Modeling and stabilization of a rotating mechanical system with elastic plates
Alexander Zuyev Julia Novikova
math.OC 93D15, 93C25, 74H40, 74K20, 35Q93, 47N20

[NO \title GIVEN]
    [NO \author GIVEN]
    December 30, 2023
======================



                
A mechanical system consisting of a rigid body and attached Kirchhoff plates under the action of three independent controls torques is considered.
The equations of motion of such model are derived in the form of a system of coupled nonlinear ordinary and partial differential equations.
The operator form of this system is represented as an abstract differential equation in a Hilbert space.
A feedback control law is constructed such that the corresponding infinitesimal generator is dissipative.









§ INTRODUCTION

Problems of the aerospace industry and robotics stimulate the development of new methods for mathematical modeling and control design for complex mechanical systems with elastic elements.
In particular, it is a well-known fact that the vibrations of flexible parts of satellites influence significantly their dynamics, so that a rigid body model is not acceptable in stability and control investigations for such distributed parameter systems (see, e.g., <cit.>).
This brings the motivation for studying the controllability and stabilization issues for infinite-dimensional mathematical models of flexible structures with strings, beams, and plates.
Without pretending to be complete, we refer to the monographs in this area <cit.>.


The stabilization problem for a thin plate with boundary control has already received attention in <cit.>
A mathematical model of a rigid body with the Kirchhoff plate has been considered in the paper <cit.>. It is assumed there that the body rotates around the fixed axis and its angular acceleration is taken as the control. The reachable sets for the linearized representation of such a system with modal coordinates have been analysed in <cit.>.

The purpose of our present paper is to derive a nonlinear model of a rotating rigid body with two Kirchhoff plates and propose a stabilizing feedback control for this model.
We will consider spatial rotations of the system and treat the three independent torques, applied to the body, as control inputs.
This framework is considered as a mathematical model of a satellite with solar panels controlled by jet thrusters or flywheels.






§ NONLINEAR MODEL OF THE ROTATIONAL MOTION


Consider a mechanical system that consists of a rigid
body and two elastic plates (Fig. 1).



Let (g_1,g_2,g_3) be the unit vectors of a fixed Cartesian frame.
Suppose that the rigid body is endowed with two Cartesian frames O_1x_1x_2x_3 and O_2x'_1x'_2x'_3,
and their basis vectors  (e_1,e_2,e_3) and (e'_1,e'_2,e'_3), respectively, are related as e'_1=-e_1,  e'_2=-e_2, and e'_3=-e_3.

Let M=f_1e_1+f_2e_2+f_3e_3 be the torque of external forces applied to the rigid body. We will treat the components (f_1,f_2,f_3)^T∈ℝ^3 as control inputs and consider the problem of defining a state feedback law in order to stabilize the moving frame (e_1,e_2,e_3) in the direction of (g_1,g_2,g_3) and to damp the vibrations of the plates.
Note that a similar problem for an absolutely rigid body was solved in the book <cit.>, and the problem of partial stabilization was considered in <cit.>.

In this paper, we assume that two rectangular plates are attached to the rigid body, so that in the undeformed state their median surfaces are located on the planes O_1 x_1 x_2 and O_2 x_1' x_2', respectively.
At time t, the coordinates of a point P on the median surface of the first plate can be represented in the frame O_1 x_1 x_2 x_3 as

    P=(x_1,x_2,w_1(x_1,x_2,t)), (x_1,x_2)∈Ω_1=[0,l_1]×[0,l_2].

Similarly, the coordinates of a point K on the median surface of the second plate are as follows (in the frame O_2 x_1' x_2' x_3'):

    K = (x'_1,x'_2,w_2(x'_1,x'_2,t)),  (x'_1,x'_2)∈Ω_2=[0,l'_1]×[0,l'_2].

Thus, the functions w_1(x_1,x_2,t) and w_2(x_1',x_2',t) define the transverse displacements for the case of small deformations of the plates.

In order to describe the motion of the considered mechanical system, we assume that the center of mass of the rigid body (point O) is fixed and expand the vectors O O_1 and O O_2 with respect to the moving frames:

    O O_1 = d_1 e_1 + d_2 e_2 +d_3 e_3,  O O_2 = d_1' e_1' + d_2' e_2' +d_3' e_3'.

