 (0, T ; U ) → C (0, T ; H) defined by ∫t (W y) (t) = y (t) = S (t − s) [F (y (s)) + Bu (s)] ds, 0 ≤ t ≤ T 0

is compact. (Here, C (0, T ; H) is the space of continuous H-valued functions on [0, T ].) Assumption 11. For every p ∈ L2 (0, T ; H), there exists q ∈cl(rng (B)) such that ∫T

∫T S (T − s) p (s) ds =

0

S (T − s) q (s) ds. 0

cF such that Assumption 12. F : H → H is such that there exists a positive constant M cF for all h ∈ H. ∥F (h)∥H ≤ M Remark 6. If assumption 9 is strengthened to “A generates a compact semigroup on H,” then assumption 10 automatically holds.


253

Remark 7. Assumption 11 is equivalent to the condition L2 (0, T ; H) =cl (rng (B)) + ker (Φ), where Φ : L2 (0, T ; H) → H the mapping is defined by ∫T S (T − s) p (s) ds.

Φ (p) = 0

The following result is established in [33]. Theorem 8. If assumptions 6, 7, and 9 – 12 are satisfied, then (11) (with y0 = 0) is approximately controllable on [0, T ]. An estimate of the diameter of the set of admissible controls is also obtained when the following additional assumption is imposed: Assumption 13. There exists a positive constant MB such that ∥u∥L2 (0,T ;U ) ≤ MB ∥Bu∥L2 (0,T ;H) for all u ∈ L2 (0, T ; U ) .

(16)

In [45] the same problem for parabolic equations with uniformly bounded linear part is studied. In tandem with the 1983 work, a slightly more general abstract semilinear system of the form { ′ y (t) = Ay (t) + F (y (t) , u (t)) + (Bu) (t) , 0 < t < T, (17) y (0) = y0 , where y : [0, T ] → H is the state trajectory and H is a Hilbert space, the control u belongs to L2 (0, T ; U ) and takes values in a Hilbert space U , A : H → H is a linear operator on H, F : H × U → H is a nonlinear operator, and B : U → H is a bounded linear operator is considered in [50]. The following assumptions are imposed in this paper: Assumption 14 B : U → H is a bounded linear operator such that there exists a positive constant M B for which ∥u∥U ≤ M B ∥Bu∥H for all u ∈ U. (18) Assumption 15. F : H × U → H is a nonlinear operator for which there exist positive constants M1 and M2 such that ∥F (y1 , u1 ) − F (y2 , u2 )∥H ≤ M1 ∥y1 − y2 ∥H + M2 ∥u1 − u2 ∥U , for all y1 , y2 ∈ H and u1 , u2 ∈ U. Assumptions 9, 14, and 15, taken together, imply that for each u ∈ L2 (0, T ; U ) and y0 ∈ H, the Cauchy problem (17) has a unique mild solution satisfying ∫t S (t − s) [F (y (s) , u (s)) + Bu (s)] ds, t0 ≤ t ≤ T.

y (t) = S (t − t0 ) y0 +

(19)

t0

The approximate controllability result for (17) is formulated using the family of associated quadratic optimal control problems given by inf Jε (u; h) ,

ε>0

where Jε (u; h) = ∥y (T ) − h∥2H + ε ∥u∥2L2 (0,T ;U ) .

(20)

It can be shown that for every h ∈ H and ε > 0, there exists a control uε ∈ L2 (0, T ; U ) such that Jε (uε ; h) = inf Jε (u; h) . (21) u∈L2 (0,T ;U )

The following theorem is established. Theorem 9. System (17) is approximately controllable on [0, T ](in the sense of Definition 2, suitably modified) if and only if for every h ∈ H, lim Jε (uε ; h) = 0.

