JOURNAL OF THEORETICAL AND APPLIED MECHANICS 43, 3, pp. 539-554, Warsaw 2005 APPROXIMATE CONSTRAINED CONTROLLABILITY OF MECHANICAL SYSTEM1 Jerzy Klamka Institute of Control Engineering, Silesian University of Technology, Gliwice e-mail: jklamka@ia.polsl.gliwice.pl In the present paper approximate constrained controllability of linear abs- tract second-order infinite-dimensional dynamical control systems is conside- red.First, fundamental definitions andnotionsare recalled.Next it is proved, using the so-called frequency-domainmethod, that approximate constrained controllability of second-order dynamical control system can be verified by the approximate constrained controllability conditions for the simplified, su- itably defined first-order linear dynamical control system. General results are then applied for approximate constrained controllability investigation of mechanical flexible structure vibratory dynamical system. Some special ca- ses are also considered. Moreover, many remarks, comments and corollaries on the relationships between different concepts of approximate controllabi- lity are given. Finally, the obtained results are applied for investigation of approximate constrained controllability for flexible mechanical structure. In this case linear second-order partial differential state equation describes the transversemotion of an elastic beamwhich occupies the givenfinite interval. Key words: linear infinite-dimensional control systems, mechanical flexible structure vibratory systems, controllability of abstract dynamical systems 1. Introduction Controllability, similarly to stability and observability is one of the fun- damental concepts in mathematical control theory (Huang, 1988). Roughly speaking, controllability generally means, that it is possible to steer a gi- ven dynamical system from an arbitrary initial state to an arbitrary final 1 This work was supported by the State Committee for Scientific Research under grant 3T11A00228 540 J.Klamka state using the control taken from the set of admissible controls. Therefo- re, controllability of dynamical system depends on the one side on the form of the strate equation and on the other side-on the set of admissible con- trols. In the literature, there are many different definitions and conditions of controllability, which depend on the class of dynamical system (Ahmed and Xiang, 1996; Huang, 1988; Klamka, 1992, 1993a,b; Kunimatsu and Ito, 1988; Narukawa, 1982; O’Brien, 1979; Triggiani, 1975b, 1977). Moreover, it should be pointed out, that for infinite-dimensional dynamical systems, it is necessary to distinguish between the notions of approximate controllabi- lity and exact controllability (Huang, 1988; Klamka, 1993a, O’Brien, 1979; Triggiani, 1975a,b, 1976, 1977, 1978; Triggiani and Lasiecka, 1991). It fol- lows directly from the fact, that in infinite-dimensional spaces there exist linear subspaces that are not closed. Finally, it should be mentioned, that most of the papers concerning different controllability problems are main- ly devoted to a study of unconstrained controllability, i.e. when the valu- es of admissible controls are unconstrained. However, in the papers (Klam- ka, 1992, 1993a,b) several necassary and sufficient conditions for constrained approximate controllability for linear dynamical systems are formulated and proved. The present paper is devoted to the study of approximate controllability of linear infinite-dimensional second-order dynamical systems with damping and with constrained set of admissible controls. For such dynamical systems direct verification of approximate constrained controllability is possible but it is