Controllability of second order infinite dimensional dynamical systems with delays

The article is devoted to analysing the approximate absolute and approximate relative controllability of a given type second order infinite dimensional system. The considered dynamical system is governed by the evolution equation with three damping terms and three terms without derivatives. Following this aim, spectral theory for linear unbounded operators is involved. At first the representation of considered infinite dimensional dynamical system by the infinite series of finite dimensional systems is given. Next, two theorems on necessary and sufficient conditions of approximate absolute and approximate relative controllability of the considered system are formulated and proved. Finally, proven theorems are applied to the analysis of the elastic beam. presented results into the case of arbitrary eigenvalues multiplicities of the state operator.


Problem statement
Let us consider the dynamical system described by the following abstract differential equation: with the initial conditions: where ( ) x t X ∈ (X is a Hilbert space) and 0, 0, 0,1, 2 . Furthermore, it is assumed that is a linear, generally unbounded, selfadjoint and positivedefinite operator with domain D(A) dense in X and compact resolvent R(λ,A) for all λ in the resolvent set ρ(A) (assumption 1).
The physical interpretation of equation (1) encompasses a broad class of real systems in this form and depends on a particular form of the A operator and of the coefficients α i and β i , i = 0,1,2.
It is well known that the operator A has the following spectral properties [1][2][3][4]: - φ form complete orthogonal system in Hilbert space X and after ortogonalization form complete orthonormal system in Hilbert space X. Hence, for every x X ∈ the following unique expansion holds true: 1 1 , -Operator A has the following spectral resolution: -The fractional power of operator A is defined as follows: -Operator A β , 0 < β < 1 is also selfadjoint and positive-definite with domain 2. The transformation of the state equation Using the spectral resolution of the state operator A and its properties (4)- (9) we can transform the infinite dimensional dynamical system, given by the abstract differential equation (1), into equivalent form of the infinite series of the finite dimensional second order linear dynamical systems with constant coefficients of the following form [5]: and x i (t) is a vector given by the equality (12): where x ij (t) denotes the (ij) th coefficient of the Fourier series of spectral representation for the element x in the state space X. The coefficients are explicitly given by the inner product between element in the state space X and the appropriate eigenfunctions φ ij of the operator A: Additionally, in the series of the equations (10) there exist constant coefficients * i α and * i β , which are defined by the equalities (14) and (15) [5] pp.295:   Basing on the infinite series of the equations (10) we can transform given system (1) to the more convenient form in the control theory, namely the form of infinite series of the set of first order finite-dimensional ordinary finite dimensional differential equations with constant coefficients (16) as follows [5]: where the state vector is given by the equality (17) [5]: The variables ( ), ( ) are defined by formula (20) as follows [5]: 3. The Jordan decomposition of the state matrix The Jordan decomposition of the state matrix (18) is investigated in paper [5] and now let us recall the results. This matrix has two distinct eigenvalues s i1 , s i2 each with the same multiplicity m i [5] pp.297: Pobrane z czasopisma Annales AI-Informatica http://ai.annales.umcs.pl Data: 06/11/2022 17:48:56

Case 3:
Obviously the following identity holds true for all the Jordan's decomposition cases: (34) Now let us verify whether the operator A i (18) is the infinitesimal generator of an analytic semigroup. Following this aim let us calculate some auxiliary limits.

Basic notions
Following the aim of analyzing the approximate controllability of infinite dimensional system with delays (1) at first let us present this notion in the case of finite dimensional systems. At first let us consider the linear stationary dynamical system, described by the differential equation without delays in control [6] pp.5:

Definition 4.1 [6]
The dynamical system (40) is said to be controllable, if and only if there exists such a control u(t), which will transfer the system from any given initial state to any final state in the control space in the finite time.

Theorem 4.2 [6] pp.16, [7] pp.70
The dynamical system (40) is controllable if and only if condition (41) holds true: Now let us consider linear stationary dynamical system, described by the differential equation with delays in control (42) [6] pp.196: where A 0 , B 0 are the constant matrices with dimensions respectively n×n, n×p. For the dynamical system of form (42) besides the instantaneous state x(t) ∈ R n , we introduce also the notion of the so-called complete state at time

Definition 4.3 [6] pp.195
Dynamical system (42) is said to be relatively controllable in [t 0 ,t 1 ], if for any initial complete state z(t 0 ) and any vector x 1 ∈ R n , there exists a control There are some known theorems for verifying the relative and absolute controllability of linear time varying systems with delays and control. Let us present two main theorems adapted to the stationary dynamical system of the form (42).

