=Paper= {{Paper |id=Vol-3746/Short_3 |storemode=property |title=Weak Chaos and Controllability Conditions in Discrete Systems |pdfUrl=https://ceur-ws.org/Vol-3746/Short_3.pdf |volume=Vol-3746 |authors=Sergey Dashkovskiy,Oleksandr Boichuk,Oleksandr Pokutnyi,Olena Polishchuk |dblpUrl=https://dblp.org/rec/conf/dsmsi/DashkovskiyBPP23 }} ==Weak Chaos and Controllability Conditions in Discrete Systems== https://ceur-ws.org/Vol-3746/Short_3.pdf 
                         Weak Chaos and Controllability Conditions in Discrete Systems
                         Sergey Dashkovskiy 1, Oleksandr Boichuk 2, Oleksandr Pokutnyi 2,3 and Olena Polishchuk 3

                         1
                                University of Wπ‘’Μˆ rzburg, Institute of mathematics, 40 Emil Fischer str., Wπ‘’Μˆ rzburg, 97074, Germany
                         2
                                Institute of Mathematics of the National Academy of Sciences, 3 Tereschenkivska Street, Kyiv-4, 01024,
                                Ukraine
                         3
                                Taras Shevchenko University, 4d Glushkova Street, Kyiv, 03022, Ukraine

                                          Abstract
                                          Report is devoted to obtaining the conditions of the input-to-state stability for the discrete linear
                                          systems in the Banach and Hilbert spaces. For the nonlinear boundary-value problem we found
                                          a necessary condition for the existence of bounded solutions. Such condition was obtained with
                                          using of the system of operator equations for generating elements. Moreover, conditions of the
                                          controllability and reachability were obtained with using of the operator matrix equation.
                                          Estimates on the norms of solutions were obtained under assumption that the corresponding
                                          linear interconnected system admits a discrete dichotomy. For the boundary-value problem
                                          conditions of solvability were obtained and examples of boundary conditions were represented.
                                          Controllability conditions were obtained in the case when the corresponding set of controls are
                                          constant. We consider the so-called resonance ill-posed problem when the uniqueness can be
                                          disturbed, and corresponding linear interconnected system can have solutions not for any right-
                                          hand sides.
                                          Keywords 1
                                          Moore-Penrose pseudoinverse operator, weakly nonlinear equation, boundary-value problem

                         1. Introduction
                              Weakly nonlinear boundary-value problems for the discrete equations plays an important role in
                         the qualitative theory of dynamical systems. We consider conditions of the solvability for such
                         interconnected systems with linear boundary conditions. Interconnected systems use as a model for
                         investigations in applied sciences (see [1]). Moreover, the notion of input-to-state stability with
                         corresponding estimates for such system is a very popular direction in the last years (see [2]). That’s
                         why we obtain input-to-state stability estimates and corresponding controllability conditions. It should
                         be noted that this work is additional. We formulate general statement of the problem in nonlinear case
                         but represent and prove the main results only in linear case. In the future works we use these results for
                         investigating of nonlinear case and try to formulate chaotic conditions (in the weaklier sense than in
                         [4]- [10]). It should be noted the papers [11], [12].

                         1.1.        Statement of the problem

                         Consider the following interconnected system of nonlinear equations
                                               π‘₯𝑖 (𝑛 + 1, πœ€) = 𝐴𝑖 (𝑛)π‘₯𝑖 (𝑛, πœ€) + 𝐡𝑖 (𝑛)𝑒𝑖 (𝑛) + β„Žπ‘– (𝑛) +
                                              + πœ€π‘…π‘– (π‘₯1 (𝑛, πœ€), … , π‘₯π‘–βˆ’1 (𝑛, πœ€), π‘₯𝑖+1 (𝑛, πœ€), … , π‘₯𝑝 (𝑛, πœ€)) , π‘›πœ–π‘, 𝑖 = Μ…Μ…Μ…Μ…Μ…
                                                                                                                        1, 𝑝                           (1)
                         with boundary conditions