rather very difficult and complicated (Klamka, 1991). Therefore, using the frequency-domainmethod (Klamka, 1993b), it is shown that approximate constrained controllability of second-order dynamical system can be verified by the approximate constrained controllability condition for suitably defined, simplified first-order dynamical system. The paper is organized as follows. Section 2 contains systems descriptions and fundamental results concerning linear self-adjoint operators. In Section 3 constrained approximate controllability problem for general linear second- order infinite-dimensional control systemswith constant coefficients is discus- sed. The Section 4 is devoted to a detailed study of constrained approximate controllability of certain flexiblemechanical control system. In this case, linear second-order partial differential state equation describes the transversemotion of an elastic beamwhich occupies the given finite interval (Kobayashi, 1992). The solution of the state equation denotes the displacement from the reference state at a given time and at a given space variable. In the state equation, the first term is introduced by accounting rotational forces, next terms with the Approximate constrained controllability... 541 first-order derivative with respect to time represent internal structural viscous damping, and the last term represents the effect of axial force on the beam (Kobayashi, 1992). Moreover, the boundary conditions correspond to hinged ends of the beam.The special attention is paid to the so-called positive appro- ximate controllability, i.e. approximate controllability with positive controls. Finally, concluding remarks are presented. 2. System description First of all let us introduce notations and concepts taken directly from the theory of linear opetarors. Let V and U denote separable Hilbert spaces. Let A : V ⊃ D(A) → V be a linear generally unbounded self-adjoint and positive-definite linear ope- rator with dense domain D(A) in V and compact resolvent R(s;A) = = (sI−A)−1 for all s in the resolvent set ρ(A). Then operator A has the following properties (Ahmed andXiang, 1996; Huang, 1988; Kobayashi, 1992; O’Brien, 1979, Triggiani, 1975b): • Operator A has only pure discrete point spectrum σp(A) consisting en- tirely of isolated real positive eigenvalues si such that 0 0, c2 ­ 0, c1 ­ 0, c0 ­ 0, d1 and d0 unrestricted in sign, d2 > 0 are given real constants. It is assumed that the operator B : U → V is linear and its adjoint operator B∗ : V →U is A 1 2-bounded (Ahmed and Xiang, 1996; Bensoussan et al., 1993; Klamka, 1993b), i.e. D(B∗)⊃D(A 1 2) and there is a positive real number M such that ‖B∗v‖U ¬M ( ‖v‖V +‖A 1 2v‖V ) for v∈D(A) Let Ω ⊂ U be a convex cone with vertex at the origin in U such that int co Ω 6= ∅. In the sequel it is generally assumed, that the admissible controls u ∈ L2loc([0,∞),Ω). For the set Ω we define the polar cone by Ωo = {w ∈ U,〈w,v〉U ¬ 0 for all v ∈ Ω}. The closure, the convex hull and the interior are denoted respectively by cl Ω, co Ω and int Ω. The linear subspace spanned by Ω is denoted by span Ω. It is well known (Bensoussan et al., 1993; Chen and Russell, 1982; Chen andTriggiani, 1989, 1990a,b) that linearabstract ordinarydifferential equation (2.1) with initial conditions v(0)∈D(A) v̇(0)∈V has for each t1 > 0 and admissible control u ∈ L2loc([0,∞),Ω) an uni- que solution v(t;v(0), v̇(0),u) ∈ C2([0, t1],V ) such that v(t) ∈ D(A) and v̇(t)∈D(A) for t∈ (0, t1]. Moreover, for v(0)∈V there exists so-called ”mild solution” for the equ- ation (2.1) in the product space W = V ×V with inner product defined as follows 〈v,w〉W = 〈[v1,v2], [w1,w2]〉W = 〈v1,w1〉V + 〈v1,w1〉V Approximate constrained controllability... 