ε→0

(22)

254


Next, [26] investigated the following variant of (17) { ′ y (t) = Ay (t) + F (t, y (t) , u (t)) + Bu (t) , 0 < t < T, y (0) = y0

(23)

where F : [0, T ] × H × U → H is a nonlinear operator and all other mappings are as in (17). Approximate controllability results for (13), as well as a stochastic variant of (23), are formulated under the basic assumption of approximate controllability of the associated linear system. The following assumptions are imposed: Assumption 16. H is a separable reflexive Banach space and U is a Hilbert space. Assumption 17. A : dom (A) ⊂ H → H generates a compact semigroup {S (t) : 0 < t ≤ T } on H. Assumption 18. F : [0, T ] × H × U → H is a continuous nonlinear operator for which there fF such that exists a positive constant M fF ∥F (t, h, u)∥ ≤ M H

for all (t, h, u) ∈ [0, T ] × H × U. Assumption 19. The linear system corresponding to (23) is approximately controllable on [0, T ]. Under these conditions, the following theorem is proved. Theorem 10. If assumptions 16 – 19 are satisfied, then (23) is approximately controllable on [0, T ]. The proof of this result involves a constructive approach for approximate controllability of semilinear evolution equations and is based on a characterization of a symmetric positive operator in terms of strong (weak) convergence of a sequence of (resolvent) operators. This result can be applied to both distributed and lumped controls. In [6] authors consider a version of (23) with finite delay. Precisely, let C = C ([−h, 0] ; H) be the space of continuous functions from [−h, 0] into H equipped with the usual supremum norm and consider the system ∫t yt (0) = y (t) = S (t) ϕ (0) + S (t − s) [F (s, ys , u (s)) + Bu (s)] ds, 0 ≤ t ≤ T, (24) t0

y0 (θ) = ϕ (θ) , −h ≤ θ ≤ 0 where S (t) is a linear semigroup on H, F : [0, T ] × C × U → H is a nonlinear operator; ϕ ∈ C; and y, u, and B are as in (23). The following assumptions are imposed: Assumption 20. {S (t) : 0 < t ≤ T } is a compact semigroup on H. Assumption 21. F : [0, T ] × C × U → H is a continuous nonlinear operator for which there exists a positive constant MF∗ such that fF , for all (t, ϕ, u) ∈ [0, T ] × C × U. ∥F (t, ϕ, u)∥H ≤ M ( ) Assumption 22. αR α, ΦT0 → 0 as α → 0+ in the strong operator topology, where R

(

α, ΦT0

)

(

= αI +

)−1 ΦT0

∫T and

ΦT0

=

S (T − s) BB ∗ S ∗ (T − s) ds.

0

Remark 8. Assumption 22 holds if and only if the associated linear system { ′ y (t) = Ay (t) + Bu (t) , 0 < t < T, y (0) = ϕ (0) is approximately controllable on [0, T ]. The following theorem is established:


255

Theorem 11. If assumptions 20 – 22 are satisfied, then (24) is approximately controllable on [0, T ]. The papers summarized above have served as springboards for related investigations in a plethora of different directions in which the growth conditions imposed on the nonlinear forcing terms are weakened to growth conditions of a non-Lipschitz type and those in which the classical initial condition is replaced by a so-called nonlocal initial condition. Motivated by physical problems, in [5] the notion of a Cauchy problem equipped with a nonlocal initial condition of the form is introduced y (0) + g (y) = y0 , (25) where g : C (0, T ; H) is a given function satisfying an appropriate growth condition. The literature regarding existence theory of solutions of abstract evolution equations equipped with such nonlocal initial conditions has since flourished. More recently, researchers have become interested in controllability issues of such systems; see [11, 31, 34, 48]. Approximate controllability results for abstract second-order equations governed by the generator of a strongly continuous cosine family have been established using similar fixed-point techniques in [30]. The extension of the theory to fractional nonlinear differential inclusions has been provided in [34] and to the case of time-dependent differential inclusions with finite delay and impulsive effects in [12]. The extension of the theory of approximate controllability of deterministic abstract systems to abstract stochastic systems driven by Brownian motion, fractional Brownian motion, and Levy jump processes continues to be a very active research direction. We refer the reader to the following papers and the references included therein: [4, 26, 29].