Then the absolute velocities of the points P and K are, respectively,

    v_P=ω× r_P+ẇ_̇1̇e_3

and

    v_K=ω× r_K-ẇ_2e_3,

where ω=ω_1 e_1 + ω_2 e_2 + ω_3 e_3 is the angular velocity vector of the rigid body,

    r_P=(x_1+d_1)e_1+(x_2+d_2)e_2+(w_1+d_3)e_3,


    r_K=-(x'_1+d'_1)e_1-(x'_2+d'_2)e_2-(w_2+d'_3)e_3.


Thus, from formulas (<ref>) and (<ref>) with taking into account (<ref>) and (<ref>), we get:

    v_P=(ω_2(d_3+w_1)-ω_3(d_2+x_2))e_1+
    
    +
    (ω_3(d_1+x_1)-ω_1(d_3+w_1))e_2+
    
    +(ω_1(d_2+x_2)-ω_2(d_1+x_1)+ẇ_1)e_3,


    v_K=-(ω_2(d'_3+w_2)-ω_3(d'_2+x'_2))e_1-
    
    -
    (ω_3(d'_1+x'_1)-ω_1(d'_3+w_2))e_2-
    
    -(ω_1(d'_2+x'_2)-ω_2(d'_1+x'_1)+ẇ_2)e_3.


According to the Kirchhoff plate model (cf. <cit.>), we write the following partial differential equations with respect to w_1 and w_2:

    ẅ_1+a_1^2( ∂^2/∂ x_1^2+∂^2/∂ x_2^2)^2 w_1   =(x_1+d_1)ω̇_2-(x_2+d_2)ω̇_1,
       for (x_1,x_2) ∈Ω_1,


    ẅ_2+a_2^2( ∂^2/∂ x_1'^2+∂^2/∂ x_2'^2)^2w_2    =(x'_1+d'_1)ω̇_2-(x'_2+d'_2)ω̇_1,
       for (x_1',x_2') ∈Ω_2,

where
a_j^2=E_jh_j^3/12ρ_j(1-ν_j^2) is the stiffness parameter of the j-th plate,
E_j is Young's modulus, ν_j is Poisson's ratio,
ρ_j  is the area density, and
 h_j is the thickness of the j-th plate.
The right-hand sides of (<ref>) and (<ref>) contains the inertia forces because of the rotational motion of the rigid body (cf. <cit.>). In the differential equations (<ref>) and (<ref>),
only linear terms with respect to the displacements, angular velocities, and derivatives of these quantities are taken into account (this is the linearized model of the plates' vibrations).

We assume that the plates are supported at their boundaries, which yields the following boundary conditions:

    w_j|_∂Ω_j=0,
       .∂^2w_j/∂ n^2|_∂Ω_j=0,   j=1,2.

Here .∂ w_j/∂ n|_∂Ω_j is the normal derivative of w_j evaluated at the boundary of Ω_j.

To derive the equations of motion of the rigid body-carrier, we exploit the angular momentum equation with respect to the fixed point O (see, e.g., <cit.>):

    K̇+ω× K=M,

where K=K_1 e_1 + K_2 e_2 + K_3 e_3 is the angular momentum of the system, and K̇ stands for the local derivative of K in the moving frame (e_1,e_2,e_3), i.e.

    K̇ = K̇_1 e_1 + K̇_2 e_2 + K̇_3 e_3.



In the sequel, we use formulas (<ref>)–(<ref>) to express the angular momentum K  for the mechanical system under consideration:

    K = Iω+  K_p1+ K_p2,

where I is the tensor of inertia of the rigid body, and

    K_p1 = ∫_Ω_1 r_P× v_Pρ_1 dx,  K_p2=∫_Ω_2 r_K× v_K ρ_2 dx'.

We will assume that (e_1,e_2,e_3) are the principal axes of inertia of the rigid body to simplify computations,
so that I=diag(I_1,I_2,I_3).

Let us now compute the terms K_p1 and K_p2 for (<ref>).
Formulas (<ref>) and (<ref>) imply that,

    K_p1=∫_Ω_1ρ_1(K_11e_1+K_12e_2+K_13e_3)dx,

where

    K_11=ω_1[(x_2+d_2)^2+(w_1+d_3)^2]-
    ω_2(x_1+d_1)(x_2+d_2)-
    -ω_3(x_1+d_1)(w_1+d_3)+ẇ_1(x_2+d_2),
    
    K_12=-ω_1(x_1+d_1)(x_2+d_2)+ω_2[(x_1+d_1)^2+(w_1+d_3)^2]-      
    
    -ω_3(x_2+d_2)(w_1+d_3)-ẇ_1(x_1+d_1),
    
    K_13=-ω_1(x_1+d_1)(w_1+d_3)-ω_2(x_2+d_2)(w_1+d_3)+            
    
    +ω_3[(x_1+d_1)^2+(x_2+d_2)^2].