543 In order to transform the second-order equation (2.1) into the first-order equ- ation in theHilbert space W , let usmake the substitution (AhmedandXiang, 1996; Bensoussan et al., 1993; Chen and Russell, 1982; Chen and Triggiani, 1989, 1990a,b; Triggiani, 1977) v(t)=w1(t) v̇(t)=w2(t) Then equation (2.1) becomes w(t)=Fw(t)+Gu(t) (2.2) where F= [ 0 I −F−10 (d2A+d1A 1 2 +d0I) −2F−10 (c2A+ c1A 1 2 + c0I) ] w(t)= [ w1(t) w2(t) ] G= [ 0 F −1 0 B ] and F0 =(e2A+e1A 1 2 +e0I). Remark 2.1. Since the operators A and A 1 2 are self-adjoint and un- der assumptions on coefficients ei (i = 0,1,2), the sequence {(e2si + e1 √ si + e0) −1 ∈ R, i = 1,2, . . .} converges towards zero, it is easy to see that operator (e2A+e1A 1 2 +e0I) −1 is self-adjoint, positive- definite and bounded on V . Taking advantage of relation 〈v1,F∗v2〉W = 〈Fv1,v2〉W , we can obtain for the operator F its adjoint operator F∗ as follows F ∗ = [ 0 −(d2A+d1A 1 2 +d0I)(e2A+e1A 1 2 +e0I) −1 I −2(c2A+ c1A 1 2 +c0I)(e2A+e1A 1 2 +e0I) −1 ] Similarly, the adjoint for operator G can be obtained as G ∗ = [ 0 B∗(e2A+e1A 1 2 +e0I) −1 ] Remark 2.2. It should be pointed out, that properties of operators F and F ∗ depend strongly on the values of coefficients ci, di, ei (i = 0,1,2) (Bensoussan et al., 1993; Chen and Russell, 1982; Chen and Triggiani, 1989, 1990a,b). In particular: 544 J.Klamka 1. If c2 = c1 = c0 =0 and additionally (a) e2 6=0 or (e2 =0 and d2 =0 and e1 6=0) or (e2 = e1 =0 and d2 = d1 = 0), then the operator F is bounded and generates an analytic group of linear bounded operators on the Hilbert space W =V ×V . (b) (e2 =0 and d2 6=0) or (d2 =0 and e2 = e1 =0 and d1 6=0), then the operator F is unbounded and generates a group of linear bounded operators on the Hilbert space W = V × V which cannot be analytic (Triggiani, 1975b). 2. If (e2 = 0 and c2 6= 0) or (e2 = e1 = 0 and (c2 6= 0 or c1 6=0)), then the operator F is unbounded and generates an ana- lytic semigroup of linear bounded operators on the Hilbert space W =V ×V . 3. Moreover, if e2 6= 0 or (e2 = e1 = 0 and c2 = c1 = 0 and d2 = d1 = 0) or (e2 = 0 and c2 = 0 and d2 = 0 and e1 6= 0), then the operator F is bounded and generates an analytic semigro- up of linear bounded operators on the Hilbert space W =V ×V . 4. If c2 = e2 = 0 and e1 6= 0 and d2 6= 0, then the operator F is unbounded and generates an C0-semigroup of linear unbounded operators on the Hilbert space W =V ×V which is not analytic. In the sequel, in addition to the second-order equation (2.1), we shall also consider the simplified first-order linear differential equation of the following form v̇(t)=−Aαv(t)+Bu(t) (2.3) where constant α ∈ (0,∞) is such that there exists solution of differential equation (2.3). In the next sections we shall also consider dynamical control system (2.1) with finite-dimensional control space U =Rm. In this special case, for conve- nience, we shall introduce the following notations B= [ b1 · · · bj · · · bm ] u(t)=          u1(t) ... uj(t) ... um(t)          where for bj ∈V for j =1,2, . . . ,m and u∈L2loc([0,∞),Ω). Approximate constrained controllability... 545 Let us observe, that in this special case linear boundedoperator B is finite- dimensional and therefore, it is a compact operator (Ahmed andXiang, 1996; Huang, 1988; Triggiani, 1975a, 1976). Using eigenvectors vik (i=1,2, . . . and k=1,2, . . . ,ni) we introduce for finite-dimensional operator B the following notation (Huang, 1988; Triggiani, 1975b) for i=1,2, . . . Bi =            〈b1,vi1〉V 〈b2,vi1〉V · · · 〈bj,vi1〉V · · · 〈bm,vi1〉V 〈b1,vi2〉V 〈b2,vi2〉V · · · 〈bj,vi2〉V · · · 〈bm,vi2〉V ... ... ... ... ... ... 