Approximate controllability of linear distributed control systems

Applied Mathematics Letters 21 (2008) 1041–1045 www.elsevier.com/locate/aml

Approximate controllability of linear distributed control systems Hern´an R. Henr´ıquez Departamento de Matem´atica, Universidad de Santiago, USACH, Casilla 307, Correo 2, Santiago, Chile Received 8 February 2005; received in revised form 10 October 2007; accepted 18 October 2007

Abstract We characterize the approximate controllability of first-order and second-order linear abstract control systems in terms of quasianalytic vectors. c 2007 Elsevier Ltd. All rights reserved.

Keywords: Controllability of distributed systems; First-order control systems; Second-order control systems; Semigroup of operators; Cosine functions of operators

1. Introduction This work is devoted to establishing some properties of approximate controllability for linear distributed control systems. In what follows, X and U will denote Banach spaces that are endowed with a norm k · k. We consider a linear operator A : D(A) ⊆ X → X that satisfies certain conditions which will be specified subsequently. We will be concerned with systems of first and second order modeled with states x(t) ∈ X and controls u(t) ∈ U . More specifically, we consider first-order systems governed by the equation x 0 (t) = Ax(t) + Bu(t),

t ≥ 0,

(1.1)

with initial condition x(0) = x0 , and second-order systems governed by the equation x 00 (t) = Ax(t) + Bu(t),

t ≥ 0, x 0 (0)

(1.2)

with initial conditions x(0) = x0 and = x1 , where B : U → X is a bounded linear operator which represents the control action, and the control function u(·) is at least locally integrable. For the system (1.1) we assume that A is the infinitesimal generator of a strongly continuous semigroup, while for the system (1.2) we assume that A is the infinitesimal generator of a cosine function of operators. The terminology and notation are those generally used in functional analysis. In particular, we will say that a set D ⊆ X is total in X if Span(D) is dense in X . We denote the dual space of a Banach space Z by Z ∗ = L(Z , K). In addition, if C is a linear operator, we denote by R(C) the range space of C, and if C is a linear operator defined in a dense subspace D(C), then C ∗ will represent the adjoint operator of C. Finally, if x ∗ ∈ X ∗ and M is a subspace of X , we will write x ∗ ⊥ M if hx ∗ , xi = 0, for all x ∈ M. E-mail address: [email protected]. c 2007 Elsevier Ltd. All rights reserved. 0893-9659/$ - see front matter doi:10.1016/j.aml.2007.10.024

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2. Approximate controllability of first-order systems Throughout this section, we will assume that the operator A generates a strongly continuous semigroup of bounded linear operators T (t) on X . For the properties of semigroups, the reader can consult for example [1]. In particular, there are constants M ≥ 1 and ω ∈ R such that kT (t)k ≤ Meωt , for all t ≥ 0. Moreover, it is well known that the unique mild solution of (1.1), with initial condition x(0) = x0 , is given by Z t T (t − s)Bu(s)ds. x(t) = T (t)x0 + 0

Considering the initial condition x0 = 0 leads us to define, for each t > 0, the map G t : L ∞ ([0, t], U ) → X by Z t T (t − s)Bu(s)ds. G t (u) = 0

It is clear that G t is a bounded linear operator. We will use the following terminology. Definition 2.1. The system (1.1) is called approximately controllable on [0, t] if the space R(G t ) is dense in X . The system (1.1) will be called approximately controllable in finite time if the space ∪t>0 R(G t ) is dense in X . In [2], the reader can find several characterizations for the approximate controllability of diverse classes of first-order distributed systems. In particular, for the sake of completeness we state the next result, which extends the well known Kalman’s controllability criterion for lumped systems to distributed systems. Henceforth we use the notation n D∞ (A) = ∩∞ n=1 D(A ), U∞ = {u ∈ U : Bu ∈ D∞ (A)},

X a = {x ∈ X : T (t)x is analytic for t > 0}, Ua = {u ∈ U : Bu ∈ X a }. Theorem 2.1. Under the previous conditions the following statements are satisfied: (a) The system (1.1) is approximately controllable on [0, a] (resp. in finite time) if and only if for every x ∗ ∈ X ∗ such that B ∗ T (t)∗ x ∗ = 0, for all 0 ≤ t ≤ a (resp. t ≥ 0), we have that x ∗ = 0. (b) If Span{An BU∞ : n ∈ N0 } is dense in X , then the system (1.1) is approximately controllable on [0, t], for all t > 0. (c) If BUa is dense in BU and the system (1.1) is approximately controllable in finite time, then the space Span{An T (t)BUa : n ∈ N0 } is dense in X , for all t > 0. The first object of this work is to extend this result. To establish our result, we introduce some notation: ( ) ∞ n X t Da (A) = x ∈ D∞ (A) : kAn xk < ∞, for some t > 0 , n! n=0 ( ) ∞ X n −1/n kA xk = +∞ , Dqa (A) = x ∈ D∞ (A) : n=1

Uqa = u ∈ U : Bu ∈ Dqa (A) . The elements of Da (A) are called analytic vectors of A, and the elements of Dqa (A) are called quasi-analytic vectors of A. In [3], A. E. Nussbaum introduces the quasi-analytic vectors in connection with the study of self-adjoint operators. Our next result establishes relationship among these sets of vectors. It is well known (see, for example, Chernoff [4]), so its proof is omitted. Proposition 2.1. The following inclusions hold: (a) Da (A) ⊆ X a . (b) T (t)X a ⊆ Da (A), for all t > 0. (c) Da (A) ⊆ Dqa (A).