Similarly, from formulas (<ref>) and (<ref>) it follows that

    K_p2=∫_Ω_2ρ_2(K_21e_1+K_22e_2+K_23e_3)dx',

where

    K_21=ω_1[(x'_2+d'_2)^2+(w_2+d'_3)^2]-
    ω_2(x'_1+d'_1)(x'_2+d'_2)-
    -ω_3(x'_1+d'_1)(w_2+d'_3)+ẇ_2(x'_2+d'_2),
    
    K_22=-ω_1(x'_1+d'_1)(x'_2+d'_2)+ω_2[(x'_1+d'_1)^2+(w_2+d'_3)^2]-      
    
    -ω_3(x'_2+d'_2)(w_2+d'_3)-ẇ_2(x'_1+d'_1),
    
    K_23=-ω_1(x'_1+d'_1)(w_2+d'_3)-ω_2(x'_2+d'_2)(w_2+d'_3)+            
    
    +ω_3[(x'_1+d'_1)^2+(x'_2+d'_2)^2].


Then the angular momentum (<ref>) with the use of (<ref>) and (<ref>) can be rewritten as follows:

    K=   (I+J)ω+∑_n=1^2∫_Ω_nẇ_n(x_2+d_2n) ρ_n dx  e_1-
       -
    ∑_n=1^2ρ_n∫_Ω_nẇ_n(x_1+d_1n)dx   e_2+ K_δ,

where J=(J_ij) is the tensor of inertia for the mechanical system with “frozen” plates (i.e. when the plates are considered as rigid bodies),

    J_11=∑_n=1^2ρ_n∫_Ω_n((x_2+d_2n)^2+d_3n^2)dx, 
    
    J_12=J_21=-∑_n=1^2ρ_n∫_Ω_n(x_1+d_1n)(x_2+d_2n)dx, 
    
    J_22=∑_n=1^2ρ_n∫_Ω_n((x_1+d_1n)^2+d_3n^2)dx, 
    
    J_23=J_32=-∑_n=1^2ρ_n d_3n∫_Ω_n(x_2+d_2n)dx, 
    
    J_33=∑_n=1^2ρ_n∫_Ω_n((x_1+d_1n)^2+(x_2+d_2n)^2)dx,
    
    J_31=J_13=-∑_n=1^2ρ_n d_3n∫_Ω_n (x_1+d_1n)dx,

and the term K_δ is of order O(ω(ẇ_1+ẇ_2)) for small ẇ_j.
By computing the local derivative (in the sense of (<ref>)) for the angular momentum K given by formula (<ref>), we get:

    K̇_1  = (J_11+I_1)ω̇_1+J_12ω̇_2
    +J_13ω̇_3+∑_n=1^2ρ_n∫_Ω_nẅ_n(x_2+d_2n)dx,


    K̇_2  =J_21ω̇_1+(J_22+I_2)ω̇_2
    +J_23ω̇_3-∑_n=1^2ρ_n∫_Ω_nẅ_n(x_1+d_1n)dx,


    K̇_3  = J_31ω̇_1+J_32ω̇_2+(J_33+I_3)ω̇_3,

where the nonlinear terms with respect to the derivatives of ω_j and w_n are omitted.


We expand the cross product ω× K= ∑_j=1^3 (ω× K)_j e_j to see that

    (ω× K)_1 =
     ω_2(J_31ω_1+J_32ω_2+(J_33+I_3)ω_3)-J_21ω_1 ω_3-
    -
    ω_3((J_22+I_2)ω_2+J_23ω_3-∑_n=1^2ρ_n∫_Ω_nẇ_n(x_1+d_1n) dx),


    (ω× K)_2 =-
    ω_1(J_31ω_1+J_32ω_2+(J_33+I_3)ω_3)+J_12ω_2ω_3+
    
    +ω_3(∑_n=1^2ρ_n∫_Ω_nẇ_n(x_2+d_2n)dx+
    (J_11+I_1)ω_1+J_13ω_3),


    (ω× K)_3 =
    ω_1(J_21ω_1+(J_22+I_2)ω_2+J_23ω_3)-
    
    -ω_2((J_11+I_1)ω_1+J_12ω_2+J_13ω_3
    )-
    
    - ∑_n=1^2ρ_n∫_Ω_nẇ_n(ω_2(x_2+d_2n)+ω_1 (x_1+d_1n))dx.