〈b1,vik〉V 〈b2,vik〉V · · · 〈bj,vik〉V · · · 〈bm,vik〉V ... ... ... ... ... ... 〈b1,vini〉V 〈b2,vini〉V · · · 〈bj,vini〉V · · · 〈bm,vini〉V            (2.4) Bi (i = 1,2, . . .) are ni ×m-dimensional constant matrices which play an important role in approximate controllability investigations (Huang, 1988; Klamka, 1991, 1993a; Triggiani, 1975b). For the special casewheneigenvalues si are simple, i.e. ni =1(i=1,2, . . .) and consequently the matrices Bi are in fact m-dimensional row vectors for i=1,2, . . . Bi = [ 〈b1,vi〉V · · · 〈bj,vi〉V · · · 〈bm,vi〉V ] (2.5) 3. Constrained approximate controllability It is well known, that for infinite-dimensional dynamical systems we may introduce two general kinds of controllability, i.e. approximate (weak) control- lability and exact (strong) controllability (Ahmed and Xiang, 1996; Huang, 1988; Klamka, 1993a; Triggiani, 1975a, 1976). However, it should be mentio- ned, that in the casewhen the linear semigroup associatedwith the dynamical system is a compact semigroup or the control operator B is compact, then dy- namical system is never exactly controllable in infinite-dimensional state space (Ahmed and Xiang, 1996; Huang, 1988; Triggiani, 1975b, 1977). Therefore, in the present paper we shall concentrate on approximate controllability for second-order dynamical system (2.1) or equivalently (2.2), and first of all we recall the basis definition.Next, we shall recall from the literature several lem- mas and controllability conditionswhichwill be used to verify the constrained approximate controllability of certain mechanical system. 546 J.Klamka Definition 3.1 (Ahmed and Xiang, 1996; Huang, 1988; O’Brien, 1979). Dy- namical system (2.1) is said to be Ω-approximately controllable if for any initial condition w(0)∈V×V , anygivenfinal condition wf ∈V×V and eachpositive real number ε, there exists a finite time t1 <∞ (depending generally on w(0) and wf) and an admissible control u∈L2([0, t1],Ω) such that ‖w(t1;w(0),u)−wf‖V×V ¬ ε Now, let us recall severalwell-known lemmas (Klamka, 1993a,b;Narukawa, 1984;O’Brien, 1979) concerning constrained approximate controllability of the first-order linear infinite-dimensional dynamical system (2.2), which will be useful in the sequel. Lemma 3.1. (Klamka, 1993b). Dynamical system (2.2) is U-approximately controllable if and only if for any complex number z, there exists no nonzero w∈D(F∗) such that [ F ∗−zI G ∗ ] w=0 (3.1) Similarly, dynamical system (2.3) is U-approximately controllable if and only if for anycomplexnumber s there existsnononzerov∈D(Aα)⊂ V such that [ A α−sI B ∗ ] v=0 Lemma 3.2. (Klamka, 1993a). Suppose that U = Rm, and the cone Ω = = {u∈Rm =U : uj(t)­ 0, for t­ 0}, then dynamical system (2.3) is Ω-approximately controllable if and only if the columns of the matrices Bi form a positive basis in the space R ni for every i=1,2, . . .. Lemma 3.3. (Narukawa, 1984). Dynamical system (2.3) is U-approximately controllable if and only if it is approximately controllable for some α∈ (0,∞). Lemma 3.4. (O’Brien, 1979). Dynamical system (2.2) is Ω-approximately controllable if and only if it is U-approximately controllable and Ker(zI−F∗)∩ (GΩ)o = {0} for every z∈R (3.2) Approximate constrained controllability... 