Approximate controllability analysis
Both theorems 4.5 and 4.6 base on the transformation of the system with delays in controls into the corresponding system without delays in controls. It can be noticed that that the state matrix in the corresponding systems (45) and (47) remains the same as in the investigated system with delays in controls (42). Let us return to the considered in this paper second order infinite dimensional dynamical system (1) in the form of the series (16). We will analyze its approximate relative and approximate absolute controllability by theorems 4.5 and 4.6, so the system series (16) in both the corresponding forms (45) and (47) has the same state matrices A i like in the system (16). In point 3 we recalled the Jordan decomposition of the state matrices A i . This form is very convenient for testing the controllability of a given dynamical system-involves only calculating the 1 term, instead of calculating the n-1 terms of the form l i i A B in the block matrix (41) necessary in case of using theorem 4.2. The general conditions of the controllability for the stationary, linear, finite dimensional without delays in the control dynamical system in the Jordan canonical form have been formulated by C.T.Chen in Chapter 5.5 of work [8] and have been recalled in paper [6] by theorem 1.5.1. In the next subchapters we use theorems 4.5 and 4.6 and the Chen's theorem to find the conditions of approximate absolute and approximate relative controllability of a given system (1).

Proof of the absolute controllability
The proof will be given as an example for case 1 of the Jordan decomposition (paragraph 3.1). We will prove the conditions of the absolute controllability of system (1) in the form of series (16) by the theorem 4.6 and mentioned Chen's theorem [6] pp. 25. At first let us calculate the matrix B for the system (16): Considering that the odd rows in the series of the matrices ik B (19) are zero from (51) we have directly (52): Now let us return to the verification of the controllability of dynamical system (1) in the form (16). Applying the Chen's theorem to it, based on formula (52), considered system (16) presented in the corresponding form without delays (47) with the input matrix ˆi B (given by (50)) is approximately controllable if and only if series (53), (54) are fulfilled: Pobrane z czasopisma Annales AI-Informatica http://ai.annales.umcs.pl Data: 06/11/2022 17:48:56

Proof of the relative controllability
The proof will also be given for an example for case 1 of the Jordan decomposition (paragraph 3.1). The proof bases on theorem 4.5 and the Chen's theorem [6]   Q.E.D.

Mechanical example
Let us consider a mechanical system described by the following linear partial differential equation: force on the beam. The boundary conditions correspond to hinged ends of the beam. More detailed description of these terms and the phenomenon they are describe can be found in papers [4,9,10,11].

The definition of the state differential operator
Let us define the linear unbounded differential operator : ( ) A D A H H ⊂ → [9,11] in the following way: It can be proved [9,11]  x A x z

The state equation
Applying operator A (62) to partial differential equation (58) with boundary conditions (61) we obtain the following abstract, ordinary second order differential equation with respect to t in the Sobolev space H: where: [ ] It is easy to see that equation (68) has form of the dynamical system (1) after introduction of the following coefficients:

The approximate absolute controllability analysis of the infinite dimensional mechanical system
In this subchapter the analysis of the approximate absolute controllability of given infinite dimensional dynamical system (58) will be performed. Also this dynamical system will be represented by the infinite series of the finite dimensional dynamical systems (16). These aims will be accomplished theorem 5.1. First, let us calculate the coefficients *

The approximate relative controllability analysis of the infinite dimensional mechanical system
Theorem 5.2 by (75) and the remarks from point 6.3 are fulfilled.

Summary of the mechanical example
The mechanical system (58) with conditions (59)-(61) is both aproximately relative and absolute controllable at any time.

Conclusions
In the article we obtained general conditions of different types of controllability for the infinite dimensional systems. It was possible thanks to making the use of the Chen's theorem. The obtained theorems of the approximate controllability without constraints, with the cone type constraints, and with delays in control hold true for the second order of the verified infinite dimensional dynamical system. This is innovative outcome in the controllability theory field.
Moreover, it should be pointed out that the presented methods can be easily adapted to the analysis of other dynamical properties of the considered nth order system, i.e. observability, attainability, stability and optimal control.
A possible way of further investigations can be the generalisation of the presented results into the case of arbitrary eigenvalues multiplicities of the state operator.