By putting together  the formulas (<ref>) and (<ref>) and expressing the values of ẅ_1 and ẅ_2 from (<ref>) and (<ref>), we write the components of the angular momentum equation (<ref>) as follows:

    (I_1+∑_n=1^2ρ_nd_3n^2l_1nl_2n)ω̇_1+J_13ω̇_3=
    
    =f_1+
    ω_3[J_21ω_1+(J_22+I_2)ω_2+J_23ω_3]-
    
    -ω_2[J_31ω_1+J_32ω_2+(J_33+I_3)ω_3]+
    
    
    +∑_n=1^2ρ_n∫_Ω_n{a_n^2(x_2+d_2n)Δ^2w_n-ω_3ẇ_n(x_1+d_1n)}dx,
    (I_2+∑_n=1^2ρ_n d_3n^2l_1nl_2n)ω̇_2+J_23ω̇_3=
    
     =f_2+
    ω_1[J_31ω_1+J_32ω_2+(J_33+I_3)ω_3]-
    
    -ω_3[(J_11+I_1)ω_1+J_12ω_2+J_13ω_3]-
    
    -
    ∑_n=1^2ρ_n∫_Ω_n{a_n^2(x_1+d_1n)Δ^2w_n+ω_3ẇ_n(x_2+d_2n)}dx,
    
    J_31ω̇_1+J_32ω̇_2+(J_33+I_3)ω̇_3=
    
    =f_3-
    ω_1[J_21ω_1+(J_22+I_2)ω_2+J_23ω_3]+
    
    +ω_2[(J_11+I_1)ω_1+J_12ω_2+J_13ω_3]+
    
    +∑_n=1^2ρ_n∫_Ω_n{(x_1+d_1n)ω_1+(x_2+d_2n)ω_2}ẇ_ndx,

where the nonlinear terms with respect to the derivatives are omitted.

In order to  rewrite the above differential equations in the normal form with respect to ω̇_j, we compute the inverse matrix J^-1= Ĵ =  (Ĵ_ij):

    Ĵ_11 = (I_2 + ∑_n=1^2ρ_n d_3n^2 l_1n l_2n)(I_3+J_33)-J_23^2/D,


    Ĵ_12 = Ĵ_21 =J_13 J_23/D,


    Ĵ_13 =Ĵ_31 =-(I_2 + ∑_n=1^2ρ_n d_3n^2 l_1n l_2n)J_13/D,


    Ĵ_22 = (I_1 + ∑_n=1^2ρ_n d_3n^2 l_1n l_2n)(I_3+J_33)-J_13^2/D,


    Ĵ_23 =Ĵ_32 =-(I_1 + ∑_n=1^2ρ_n d_3n^2 l_1n l_2n)J_23/D,


    Ĵ_33 = (I_1 + ∑_n=1^2ρ_n d_3n^2 l_1n l_2n)(I_2 + ∑_n=1^2ρ_n d_3n^2 l_1n l_2n)/D,


    D = (I_1 + ∑_n=1^2ρ_n d_3n^2 l_1n l_2n)(I_2 + ∑_n=1^2ρ_n d_3n^2 l_1n l_2n)(I_3+J_33)-
    -J_13^2 (I_2 + ∑_n=1^2ρ_n d_3n^2 l_1n l_2n)-J_23^2 (I_1 + ∑_n=1^2ρ_n d_3n^2 l_1n l_2n).


Note that the denominator D in formulas (<ref>) is strictly positive at least for sufficiently small moments of inertia of the plates J_ik compared with the moments of inertia of the carrier body I_i.
In particular, this condition is satisfied for sufficiently small area densities ρ_j (i.e., for sufficiently thin plates). Thus, we assume that D≠ 0 in the sequel. Then the differential equations (<ref>) can be written in the form

    [ ω̇_1; ω̇_2; ω̇_3 ] = Ĵ[ ϕ_1; ϕ_2; ϕ_3 ],

where ϕ_i denotes the right-hand side of the i-th equation in (<ref>).

We write the Poisson kinematic equations to ensure the condition that the frame (g_1,g_2,g_3) is fixed in the inertial space:

    ġ_i=-ω× g_i,      i=1,3.

Let g_i=g_i1e_1+g_i2e_2+g_i3e_3, then system (<ref>) takes the following coordinate form:

    ġ_i1=ω_3g_i2-ω_2g_i3, 
       ġ_i2=ω_1g_i3-ω_3g_i1, 
       ġ_i3=ω_2g_i1-ω_1g_i2,      i=1,3.