                         Dynamical System Modeling and Stability Investigation (DSMSI-2023), December 19-21, 2023, Kyiv, Ukraine
                         EMAIL:      sergey.dashkovskiy@uni-wuerzburg.de         (A.1);  boichuk.aa@mail.com       (A.2);     sherlock_int@ukr.net   (A.3),
                         olenap0934134133qaz@gmail.com (A.4)
                         ORCID: 0000-0001-7049-012X (A.1); 0000-0002-5020-9617 (A.3)
                                       ©️ 2023 Copyright for this paper by its authors.
                                       Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
                                       CEUR Workshop Proceedings (CEUR-WS.org)


CEUR
Workshop
                  ceur-ws.org                                                                                                                          96
              ISSN 1613-0073
Proceedings
                                           𝑙𝑖 π‘₯𝑖 (βˆ™, πœ€ ) = 𝛼𝑖 ,                                       (2)
where 𝐴𝑖 (𝑛), 𝐡𝑖 (𝑛) ∢ 𝐡 β†’ 𝐡 - are a set of bounded operators, from the Banach space B into itself.
  Assume that
                         𝐴𝑖 = (𝐴𝑖 (𝑛))π‘›βˆˆπ‘ ∈ π‘™βˆž (𝑍, 𝐿(𝐡)), 𝐡𝑖 = (𝐡𝑖 (𝑛))π‘›βˆˆπ‘ ∈ π‘™βˆž (𝑍, 𝐿(𝐡)),
                                          β„Žπ‘– = (β„Žπ‘– (𝑛))π‘›βˆˆπ‘ ∈ π‘™βˆž (𝑍, 𝐡).
It means that:
            |||𝐴𝑖 ||| = π‘ π‘’π‘π‘›βˆˆπ‘ ||𝐴𝑖 (𝑛)|| < + ∞, |||β„Žπ‘– ||| = π‘ π‘’π‘π‘›βˆˆπ‘ ||β„Žπ‘– (𝑛)|| < +∞,

                                             𝑙𝑖 : π‘™βˆž (𝑍, 𝐡) β†’ 𝐡𝑆
are the linear and bounded operators which translates bounded solutions of (1) into the Banach space
BS, Ξ±i are the elements of Banach space BS. We find conditions of the existence of bounded solutions
(1), (2) which turn (Ξ΅ = 0) in one of the bounded solutions of generating boundary-value problem
                        π‘₯𝑖0 (𝑛 + 1) = 𝐴𝑖 (𝑛)π‘₯𝑖0 (𝑛) + 𝐡𝑖 (𝑛)𝑒𝑖 (𝑛) + β„Žπ‘– (𝑛), 𝑛 ∈ 𝑍, 𝑖 = Μ…Μ…Μ…Μ…Μ…
                                                                                        1, 𝑝          (3)
                                                  𝑙𝑖 π‘₯𝑖0 (βˆ™) = 𝛼𝑖 .                                   (4)
   First, we formulate the conditions for the existence of bounded solutions of the equation (3) and
boundary value problem (3), (4). The corresponding homogeneous system of difference equations has
the following form:
                                   π‘₯𝑖 (𝑛 + 1) = 𝐴𝑖 (𝑛)π‘₯𝑖 (𝑛).                                      (5)
It should be noted that an arbitrary solution of a homogeneous system can be represented as: π‘₯𝑖 (π‘š) =
 Φ𝑖 (π‘š, 𝑛)π‘₯𝑖 (𝑛), π‘š β‰₯ 𝑛, where:
                                    𝐴 (π‘š βˆ’ 1)𝐴𝑖 (π‘š βˆ’ 2) … 𝐴𝑖 (𝑛), 𝑖𝑓 π‘š > 𝑛
                       Φ𝑖 (π‘š, 𝑛) = { 𝑖                                     .
                                    𝐼,                            𝑖𝑓 π‘š = 𝑛
   It is clear, that
                              Φ𝑖 (π‘š, 0 ) = 𝐴𝑖 (π‘š βˆ’ 1)𝐴𝑖 (π‘š βˆ’ 2) … 𝐴𝑖 (0).
   Also, we denote
                                    π‘ˆπ‘– (π‘š ): = Φ𝑖 (π‘š, 0) and π‘ˆπ‘– (0) = 𝐼.
    Traditionally [12], the mappings Φ𝑖 (π‘š, 𝑛) are called the evolutionary operators of the problem
(5). Suppose that the system (5) is exponentially dichotomous [4, 12] on the semiaxes 𝑍+ and π‘βˆ’ with
projectors 𝑃𝑖 and 𝑄𝑖 in the space B respectively, which means that there are projectors
                                    𝑃𝑖 (𝑃𝑖2 = 𝑃𝑖 ) and 𝑄𝑖 (𝑄𝑖2 = 𝑄𝑖 ),
constants
                                         𝑖                   (𝑖)
                                        π‘˜1,2 β‰₯ 1, 0 < πœ†1,2 < 1
such that
                                                                        π‘›βˆ’π‘š
                                                                  (𝑖)
                               ||π‘ˆπ‘– (𝑛)𝑃𝑖 π‘ˆπ‘–βˆ’1 (π‘š)|| ≀ π‘˜1𝑖 (πœ†1 )              ,𝑛 β‰₯ π‘š
                                                                        (𝑖) π‘šβˆ’π‘›            ,
                           ||π‘ˆπ‘– (𝑛)(𝐼 βˆ’ 𝑃𝑖 )π‘ˆπ‘–βˆ’1 (π‘š )|| ≀ π‘˜1𝑖 (πœ†1 )               ,π‘š β‰₯ 𝑛
                          {
for arbitrary π‘š, 𝑛 ∈ 𝑍+ (dichotomy on 𝑍+ ).
                                                                        π‘›βˆ’π‘š
                                                                  (𝑖)
                               ||π‘ˆπ‘– (𝑛)𝑄𝑖 π‘ˆπ‘–βˆ’1 (π‘š)|| ≀ π‘˜2𝑖 (πœ†2 )              ,𝑛 β‰₯ π‘š
                                                                (𝑖) π‘šβˆ’π‘›        ,
                           ||π‘ˆπ‘– (𝑛)(𝐼 βˆ’ 𝑄𝑖 )π‘ˆπ‘–βˆ’1 (π‘š )|| ≀ π‘˜2𝑖 (πœ†2 )     ,π‘š β‰₯ 𝑛
                        {
for arbitrary π‘š, 𝑛 ∈ π‘βˆ’ (dichotomy on π‘βˆ’ ).