547 Remark 3.1. Since the linear operator A is selfadjoint then from Lemmas 3.1, 3.3, and 3.4 directly follows that the dynamical system (2.3) is Ω-approximately controllable if and only if Ker(sI−Aα)∩ (BΩ)o = {0} for every s∈R (3.3) Proposition 3.1. Dynamical system (2.3) is Ω-approximately controllable if and only if it is Ω-approximately controllable for some α∈ (0,∞). Proof. Since the operator A is selfadjoint and positive definite, then for any real number α∈ (0,∞) Ker(sI−A)=Ker(sαI−Aα)=Ker(zI−Aα) where z= sα is a homeomorphizm. Hence our proposition follows. Now, using the frequency-domain method (Klamka, 1993b) we shall for- mulate the necessary and sufficient condition for approximate controllability of dynamical system (2.1), which is proved by Klamka (1993a). Theorem 3.1. (Klamka, 1993a).Dynamical system(2.1) is Ω-approximately controllable if and only if dynamical system (2.3) is Ω-approximately controllable for some α∈ (0,∞). FromTheorem 3.1 follow several Corollaries, which are necessary and suf- ficient conditions for constrained approximate controllability for different spe- cial cases of dynamical system (2.1). Corollary 3.1. Suppose that Ω = {u ∈ Rm = U : uj(t) ­ 0, for t ­ 0}. Then the dynamical control system (2.1) is Ω-approximately controlla- ble, i.e. with positive controls if and only if columns of the matrices Bi form a positive basis in the space Rni for every i=1,2, . . .. Proof. If the columns of the matrices Bi form a positive basis in the space Rn for every i=1,2, . . . and Ω is a positive cone in the space Rm, then image BΩ is the whole space Rni for every i= 1,2, . . .. Therefore our Corollary 3.1 follows. Corollary 3.2. Suppose that c21+ c 2 2 > 0 and Ω =U. Then dynamical sys- tem (2.1) is U-approximately controllable, i.e. without control constra- ints in any time interval [0, t1] if and only if dynamical system (2.3) is U-approximately controllable in finite time. 548 J.Klamka Proof. Since for the case when c21 + c 2 2 > 0 operator F generates analytic semigroup, then approximate controllability of dynamical system (2.2) and hence also of dynamical system (2.1) is equivalent to its approxima- te controllability in any time interval [0, t1] (Klamka, 1993a; Triggiani, 1977). Therefore, fromTheorem 3.1 immediately follows Corollary 3.2. Corollary 3.3. Suppose that c21 + c 2 2 > 0, Ω = U, and the space of control values is finite-dimensional, i.e. U = Rm. Then the dynamical system (2.1) is U-approximately controllable, i.e. without control constraints in any time interval [0, t1] if and only if rankBi =ni for i=1,2, . . . Proof. Corollary 3.3 is a direct consequence of the Theorem 3.1, Corolla- ry 3.2 and well-known results (Huang, 1988; Triggiani, 1975a,b, 1976) concerning approximate controllability of infinite-dimensional dynami- cal systems with finite-dimensional controls. Corollary 3.4. Suppose that c21+c 2 2 > 0,Ω=U, the space of control values is finite-dimensional, i.e. U = Rm, and moreover, multiplicities ni = 1 for i=1,2, . . ..Thendynamical control system(2.1) is U-approximately controllable, i.e. without control constraints in any time interval [0, t1] if and only if m ∑ j=1 〈bj,vi〉2V 6=0 for i=1,2, . . . Proof. From Corollary 3.3 immediately follows that for the case when multiplicities ni = 1 for i = 1,2, . . . dynamical system (2.1) is U-approximately controllable in any time interval if and only if m-dimensional row vectors for i=1,2, . . . Bi = [ 〈b1,vi〉V 〈b2,vi〉V . . . 〈bj,vi〉V . . . 〈bm,vi〉V ] Thus, Corollary 3.4 follows. In the next section we shall use the general controllability results given above to verify approximate constrained controllability of a certain vibratory dynamical systemmodeling mechanical flexible structure. Approximate constrained controllability... 