For the Cartesian frames (g_1,g_2,g_3) and (e_1,e_2,e_3) of the same orientation, the system of
differential equations
(<ref>), (<ref>), (<ref>), (<ref>), (<ref>)
has the following particular solution with f_1=f_2=f_3=0:

    w(x,t)=0,   ω_i(t)=0,    g_ij(t)=δ_ij,       i,j=1,3,

where δ_ij is the Kronecker symbol.

To study the stabilization problem for the equilibrium (<ref>),
we introduce new variables g_ij(t)=g_ij(t)-δ_ij and consider the equations
of perturbed motion for (<ref>):

    ġ_11=ω_3g_12-ω_2g_13,   ġ_12=ω_1g_13-ω_3(g_11+1), 
       ġ_13=ω_2(g_11+1)-ω_1g_12,
       ġ_21=ω_3(g_22+1)-ω_2g_23,   ġ_22=ω_1g_23-ω_3g_21,
       ġ_23=ω_2g_21-ω_1(g_22+1),
       ġ_31=ω_3g_32-ω_2(g_33+1), ġ_32=ω_1(g_33+1)-ω_3g_31,
       ġ_33=ω_2g_31-ω_1g_32.


We consider a modified energy functional

    E =T+U+1/2∑_i,j=1^3α_ig_ij^2

with positive parameters α_i, where

    T=1/2 (I_1ω_1^2+I_2ω_2^2+I_3ω_3^2+∫_Ω_1 v_P^2 ρ_1dx+ ∫_Ω_2 v_K^2 ρ_2 dx )
 is the
kinetic energy of the system, and

    U = 1/2∑_n=1^2∫_Ω_n( w_n(x,t))^2 a_n^2 ρ_n  dx
 is the potential energy of elastic deformations according to the Kirchhoff model.
Here Δ = ∂^2/∂ x_1^2+ ∂^2/∂ x_2^2 is the Laplace operator.
For future use, we introduce the Lyapunov functional V as a quadratic approximation of E:

    2V=(I_1+J_11)ω_1^2+(I_2+J_22)ω_2^2+(I_3+J_33)ω_3^2+2J_12ω_1ω_2 +


    +
    2J_13ω_1ω_3
    +2 J_23ω_2ω_3+∑_i,j=1^3 α_i g̃_ij^2+∑_n=1^2ρ_n ∫_Ω_nẇ_n^2 dx+


    +
    ∑_n=1^2ρ_n ∫_Ω_n{2ẇ_n [ω_1(d_2n+x_2)-ω_2(d_1n+x_1)]+a_n^2 (Δ w_n)^2}dx.



Let us compute the time derivative of the functional (<ref>) along the trajectories of (<ref>), (<ref>), (<ref>):

    V̇ = (K̇_1 - α_2 g̃_23+α_3 g̃_32)ω_1 + (K̇_2 + α_1 g̃_13-α_3 g̃_31)ω_2 +


    +(K̇_3 + α_2 g̃_21-α_1 g̃_12)ω_3+


    +∑_n=1^2∫_Ω_n{Δ w_n Δẇ_n - ẇ_nΔ^2 w_n}a_n^2 ρ_n dx,

where the expressions for K̇_i are given by (<ref>).
If the partial derivatives of w_i(x,t) of the fourth order in x and the first order in t are continuous and the boundary conditions (<ref>) are satisfied, then the integration by parts in formula (<ref>) leads to the following identities:

    ∫_Ω_n{Δ w_n Δẇ_n - ẇ_nΔ^2 w_n}dx =0,   n=1,2.

Using these identities and expressing K̇_i from the equation (<ref>), we rewrite formula (<ref>) as

    V̇ = (f_1 - (ω× K)_1 - α_2 g̃_23+α_3 g̃_32)ω_1 +


    +(f_2 - (ω× K)_2 + α_1 g̃_13-α_3 g̃_31)ω_2 +


    +(f_3 - (ω× K)_3 + α_2 g̃_21-α_1 g̃_12)ω_3,

where (ω× K)_i are given by (<ref>).

To stabilize the trivial solution of the system (<ref>),(<ref>), (<ref>), (<ref>), (<ref>), we define a feedback control from the condition

    V̇ = -k(ω_1^2+ω_2^2+ω_3^2)≤ 0,

where k is a positive constant. It is easy to see that condition (<ref>) corresponds to the following choice of controls:

    f_1 = -kω_1 +(ω× K)_1 + α_2 g̃_23-α_3 g̃_32,


    f_2 = -kω_2 +(ω× K)_2 -α_1 g̃_13+α_3 g̃_31,


    f_3 = -kω_3 +(ω× K)_3+α_1 g̃_12-α_2 g̃_21.