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At this point, it is worthwhile mentioning that, in general, Dqa (A) is not a subspace. We next recall some concepts of quasi-analytic functions (see [5]). Let Ω ⊆ R be a non-empty connected open set. A collection C ⊆ C ∞ (Ω ) is said to be a class of quasi-analytic functions if for every f, g ∈ C such that f (n) (t0 ) = g (n) (t0 ), for some t0 ∈ Ω and all n ∈ N0 , then f = g. If (Mn )n is a sequence of positive numbers, we denote by C{Mn } the class consisting of the functions f ∈ C ∞ (Ω ) such that k f (n) k∞ ≤ C L n Mn , for all n ∈ N0 , where C, L are positive constants independent of n. We next apply repeatedly the Denjoy–Carleman theorem, which characterizes the quasi-analytic classes C{Mn } when (Mn )n is a log-convex sequence (see [5]). Theorem 2.2. If BUqa is total in BU and the system (1.1) is approximately controllable in finite time, then the space Span{An T (t)BUqa : n ∈ N0 } is dense in X , for all t ≥ 0. Proof. Fix t1 ≥ 0. Let x ∗ ∈ X ∗ be such that x ∗ ⊥ Span{An T (t1 )BUqa : n ∈ N0 }. For u ∈ U , define the function f (t) = hx ∗ , T (t)Bui. We set x = Bu. Assume initially that u ∈ Uqa . Then x ∈ Dqa (A), the function f is of class C ∞ and f (n) (t) = hx ∗ , An T (t)xi. Therefore, | f (n) (t)| ≤ Meωt kx ∗ kkAn xk. Next fix a > t1 . We study the functions f considered above with domain Ω = (0, a). Let Mn = kAn xk, n ∈ N0 . It P∞ follows from the above inequality that f ∈ C{Mn }. Since n=1 kAn xk−1/n = +∞, there is a log-convex sequence P ¯ −1/n = +∞. It follows from the Denjoy–Carleman theorem that ( M¯ n )n such that C{Mn } = C{ M¯ n } and ∞ n=1 Mn C{ M¯ n } is quasi-analytic. Since f (n) (t1 ) = 0, for all n ∈ N0 , we have f = 0. Now if we assume that x ∈ Span{BUqa }, Pk then we may represent x as x = i=1 αi Bu i , where u i ∈ Uqa . From our preceding result we conclude that the function k X f (t) = hx ∗ , T (t)xi = αi hx ∗ , T (t)Bui = 0, i=1

for all t ∈ (0, a). Now consider an arbitrary element u ∈ U . Then Bu ∈ Span{BUqa }. Using the preceding property and the strong continuity of the semigroup T , we see that the function f (t) = hx ∗ , T (t)Bui = 0, for all t ≥ 0. This implies B ∗ T (t)∗ x ∗ = 0, for all t ≥ 0. Since the system (1.1) is approximately controllable in finite time, it follows from Theorem 2.1 that x ∗ = 0. We infer that, as a consequence of the Hahn–Banach theorem, Span{An T (t1 )BUqa : n ∈ N0 } is dense in X . 3. Approximate controllability of second-order systems Throughout this section, we will assume that the operator A generates a strongly continuous cosine function of bounded linear operators C(t) on X . For the theory of cosine functions of operators we refer the reader to [6,7]. We next mention only a few concepts and properties related to the second-order abstract Cauchy problem. We denote by S(t) the sine function associated with C(t) that is defined by Z t S(t)x = C(s)xds, x ∈ X, t ∈ R. 0

The notation E stands for the space consisting of the vectors x ∈ X for which C(·)x is of class C 1 on R. Existence of solutions for the second-order abstract Cauchy problem (1.2) is discussed in [8]. We mention that the function x(·) given by Z t x(t) = C(t)x0 + S(t)x1 + S(t − s)Bu(s)ds, t ∈ I, (3.1) 0

is called a mild solution of (1.2) with initial conditions x(0) = x0 and x 0 (0) = x1 . When x0 ∈ E, the function x(·) is continuously differentiable and Z t x 0 (t) = AS(t)x0 + C(t)x1 + C(t − s)Bu(s) ds. (3.2) 0

Furthermore, A generates an analytic semigroup which we denote by T (t).