                                                                                                       97
2. Main results. Linear case. Banach space case

    In this part we obtain the necessary and sufficient conditions of the existence of the sets of bounded
solutions for the linear generating boundary-value problems (3), (4) and controllability conditions.

2.1. Bounded solutions. Linear case

   We formulate an auxiliary lemma which we will use when obtaining the main results (first lemma
directly follows from the well-known results of [4], [12]).
   Lemma 1. Suppose that a homogeneous system is dichotomous on the semi-axes 𝑍+ and π‘βˆ’ with
projectors 𝑃𝑖 and 𝑄𝑖 respectively, and the operators
                                                    𝐷𝑖 = 𝑃𝑖 βˆ’ (𝐼 βˆ’ 𝑄𝑖 ): 𝐡 β†’ 𝐡
are generalized invertible [5]. The solutions of the equation (1) bounded on the entire axis 𝑍 exist if
and only if the following conditions are satisfied:
                                        βˆ‘+∞
                                         π‘˜= βˆ’βˆž 𝐻𝑖 (π‘˜ + 1)(β„Žπ‘– (π‘˜ ) + 𝐡𝑖 (π‘˜ )𝑒𝑖 (π‘˜ )) = 0.                                                        (6)
If the conditions (6) hold, the set of bounded solutions has the following view:
                            π‘₯𝑖0 (𝑛, 𝑐𝑖 ) = π‘ˆπ‘– (𝑛)𝑃𝑖 𝑃𝑁(𝐷𝑖 ) 𝑐𝑖 + (𝐺𝑖 [β„Žπ‘– + 𝐡𝑖 𝑒𝑖 ])(𝑛), 𝑐𝑖 ∈ 𝐡                                                  (7)
where 𝐺𝑖 are generalized inverse Green’s operators [12, 13] on 𝑍 with the following properties:
                       (𝐿𝑖 𝐺𝑖 [β„Žπ‘– + 𝐡𝑖 𝑒𝑖 ])(𝑛) = 𝐡𝑖 (𝑛)𝑒𝑖 (𝑛) + β„Žπ‘– (𝑛), 𝑛 ∈ 𝑍,
where
(𝐿𝑖 π‘₯𝑖 )(𝑛) ≔ π‘₯𝑖 (𝑛 + 1) βˆ’ 𝐴𝑖 (𝑛)π‘₯𝑖 (𝑛): π‘™βˆž (𝑍, 𝐡) β†’ π‘™βˆž (𝑍, 𝐡),
𝐻𝑖 (π‘˜ + 1) = 𝑃𝐡𝐷 𝑄𝑖 π‘ˆπ‘–βˆ’1 (π‘˜ + 1),
π·π‘–βˆ’ are generalized inverse to the operators 𝐷𝑖 ,
projectors 𝑃𝑁(𝐷𝑖 ) = 𝐼 βˆ’ π·π‘–βˆ’ 𝐷𝑖 and 𝑃𝐡𝐷 = 𝐼 βˆ’ 𝐷𝑖 π·π‘–βˆ’ (see [3], [13]),
                                                              𝑖