Note that the time-derivative (<ref>) is not negative definite, and the finite-dimensional method described in <cit.> is not directly applicable
to establish strictly decreasing behavior of V along the trajectories of the closed-loop system.




§ OPERATOR FORM OF THE DYNAMICAL EQUATIONS


To formulate our main result, we rewrite the equations of motion of the mechanical system under consideration in the operator form. We introduce the real linear space H= H^2(Ω_1)× H^2(Ω_2)× L^2(Ω_1)× L^2(Ω_2)×ℝ^12, whose elements are denoted as

    ξ=[ u_1; v_1; u_2; v_2;   ω;  g̃ ]:
    u_n∈ H^2(Ω_n),  v_n∈ L^2(Ω_n),  n=1,2,
    ω = [ ω_1; ω_2; ω_3 ]∈ℝ^3, g̃ = [ g̃_11; g̃_12;     ⋮; g̃_33 ]∈ℝ^9.

Here H^2(Ω_n) is the Sobolev space of the functions u∈ H^2(Ω_n) having zero trace on ∂Ω_n.
The inner product of

    ξ^1=[ u_1^1; v_1^1; u_2^1; v_2^1;   ω^1;  g̃^1 ]∈ H  and  ξ^2=[ u_1^2; v_1^2; u_2^2; v_2^2;   ω^2;  g̃^2 ]∈ H

is defined as

    <ξ^1,ξ^2>_H = ∑_n=1^2ρ_n ∫_Ω_n{a_n^2 Δ u_n^1(x)  Δ u_n^2(x) + v_n^1(x) v_n^2(x) +


    + (ω^2_1 v_n^1(x)+ω^1_1 v_n^2(x))(d_2n+x_2)-


    -(ω^2_2 v_n^1(x)+ ω^1_2 v_n^2(x))(d_1n+x_1)
    }dx+


    +((I+J)ω^1,ω^2)+∑_i,j=1^3α_i g̃^1_ijg̃^2_ij.

Using the Cauchy–Schwarz and Friedrichs' inequalities, it can be shown that the norm ξ_H=√(<ξ,ξ>_H) is equivalent to the standard norm in H^2(Ω_n)× L^2(Ω_n)×ℝ^12.
Thus, (H, <·,·>_H) is a Hilbert space.

We define an unbounded operator A: D(A)→ H and a bounded linear operator B:ℝ^3→ H in the following way:

    A: ξ=[ u_1; v_1; u_2; v_2;   ω;  g̃ ]↦
    Aξ = [ u_1^ξ; v_1^ξ; u_2^ξ; v_2^ξ;   ω^ξ;  g̃^ξ ]∈ H,