4. Fractional approximate controllability The concept of a non-integral derivative arises frequently in the mathematical modeling of phenomena across disciplines. Indeed, fractional derivatives arise in the mathematical modeling of phenomena in areas such as aerodynamics, chemistry, control theory, electrodynamics of complex media, engineering, physics, and porous media; see the references in [38] and [48]. Fractional derivatives can more effectively used to describe memory and the hereditary properties of certain materials and processes. In [14] authors showed that fractional derivatives of quantities had actual physical meaning in viscoelasticity. Further, [44] points out that fractional diffusion equations more accurately model anomalous diffusion in which a plume of particles spreads in a manner different from what the classical diffusion equation predicts. A specific third-order dispersion equation is investigated in [36]. We begin by considering abstract fractional nonlinear control systems of the form { C q Dt y (t) = Ay (t) + Bu (t) + f (t, y (t)) , 0 < t < T, (26) y (0) = y0 , where the state function y : [0, T ] → H and H is a Hilbert space; the control function u belongs to L2 (0, T ; U ) where U is a another Hilbert space ; A : dom (A) ⊂ H → H is a linear operator on H that generates a strong continuous semigroup {S (t) : t > 0} on H; B : V → H is a bounded linear operator; f : [0, T ]×H → H is a given function; y0 ∈ H ; and C Dq is the Caputo fractional derivative of order 0 < q < 1 defined as follows: Definition 3. i) The fractional integral of order α > 0 with lower limit 0 for a function g : [0, ∞) → R is defined as 1 I g (t) = Γ (α)

∫t

α

0

g (s) ds, (t − s)1−α

t > 0,

provided the right-side is pointwise defined on [0, ∞), where Γ is the gamma function.

(27)

256


ii) The Riemann-Liouville derivative of order α > 0 with lower limit 0 for a function g : [0, ∞) → R is given by L

1 dn D g (t) = Γ (α) dtn

∫t

α

0

g (n) (s) ds, (t − s)α+1−n

t > 0, n − 1 < α < n.

iii) The Caputo derivative of order α > 0 with lower limit 0 for a function g : [0, ∞) → R is given by ( ) n−1 ∑ tk C α L α (k) D g (t) = D g (t) − g (0) , t > 0, n − 1 < α < n. k! k=0

The integral equation y (t) = ψbq (t) y0 +

∫t (t − s)q−1 ψq (t − s) f (s, y (s)) ds 0

∫t (t − s)q−1 ψq (t − s) Bu (s) ds,

+

(28)

0

where ψbq (t) =

∫∞ ξq (θ) S (tq θ) dθ, 0

∫∞ θξq (θ) S (tq θ) dθ,

ψq (t) = q 0

( 1) 1 −1− 1q − ξq (θ) = θ wq θ q ≥ 0, q ∞ Γ (nq + 1) 1∑ (−1)n−1 θ−nq−1 sin (nπq) , θ > 0 wq (θ) = π n! n=1

is associated with (26) in order to define the following notion of mild solution of (26) for a given control u. Definition 4. A continuous function y : [0, T ] → H is a mild solution of (26) if y satisfies the integral equation (28). The notion of approximate controllability of the linear fractional control system { C q Dt y (t) = Ay (t) + Bu (t) , 0 < t < T, (29) y (0) = y0 is a natural generalization of the notion of approximate controllability for the linear control system (1). The first of two main results established in (Sakthivel, et al., [38]) concerns the approximate controllability of (26) and is formulated under the following assumptions: Assumption 23. A : dom (A) ⊂ H → H generates a compact semigroup {S (t) : t > 0} on H. Assumption 24. The function f : [0, T ] × H → H is such that i): for every t ∈ [0, T ], the function f (t, ·) is continuous; ii): for every h ∈ H, the function f (·, h) is strongly measurable; iii): there exists a positive constant Mf for which ∥f (t, h)∥H ≤ Mf ,

for (t, h) ∈ [0, T ] × H;