which project space B on the kernels 𝑁(𝐷𝑖 ) of the operators 𝐷𝑖 and the subspaces 𝐡𝐷𝑖 ≔ 𝐡 ⊝ 𝑅(𝐷𝑖 )
respectively (𝐡 = 𝐡𝐷𝑖 ⨁𝑅(𝐷𝑖 ) ).
      Remark 1. We have the following estimates for the norm of the solutions:
                                𝑛                                      𝑛              π‘˜ 𝑖 πœ†π‘–       π‘˜2π‘–πœ†π‘–2
||π‘₯𝑖0 (𝑛, 𝑐𝑖 )|| ≀ π‘˜1𝑖 (πœ†π‘–1 ) ||𝑃𝑁(𝐷𝑖) 𝑐𝑖 || + π‘˜1𝑖 (πœ†1𝑖 ) ||π·π‘–βˆ’ || ( 1 1𝑖 +                                 ) (|||β„Žπ‘– ||| + |||𝐡𝑖 ||| |||𝑒𝑖 |||) +
                                                                                     1βˆ’ πœ†1         1βˆ’ πœ†π‘–2
                   𝑛
      (1+ πœ†π‘–1 βˆ’(πœ†π‘–1) )
π‘˜1𝑖       1βˆ’ πœ†π‘–1
                         (|||β„Žπ‘– ||| + |||𝐡𝑖 ||| |||𝑒𝑖 |||) , 𝑛 β‰₯ 0,                                                                             (8)
                                βˆ’π‘›                                         βˆ’π‘›             π‘˜ 𝑖 πœ†π‘–       π‘˜2𝑖 πœ†π‘–2
||π‘₯𝑖0 (𝑛, 𝑐𝑖 )|| ≀ π‘˜2𝑖 (πœ†π‘–2 )        ||𝑃𝑁(𝐷𝑖 ) 𝑐𝑖 || + π‘˜2𝑖 (πœ†π‘–2 )               ||π·π‘–βˆ’ || ( 1 1𝑖 +                ) (|||β„Žπ‘– ||| +
                                                                                          1βˆ’ πœ†1       1βˆ’ πœ†π‘–2
                                                   βˆ’π‘›+1
                                 (1+ πœ†π‘–2 βˆ’(πœ†π‘–2 )          )
 |||𝐡𝑖 ||| |||𝑒𝑖 |||) + π‘˜2𝑖                                   (|||β„Žπ‘– ||| + |||𝐡𝑖 ||| |||𝑒𝑖 |||) , 𝑛 ≀ 0.                                        (9)
                                        1βˆ’ πœ†π‘–2
From these inequalities follows the estimates
                                                              π‘˜ 𝑖 πœ†π‘–   π‘˜2𝑖 πœ†π‘–2
|||π‘₯𝑖0 ||| ≀ 𝐾𝑖 ||𝑃𝑁(𝐷𝑖) 𝑐𝑖 || + 𝐾𝑖 ||π·π‘–βˆ’ || ( 1 1𝑖 +                            ) (|||β„Žπ‘– ||| + |||𝐡𝑖 ||| |||𝑒𝑖 |||) +
                                                          1βˆ’ πœ†1        1βˆ’ πœ†π‘–2
        1+ Λ𝑖
+ 𝐾𝑖 1βˆ’ 𝛬𝑖1 (|||β„Žπ‘– ||| + |||𝐡𝑖 ||| |||𝑒𝑖 |||),                                                                                                (10)
             2