    B: f=[ f_1; f_2; f_3 ]↦ Bf = [ u_1^f; v_1^f; u_2^f; v_2^f;   ω^f;  g̃^f ]∈ H,

where

    ω^ξ_i = (Ĵ_i1ω_3-Ĵ_i3ω_1)[J_21ω_1+(J_22+I_2)ω_2+J_23ω_3]+


    +(Ĵ_i2ω_1-Ĵ_i1ω_2)[J_31ω_1+J_32ω_2+(J_33+I_3)ω_3]+


    +(Ĵ_i3ω_2-Ĵ_i2ω_3)[(J_11+I_1)ω_1+J_12ω_2+J_13ω_3]+


    +∑_n=1^2ρ_n ∫_Ω_n(Ĵ_i3[(x_1+d_1n)ω_1 + (x_2+d_2n)ω_2]. -


    .
    -[Ĵ_i1(x_1+d_1n)+Ĵ_i2(x_2+d_2n)]ω_3) v_n(x)dx+


    +∑_n=1^2ρ_n a_n^2∫_Ω_n(Ĵ_i1 (x_2+d_2n)-Ĵ_i2 (x_1+d_1n))Δ^2 u_n(x)dx,


    v_n^ξ(x) = -a_n^2Δ^2 u_n(x)+(x_1+d_1n)ω^ξ_2 - (x_2+d_2n)ω^ξ_1,


    u_n^ξ(x) = v_n(x),   n=1,2,


    g̃^ξ_11 = ω_3g_12-ω_2g_13, g̃^ξ_12 =ω_1g_13-ω_3(g_11+1),


    g̃^ξ_13 =ω_2(g_11+1)-ω_1g_12,


    g̃^ξ_21 =ω_3(g_22+1)-ω_2g_23, g̃^ξ_22 =ω_1g_23-ω_3g_21,


    g̃^ξ_23 =ω_2g_21-ω_1(g_22+1),


    g̃^ξ_31 =ω_3g_32-ω_2(g_33+1), g̃^ξ_32 =ω_1(g_33+1)-ω_3g_31,


    g̃^ξ_33 =ω_2g_31-ω_1g_32,


    v_n^f(x) =∑_k=1^3(Ĵ_2k(x_1+d_1n)-Ĵ_1k(x_2+d_2n))f_k,


    u_n^f(x)=0, g̃^f =0, ω^f_i = ∑_k=1^3Ĵ_ikf_k,   i=1,2,3,

and the coefficients Ĵ_ik are given in (<ref>).
The domain of definition of the nonlinear operator A has the form

    D(A)={ξ∈ H   |  [                     u_n∈ H^4(Ω_n),   v_n ∈ H^2(Ω_n),; .∂^2u_n/∂ x_1^2|_∂Ω_n=.∂^2u_n/∂ x_2^2|_∂ _n=0, n=1,2 ].}.





§ MAIN RESULT


We consider a nonlinear control system governed by the following abstract differential equation in H:

    d/dtξ(t) = Aξ(t) + B f,

where ξ(t)∈ H is the state state, f∈ℝ^3 is the control, and the operators A and B are given by (<ref>) and (<ref>).
If the functions w_1(x,t), w_2(x,t), ω(t), g̃(t) define a classical solution of  system (<ref>), (<ref>), (<ref>), (<ref>), (<ref>) with a control  f=f(t) on the interval t∈ I=[t_0,T), T≤ +∞,
then by direct substitution we verify that the corresponding function

    ξ(t) = [  w_1(·,t); ẇ_1 (·,t);  w_2(·,t); ẇ_2 (·,t);      ω(t);     g̃(t) ]∈ D(A)⊂ H

satisfies the equation (<ref>) with f=f(t) on t∈ I.
Thus, we consider the differential equation (<ref>) as a mathematical model of the considered mechanical system.

Let us represent the feedback control (<ref>) by an operator G:H→ℝ^3 defined on the state space of system (<ref>):

    G: ξ=[ u_1; v_1; u_2; v_2;   ω;  g̃ ]↦ f=Gx=[ f_1^ξ; f_2^ξ; f_3^ξ ],



    f_1^ξ = -
    ω_3(J_21ω_1+(J_22+I_2)ω_2+J_23ω_3)-


    -kω_1 +α_2 g̃_23-α_3 g̃_32 + ω_2(J_31ω_1+J_32ω_2+(J_33+I_3)ω_3)+


    +
    ω_3∑_n=1^2ρ_n ∫_Ω_n(x_1+d_1n)v_n(x) dx,


    f_2^ξ =
    -
    ω_1(J_31ω_1+J_32ω_2+(J_33+I_3)ω_3)
    -


    -kω_2  -α_1 g̃_13+α_3 g̃_31
    +ω_3(
    (J_11+I_1)ω_1+J_12ω_2+J_13ω_3)+


    +ω_3∑_n=1^2ρ_n ∫_Ω_n(x_2+d_2n)v_n(x)dx,


    f_3^ξ = -ω_2((J_11+I_1)ω_1+J_12ω_2+J_13ω_3
    )-


    -kω_3 +α_1 g̃_12-α_2 g̃_21+ω_1(J_21ω_1+(J_22+I_2)ω_2+J_23ω_3)-


    -∑_n=1^2ρ_n ∫_Ω_n
    [(x_1+d_1n)ω_1+(x_2+d_2n)ω_2]v_n(x)dx,


where k and α_i are arbitrary positive constants.
Then the closed-loop system (<ref>) with  f=Gξ takes the form

    d/dtξ(t)= Fξ(t),   F=A+BG,

where the domain of definition of the unbounded nonlinear operator F:D(F)→ H is dense in H, D(F)=D(A).

As it was noted above, the classical solutions of system (<ref>), (<ref>), (<ref>), (<ref>), (<ref>) correspond to the functions ξ(t)∈ D(A) according to the rule (<ref>).
Then the condition V̇≤ 0 for the time derivative of the functional V along the trajectories of the closed-loop system can be rewritten as

    <Fξ,ξ>_H≤ 0

for the corresponding element ξ∈ D(F)=D(A). This is a consequence of the definition of the functional V in (<ref>) and the inner product <·,·>_H in (<ref>).
Thus, inequality (<ref>) implies the following result concerning the operator F of the closed-loop system (<ref>).