549 4. Approximate constrained controllability of vibratory system In this section we shall consider a vibratory dynamical system described by the following linear partial differential state equation (Kobayashi, 1992) e1vttxx(t,x)+e0vtt(t,x)+2c1vtxx(t,x)+2c2vtxxxx(t,x)+ (4.1) +d1vxx(t,x)+d2vxxxx(t,x)= m ∑ j=1 bj(x)uj(t) defined for x∈ [0,L] and t∈ [0,∞), where the subscript t represents partial derivative with respect to time variable, while x denotes partial derivative with respect to spatial coordinate. The initial conditions for the equation (4.1) are given by v(0,x) = v0(x) and vt(0,x) = v1(x) for x∈ [0,L] (4.2) and boundary conditions are as follows v(t,0)= v(t,L) = vxx(t,0)= vxx(t,L)= 0 for t∈ [0,∞) (4.3) Let Ω be the positive cone Ω = {u∈Rm =U : uj(t)­ 0, for t­ 0}, i.e. in the sequel we shall consider mechanical system with positive controls. It should be stressed, that the partial differential state equation (4.1) de- scribes the transverse motion of an elastic beam which occupies the interval [0,L] in the reference and stress-free state. The function v(t,x) denotes the di- splacement from the reference state at time t and position x. In the left-hand side of the equation (4.1), the first term is introduced by accounting rotatio- nal forces, terms with the first-order derivative with respect to time represent internal structural viscous damping, and the fifth term represents the effect of axial force on the beam (Kobayashi, 1992). The boundary conditions (4.3) correspond to hinged ends of the beam. Let V = L2[0,L] be a separable Hilbert space of all square integrable functions on [0,L] with the standard norm and inner product (Ahmed and Xiang, 1996; Huang, 1988). In order to regard the vibratory system (4.1), (4.2) and (4.3) in the general framework considered in the previous sections, let us define linear unbounded differential operator A : V ⊃ D(A) → V by Kobayashi (1992) v(x)= vxxxx(x) for v(x)∈D(A) (4.4) D(A)= { v(x)∈H4[0,L]; v(0)= v(L)= vxx(0)= vxx(L)= 0 } 550 J.Klamka where H4[0,L] denotes the fourth-order Sobolev space on [0,L]. The linear unbounded operator A has the following properties (Ahmed and Xiang, 1996; Huang, 1988; Kobayashi, 1992; O’Brien, 1979): • Operator A is self-adjoint andpositive-definitewithdensedomain D(A) in the space V . • There exists a compact inverse A−1, and consequently the resolvent R(s;A) of A is a compact operator for all s∈ ρ(A). • Operator A has a spectral representation Av= ∞ ∑ i=1 si〈v,vi〉Vvi for v∈D(A) where si > 0 (i = 1,2, . . .) are simple eigenvalues (i.e. ni = 1) and vi ∈ D(A) (i = 1,2, . . .) are the corresponding eigenfunctions of A. Moreover, for x∈ [0,L] si = (πi L )4 vi(x)= √ 2 L sin πix L and the set {vi(x), i= 1,2, . . .} forms a complete orthonormal system in V . • Fractional powers Aα, 0<α¬ 1 can be defined by A αv= ∞ ∑ i=1 sαi 〈v,vi〉Vvi for v∈D(A) (0<α¬ 1) which is also a self-adjoint and positive-definite operator with a dense domain in V . In particular, for we have A 1 2v=−vxx with the domain D(A 1 2)= {v∈H2[0,L] : v(0)= v(L)}. Now, we can consider the partial differential equation (4.1) with condi- tions (4.2) and boundary conditions (4.3) as a special case of the second-order abstract evolution equation (2.1) in the Hilbert space V . (e1A 1 2 +e0)ẅ(t)+2(c2A+ c1A 1 2)ẇ(t)+(d2A+d1A 1 2)w(t)=Bu(t) (4.5) Approximate constrained controllability... 551 where w(t) = v(t, ·)∈V ẇ(t)= vt(t, ·)∈V ẅ(t)= vtt(t, ·)∈V bj = bj(·)∈V (j =1,2, . . . ,m) Let the initial conditions be of the following form w(0)=w0 ∈D(A) ẇ(0)=w1 ∈V Then there exists a unique solution of the partial differential equation (4.1) (Kobayashi, 1992). Now, using the results given in Section 3 we shall formulate and prove the necessary and sufficient condition for approximate controllability of the vibratory dynamical control system (4.1), which is the main result of the present paper. Theorem 4.1. Vibratory dynamical control system (4.1) is Ω-approximately controllable, i.e. with