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Both the exact and the approximate controllability of systems (1.2) have been studied by several authors. For works that are directly related to systems modeled by the Eq. (1.2), see, for example, [9–13]. Also, there is an extensive literature related to functional systems. (See, for example, [14,15].) Roughly speaking, though exact controllable distributed systems are rather scarce; most systems are approximate controllable. In what follows, we will take the initial conditions x0 = x1 = 0. For each t > 0, we define the linear operator Λt : L ∞ ([0, t], U ) → X × X by Z t Z t Λt (u) = C(t − s)Bu(s)ds . S(t − s)Bu(s)ds, 0

0

The approximate controllability of system (1.2) was first studied by Fattorini [9]. We will use the following terminology. Definition 3.1. The system (1.2) is called approximately controllable on [0, t] if the space R(Λt ) is dense in X × X . The system (1.2) is called approximately controllable in finite time if the space ∪t>0 R(Λt ) is dense in X × X . The following result has been established by Fattorini [9] and Triggiani [10,11]. We introduce the two sets X 0 = ∪t>0 T (t)(X ) and U0 = {u ∈ U : Bu ∈ X 0 }. Theorem 3.1. The following conditions are satisfied: (a) The system (1.2) is approximately controllable on [0, a] (respectively, in finite time) if and only if for every x ∗ , y ∗ ∈ X ∗ such that B ∗ S(t)∗ x ∗ + B ∗ C(t)∗ y ∗ = 0, for 0 ≤ t ≤ a (respectively, for t ≥ 0), we have that x ∗ = y ∗ = 0. (b) If Span{An BU∞ : n ≥ 0} is dense in X , then the system (1.2) is approximately controllable on [0, a], for every a > 0. (c) If BU0 is dense in BU and the system (1.2) is approximately controllable in finite time, then Span{An BU0 : n ≥ 0} is dense in X . Since A is also the infinitesimal generator of the analytic semigroup T (t), we may consider the first-order control system (1.1) as well as compare the approximate controllability of the two systems. The following result relates the approximate controllability of systems (1.1) and (1.2), and was established by Fattorini [9] and Triggiani [10,11]. Lemma 3.1. Under the previous assumptions, the following statements hold: (a) If the system (1.2) is approximately controllable at finite time, then the system (1.1) is approximately controllable in finite time as well. (b) Assume that BU0 is dense in BU . If the system (1.1) is approximately controllable in finite time, then (1.2) is approximately controllable on [0, t], for every t > 0. The object of this section is to extend these results. We begin by introducing the following two sets which are related to the control system (1.2): ( ) ∞ X Ds (A) = x ∈ D∞ (A) : kAn xk−1/(2n) = +∞, n=1

Us = {u ∈ U : Bu ∈ Ds (A)}. The elements of Ds (A) are called Stieltjes vectors of A. It is well known that X 0 ⊆ Da (A) ⊆ Dqa (A) ⊆ Ds (A). Our first result generalizes Theorem 3.1. Theorem 3.2. Assume that BUs is total in BU . If (1.2) is approximately controllable in finite time, then the space Span{An B(Us ) : n ≥ 0} is dense in X . Proof. Let x ∗ ⊥ Span{An B(Us ) : n ≥ 0}. For each u ∈ Us , we define the function f (t) = hx ∗ , C(t)Bui. Since x = Bu ∈ Ds (A), we have that f is a function of class C ∞ and f (2n) = hx ∗ , An C(t)xi, f (2n−1) = hx ∗ , An S(t)xi.


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Hence we can estimate | f (2n) | ≤ kx ∗ kkC(t)kkAn xk, | f (2n−1) | = kx ∗ kkS(t)kkAn xk. Defining the constants Mn by M2n = M2n−1 = kAn xk, we obtain ∞ X

−1/n

Mn

≥

n=1

∞ X

kAn xk−1/(2n) = +∞.

n=1

Therefore, the functions f ∈ C{Mn }. From the above estimate and the Denjoy–Carleman theorem [5] we conclude that C{Mn } is a class of quasi-analytic functions in every interval (−a, a). Since f (n) (0) = 0, for every n ≥ 0, we infer that f = 0. Since BUs is total in BU , we conclude that hx ∗ , C(t)Bui = 0, for all u ∈ U and t ≥ 0. Applying Theorem 3.1, we deduce that x ∗ = 0 which in turn implies the assertion. From the preceding result we obtain the following consequence. Proposition 3.1. Assume that BUqa is total in BU . If the system (1.1) is approximately controllable in finite time, then the system (1.2) is approximately controllable on [0, a], for every a > 0. Proof. According to Theorem 2.2, the space Span{An BUqa : n ≥ 0} is dense in X . Since Uqa ⊆ U∞ , then Span{An BU∞ : n ≥ 0} is also dense in X . Applying Theorem 3.1(b), it follows that the system (1.2) is approximately controllable on [0, a], for every a > 0. Acknowledgement This research was supported by FONDECYT, Grant 1050314. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]

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