Theorem 1.
The operator F:D(F)→ H is dissipative and D(F)=H.

We denote by I_H the identity operator on H.
If F is closed and the image of  I_H-λ F  coincides with H for λ>0, then Theorem 1 implies that F  is the infinitesimal generator of a strongly continuous semigroup of nonlinear operators
{S(t)}_t≥0 on H
because of the Crandall–Liggett theorem (cf. <cit.>).
 Then the mild solution of the Cauchy problem for (<ref>)
with the initial condition ξ(0)=ξ^0 is defined by the formula

    ξ(t)=S(t)ξ^0,   t≥ 0,

for any ξ^0∈ H. Under these assumptions, the trivial solution of the abstract differential equation (<ref>) is stable in the sense of Lyapunov because of the dissipativity inequality <Fξ,ξ>_H≤ 0 (V̇≤ 0).
Note that the mild solutions given by formula (<ref>) are classical if ξ^0∈ D(F).



§ NUMERICAL SIMULATIONS

In order to illustrate the transient behaviour of the proposed controller, we perform a numerical simulation of the closed-loop system (<ref>).
For this purpose we consider the case of identical rectangular domains Ω_1=Ω_2=[0,l_1]× [0,l_2] and introduce finite-dimensional approximations of the displacements
w_j(x_1,x_2,t)=q_j(t)W_1(x_1)W_2(x_2), j=1,2, for equations (<ref>) and (<ref>), respectively.
Here W_j(x)=sin(π m_j x/l_j), x∈ [0,l_j], m_j∈ℕ, are taken as eigenfunctions of the Sturm–Liouville problem with the boundary conditions
W_j(0)=W_j(l_j)=0, j=1,2. Let us consider the first flexible mode only (m_1=m_2=1) and apply the Ritz–Galerkin method (cf. <cit.>) for the nonlinear closed-loop system (<ref>)–(<ref>), (<ref>), (<ref>), (<ref>) to derive its finite-dimensional approximation of the form

    Ẋ (t) = Φ(X(t)),   X(t) ∈ℝ^16,

whose state vector is

    X = (g̃_11, g̃_12, ..., g̃_33 , ω_1,ω_2,ω_3, q_1,q_2,q̇_1,q̇_2 ).

We choose the following initial data and parameters for the simulation (the dimensions of physical quantities are omitted to simplify notations):

    X(0)=(0,0,0, 0,-1,1, 0,-1,-1, 0, ..., 0),


    l_1=l_2/2=1,  d_1=d_1'=0,  d_2=d_2'=1,  d_3=d_3'=0, 
       ρ_1=ρ_2=1,  a_1=a_2 = 1/2,  J=I, α_1 = α_2=α_3=1.

The above X(0) corresponds to an equilibrium of the considered mechanical system with g_1=e_1, g_2 = e_3, and g_3=-e_2.
In this case, the considered stabilization problem (steering the closed-loop system to its trivial equilibrium) means rotation about the x_1-axis by the angle π/2
with simultaneous damping of the vibration modes.

  


Fig. <ref> shows the behavior of solutions to the Cauchy problem (<ref>), (<ref>) for different values of the tuning parameter k>0 appearing in the feedback law (<ref>).
We see that the proposed controller can be used to steer the state of system (<ref>) to zero as t→ +∞, and the higher value of k (k=5) leads to faster convergence of X(t) to X=0.




§ CONCLUSIONS


In this paper, a new mathematical model of a controlled mechanical system consisting of a rotating body and elastic Kirchhoff plates has been introduced.
The model is described by the system of nonlinear ordinary and partial differential equations (<ref>)–(<ref>), (<ref>), (<ref>), or, equivalently,
by the abstract differential equation (<ref>) in the Hilbert space H.
A state feedback control has been proposed explicitly in the form (<ref>) to ensure that the time derivative of a Lyapunov functional is non-positive.

The main theoretical contribution of this work establishes the dissipativity property for the infinitesimal generator F in (<ref>).
Although the results of numerical simulations illustrate the efficiency of the proposed controller,
the question about asymptotic stability (or even partial asymptotic stability in the sense of <cit.>) remains open for the infinite-dimensional closed-loop dynamics.


                                                     

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