257

1

iv): there exists a constant q1 ∈ [0, q] and m ∈ L q ([0, T ] ; (0, ∞)) such that , for all h ∈ H and almost all t ∈ [0, T ]. Assumption 25: The linear system (29) is approximately controllable on [0, T ]. Using the technique introduced in [26], Sakthivel et al. establishes the following result: Theorem 12. If assumptions 23 – 25 are satisfied, then the fractional system (26) is approximately controllable on [0, T ]. The second main result in this paper concerns a nonlocal variant of (26), namely { C q Dt y (t) = Ay (t) + Bu (t) + f (t, y (t)) , 0 < t < T, (30) y (0) + g (y) = y0 where g : C (0, T ; H) → H is a given function satisfying the following global Lipschitz growth condition: Assumption 26: There exists a positive constant Mg such that for all z1 , z2 ∈ C (0, T ; H), ∥g (z1 ) − g (z2 )∥H ≤ Mg ∥z1 − z2 ∥C(0,T ;H) . The second main result established in [38] is Theorem 13. If assumptions 23 – 26 are satisfied, then the nonlocal fractional system (30) is approximately controllable on [0, T ]. Next, in [23] the following fractional control system related to (26) is investigated : { L q Dt y (t) = Ay (t) + Bu (t) + f (t, y (t)) , 0 < t < T, (31) It1−q y (t) |t=0 = y0 . Here, A, f, B, and y0 are as in [38], with the exception that the Hilbert space V is specifically taken to be Lp (0, T ; U ), p > 1, (the space of p-integrable U -valued functions on (0, T )), the initial condition is specified for It1−q y (t) |t=0 (cf. (27)) instead of y(0), and the Caputo fractional derivative C Dtq in (26) is replaced by the Riemann-Liouville fractional derivative L Dtq . The integral equation associated with (31) (similar to (28)) is given by ∫t y (t) = t

q−1

(t − s)q−1 ψq (t − s) f (s, y (s)) ds

ψq (t) y0 + 0

∫t (t − s)q−1 ψq (t − s) Bu (s) ds.

+

(32)

0

This leads to the following definition of a mild solution of (31): Definition 5. A y : [0, T ] → H is{a mild solution of (31) if }y satisfies the integral equation (32) and belongs to C1−q (0, T ; H) = y {: t1−q y (t) ∈ C (0, T ; H) }, which is a Banach space when equipped with the norm ∥y∥C1−q = sup t1−q ∥y (t)∥H : t ∈ [0, T ] . The results established in this paper are motivated by those in [38], but the underlying assumptions imposed on the semigroup generated by A is weakened and the growth conditions imposed on the nonlinear term f are different. Precisely, the following conditions are imposed: Assumption 27. A : dom (A) ⊂ H → H generates a differentiable semigroup {S (t) : 0 < t ≤ T } on H. Assumption 28 The function f : [0, T ] × H → H is such that i): there exists a function ϕ ∈ Lp (0, T ; (0, ∞)) (where p > 1q ) and a positive constant Mf such that ∥f (t, h)∥H ≤ ϕ (t) + Mf t1−q ∥h∥H , for almost every t ∈ [0, T ] and all h ∈ H ; and

258


ii): there exists a positive constant M f such that ∥f (t, h1 ) − f (t, h2 )∥H ≤ M f t1−q ∥h1 − h1 ∥H , for almost every t ∈ [0, T ] and all h1 , h2 ∈ H. Assumption 29. For every ε > 0 and φ ∈ Lp (0, T ; H), there exists a control u ∈ Lp (0, T ; U ) for which

∫T

q−1 i): (T − s) ψq (T − s) (φ (s) − Bu (s)) ds < ε;

0

H ii): there exists a positive constant MB (independent of φ ) such that ∥Bu∥Lp < MB ∥φ∥Lp ; and ( ) p−1 ) ( p M M f MB 1− 1 p−1 iii): SΓ(q) T p Eq MS M f T < 1 ,where MS = sup {∥S (t)∥H : 0 ≤ t ≤ T } . pq−1 It can be shown that under assumptions 27 and 28, for each control function , the control system (31) has a unique mild solution in the sense of Definition 5. Under these assumptions, in [23] authors prove the following theorem: Theorem 14. If assumptions 27 – 29 are satisfied, then the fractional system (31) is approximately controllable on [0, T ]. In [44] authors investigate the related fractional partial differential system { C q Dt y (z, t) = Ay (z, t) + Bu (z, t) + f (t, y (z, t)) , 0 ≤ z ≤ 2π, 0 < t < T, (33) y (z, 0) = 0 where the state function y (·, t) and the control function u (·, t) take values in L2 (0, 2π); A : dom (A) ⊂ L2 (0, 2π) → L2 (0, 2π) is a linear operator; and B : L2 (0, 2π) → L2 (0, 2π) is a bounded linear operator. ( ) Definition 6. A function y (z, t) ∈ L2 0, T ; L2 (0, 2π) is a mild solution of (33) if y satisfies the integral equation ∫t (t − s)q−1 ψq (t − s) (Bu (z, s) + f (s, y (z, s))) ds,