positive controls if and only if for each i=1,2, . . . m-dimensional row vectors Bi = [bi1,bi2, . . . ,bij, . . . ,bim] contain at le- ast two coefficients with different signs, where bij = L ∫ 0 √ 2 L bj(x)sin πix L dx i=1,2, . . . j =1,2, . . . ,m (4.6) Proof. Let us observe, that dynamical system (4.1) satisfies all the assump- tions of Corollary 3.1. Therefore, taking into account the analytic for- mula for the eigenvectors vi(x), i = 1,2, . . . and the form of the inner product in the separable Hilbert space L2([0,L],R), from relation (4.1) we directly obtain inequalities (4.6). Hence, Theorem 4.1 immediately follows. Corollary 4.1. Vibratory dynamical control system (4.1) is U-approxi- mately controllable, i.e. without control constraints in any time interval [0, t1] if and only if m ∑ j=1 ( L ∫ 0 √ 2 L bj(x)sin πix L dx )2 6=0 for i=1,2, . . . (4.7) 552 J.Klamka Proof. Let us observe, that dynamical system (4.1) satisfies all the assump- tions of Corollary 3.4. Therefore, taking into account the analytic for- mula for the eigenfunctions vi(x) (i=1,2, . . .) and the formof the inner product in the separable Hilbert space L2([0,L],R), from Lemma 3.4 we directly obtain inequalities (4.6). Hence, Theorem 4.1 follows imme- diately. 5. Conclusions The present paper contains results concerning approximate controllabili- ty of second-order abstract infinite-dimensional dynamical systems. Using the frequency-domain method (Klamka, 1993b) and the methods of functional analysis, especially the theory of linear unbounded operators, necessary and sufficient conditions for approximate controllability in any time interval are formulated andproved.Moreover, some special cases are also investigated and discussed.Then, the general controllability conditions are applied to investiga- te approximate controllability of vibratorydynamical systemmodeling flexible mechanical structure. The results presented in the paper are generalization of the controllability conditions given in the literature (Ahmed and Xiang, 1996; Klamka, 1991, 1993b; Narukawa, 1984; O’Brien, 1979; Triggiani, 1975a,b) to second-order abstract dynamical systemswith damping terms. 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Triggiani R., Lasiecka I., 1991,Exact controllability and uniform stabiliza- tion ofKirchhoffplateswith boundary control only on ∆w|Σ andhomogeneous boundary displacement, Journal of Differential Equations, 93, 62-101 Przybliżona ograniczona sterowalność układu mechanicznego Streszczenie W artykule rozpatrywana jest przybliżona ograniczona sterowalność liniowego abstrakcyjnego nieskończenie-wymiarowego układu dynamicznego drugiego rzędu. W pierwszej kolejności przedstawiono podstawowe definicje i pojęcia. Następnie, wy- korzystując metodę częstotliwościową, wykazano, że przybliżona ograniczona stero- walność układu dynamicznego drugiego rzędu może być weryfikowana poprzez ba- danie przybliżonej ograniczonej sterowalności odpowiednio zdefiniowanegouproszczo- nego układu dynamicznego pierwszego rzędu. Ogólne metody zastosowano do ba- dania przybliżonej ograniczonej sterowalności mechanicznego układu oscylacyjnego o elastycznej strukturze. Rozpatrzono również pewne przypadki szczególne. Ponadto podano wiele uwag, komentarzy i wniosków dotyczących relacji między różnymi ro- dzajami przybliżonej sterowalności. Jako przykład zastosowań sformułowanowarunki przybliżonej ograniczonej sterowalności w odniesieniu do elastycznego układu me- chanicznego.W tymwypadku liniowe równanie różniczkowe cząstkowe stanu opisuje odchylenie elastycznej belki o danej długości. Manuscript received January 26, 2005; accepted for print March 8, 2005