y (z, t) = 0

for all 0 ≤ z ≤ 2π and 0 < t < T . They establish the approximate controllability of (33) under the following conditions: Assumption 30. A : dom (A) ⊂ L2 (0, 2π) → L2 (0, 2π) generates a compact analytic semigroup {S (t) : 0 < t ≤ T } on L2 (0, 2π); Assumption 31. The function f : [0, T ] × L2 (0, 2π) → L2 (0, 2π) is such that i): for every t ∈ [0, T ], the function f (t, ·) : L2 (0, 2π) → L2 (0, 2π) is continuous; ) 1 ( ii): there exists a function η ∈ L j 0, T ; L2 (0, 2π) , where 0 < j < q, such that ∥f (t, h1 ) − f (t, h2 )∥L2 (0,2π) ≤ η (t) ∥h1 − h2 ∥L2 (0,2π) ,

∫T

for all (t, h1 ) , (t, h2 ) ∈ [0, T ] × L2 (0, 2π); and cf such that ∥f (t, h)∥ 2 c iii): there exists a positive constant M L (0,2π) ≤ Mf , for all (t, h) ∈ 2 [0, T ] × L (0, 2π). ( )−1 Assumption 32. λ λI + ΦT0 → 0 as λ → 0+ in the strong operator topology. Here, ΦT0 = (T − s)q−1 ψq (T − s) BB ∗ ψq∗ (T − s) ds, where B ∗ and ψq∗ are the adjoints of the operators B

0

and ψq , respectively.

( )1−j MS q Assumption 33. Γ(1+q) ∥η∥1/j 1−j T q−1 < 1 , where MS = sup {∥S (t)∥H : 0 ≤ t ≤ T } . q−j Under these assumptions, in [44] authors prove the following theorem:


259

Theorem 15. If assumptions 30 – 33 are satisfied, then the fractional partial differential system (33) is approximately controllable on [0, T ]. The theory of control systems with impulses has been studied with increasing vigor over the past decade. Impulse control problems arise, for instance, when one considers models of chaotic systems and investment decisions in financial mathematics. Sudden jumps at certain time points in the evolution of processes arising in biotechnology, pharmacokinetics, population dynamics, and radiation of electromagnetic waves, for instance, are examples of such impulsive behavior. The first study on impulsive systems that we consider is the work in [11]. They consider a nonlocal variant of (26) equipped with jumps at finitely many time points. Specifically, they study  q  C Dt y (t) = Ay (t) + Bu (t) + f (t, y (t)) , 0 < t < T, t ̸= ti , (34) y (0) = y0 ,  ∆y |t̸=ti = Ji (y (ti )) , i = 1, ..., p, ( ) ( −) ( +) ( ) where 0 < t1 < ... < tp < T and ∆y |t̸=ti = y t+ and y t− represent the i − y ti , where y ti i left- and right-limits of y(t) at t = ti . The operators A and B are as above and the functions f, g, and Ji are suitable mappings on which conditions will be imposed later in the manuscript. The collection of functions given by } { y (y) is continuous at t ̸= ti , left continuous at ti P C (0, T ; H) = y : [0, T ] → H | and lim + y (t) exists for i = 1, ..., p (35) t→ti

equipped with the norm ∥y∥P C = {∥y (t)∥ : 0 ≤ t ≤ T } is a Banach space. Also, for any R > 0, the following set is needed in the formulation of Ge et al.’s main results: ωR = {y ∈ P C (0, T ; H) : ∥y∥P C ≤ R} . The work in [11] seems to have been inspired by the final remark in [38]. In addition to assumptions 23 and 24, they impose the following restriction on the jump functions Ji : Assumption 34. The functions Ji : H → H are continuous. The authors establish two main approximate controllability results for (34) depending on the growth condition imposed on the nonlocal function g. For the first result, they assume: Assumption 35. The mapping g : P C (0, T ; H) → H is such that g(0) = 0 and there exists a positive constant Mg for which ∥g (h1 − g (h2 ))∥H ≤ Mg ∥h1 − h2 ∥P C(0,T ;H) , for all h1 , h2 ∈ P C (0, T ; H). Under assumptions 23, 24 (suitably modified so that condition (iii) is applied in the second variable in f (t, h) to the closed ball BR , for each R > 0, rather than on the entire space), 34, and 35, the authors use a standard compactness argument to show that for each bounded control u ∈ L2 (0, T ; U ), the fractional impulsive system (34) has at least one mild solution on [0, T ] provided that the data are sufficiently small. Remark 9. (i). Uniqueness is no longer guaranteed because of the weakened growth restriction on f . (ii)By the phrase “the data are sufficiently small,” we simply mean that an inequality is imposed where a (typically rather technical-looking) quantity involving the growth constants and parameters arising in the Cauchy problem is assumed to be less than 1. Imposing such a condition is standard practice, is usually done so in order to ensure that fixed-point methods can successfully be used to establish the result. In order to obtain the approximate controllability result for (34), the authors further assume that assumption 32 holds, as well as the following enhancement of assumptions 34 and 35: Assumption 36 The functions g : P C (0, T ; H) → H and Ji : H → H are uniformly bounded. The following result is then established: Theorem 16. If assumptions 23, 24 (suitably modified) and 34 – 36 are satisfied, then (34) is approximately controllable on [0, T ], provided that the data are sufficiently small.

260


The second main result in [11] is obtained by weakening the growth condition on the nonlocal initial condition g. Specifically, assumption 35 is weakened to the following: Assumption 37. The function g : P C (0, T ; H) → H is continuous and for any R > 0 and any x, y ∈ ωR , there exists δ ∈ (0, t1 ) such that x (s) = y (s) ,

for all s ∈ [δ, T ] ⇒ g (x) = g (y) .

As before, under this assumption, provided that the data are sufficiently small, the authors are able to establish the existence of a mild solution of (34), and subsequently the approximate controllability of (34) under the additional assumption 36 using a similar compactness argument. Next, in [48] the following nonlocal impulsive fractional system similar to (34) is studied:  q  C Dt y (t) = Ay (t) + Bu (t) + f (t, y (t) , Gy (t)) , 0 < t < T, t ̸= ti , (36) y (0) = g (y) + y0 ,  ∆y |t̸=ti = Ji (y (ti )) , i = 1, ..., p. Here, A : (domA) ⊂ H → H is a linear operator that generates an analytic semigroup on H. As such, the space H used in the domains and ranges of the mappings B, f, g, etc, is often replaced by the domain of the fractional power of A, dom (Aα ) (0 < α < 1 ), which is known to be a Banach space when equipped with the graph norm of Aα , denoted by ∥·∥α . More specifically, B : V → dom (Aα ) is a bounded linear operator; Ji : dom (Aα ) → dom (Aα ) , f : [0, T ] × dom (Aα ) × dom (Aα ) → H , and g : P C ([0, T ] ; dom (Aα )) → dom (Aα ). The mapping G defined by ∫t Gy (t) = K (t, s) y (s) ds (37) 0

is a Volterra integral operator with kernel K ∈ C (∆; [0, ∞)), where ∆ = {(t, s) : 0 ≤ s ≤ t ≤ T }. The space in (35) is naturally modified for studying (36) by replacing H by dom (Aα ); we denote this space by P C ([0, T ] ; dom (Aα )). A mild solution of (36) is then naturally defined as follows: Definition 7. A function y ∈ P C ([0, T ] ; dom (Aα )) is a mild solution of (36) if for any u ∈ L2 (0, T ; U ), the integral equation y (t) = ψbq (t) (y0 + g (y)) + +

∑

∫t (t − s)q−1 ψq (t − s) (u (s) + f (s, y (s) , G (s))) ds

(38)

0

( ( )) ψbq (t − ti ) Ji y t− i

0