where 𝐾𝑖 = π‘šπ‘Žπ‘₯{π‘˜1𝑖 , π‘˜2𝑖 }, 𝛬1𝑖 = π‘šπ‘Žπ‘₯{πœ†π‘–1 , πœ†π‘–2 }, 𝛬𝑖2 = π‘šπ‘–π‘›{πœ†π‘–1 , πœ†π‘–2 }.
      Remark 2. It should also be noted that if bounded solutions are united together at zero as follows:
                                       π‘₯𝑖 (0 + ) = π‘₯𝑖 (0 βˆ’ ) + 𝑐𝑖 ,



                                                                                                                                                98
where 𝑐𝑖 are the elements of Banach space, then we obtain a bounded solutions for the problem with a
jump.
   Let us find the solvability condition of the generating boundary-value problem (3), (4). Suppose that
condition (6) is satisfied. Substitute expression (7) in the boundary condition (2). Then we obtain the
operator equations which can be represented in the following form:
                                            𝑉𝑖 𝑐𝑖 = 𝛼𝑖 βˆ’ 𝑙𝑖 (𝐺𝑖 [β„Žπ‘– + 𝐡𝑖 𝑒𝑖 ])(βˆ™),                     (11)
where 𝑉𝑖 = 𝑙𝑖 π‘ˆπ‘– (βˆ™)𝑃𝑖 𝑃(𝑁(𝐷𝑖)) : 𝐡 β†’ 𝐡𝑆. If the operators 𝑉𝑖 are generalized invertible [6], then equation
(11) is solvable [6] if and only if the following conditions are satisfied:
                                      𝑃𝑉 𝑖 (𝛼𝑖 βˆ’ 𝑙𝑖 (𝐺𝑖 [β„Žπ‘– + 𝐡𝑖 𝑒𝑖 ])(βˆ™)) = 0,                         (12)
                                         𝐡1

where 𝑃𝑉 𝑖 = 𝐼 βˆ’ 𝑉𝑖 π‘‰π‘–βˆ’ (𝐡𝑆 = 𝑅(𝑉𝑖 )⨁𝑉𝐡𝑖 ). Under conditions (12) the sets of solutions of the system
         𝐡𝑆                                   𝐡𝑆
(11) have the following forms
                               𝑐𝑖 = π‘‰π‘–βˆ’ (𝛼𝑖 βˆ’ 𝑙𝑖 (𝐺𝑖 [β„Žπ‘– + 𝐡𝑖 𝑒𝑖 ])(βˆ™)) + 𝑃𝑁(𝑉𝑖) 𝑐̅𝑖 ,        𝑐̅𝑖 πœ–π΅.
Thus, we obtain the following theorem.
   Theorem 1. Under condition (6) boundary-value problem (3), (4) has bounded solutions if and
only if the conditions (12) are satisfied. The sets of bounded solutions have the following form:
                    π‘₯𝑖 (𝑛, 𝑐̅𝑖 ) = π‘ˆπ‘– (𝑛)𝑃𝑖 𝑃𝑁(𝐷𝑖 ) 𝑃𝑁(𝑉𝑖) 𝑐̅𝑖 + (𝐺̅𝑖 [β„Žπ‘– + 𝐡𝑖 𝑒𝑖 , 𝛼𝑖 ])(𝑛),     (13)
where (𝐺̅𝑖 [β„Žπ‘– + 𝐡𝑖 𝑒𝑖 , 𝛼𝑖 ])(𝑛) are generalized Green’s operators in the form
                         (𝐺̅𝑖 [β„Žπ‘– + 𝐡𝑖 𝑒𝑖 , 𝛼𝑖 ])(𝑛) = (𝐺𝑖 [β„Žπ‘– + 𝐡𝑖 𝑒𝑖 ])(𝑛) +
                        + π‘ˆπ‘– (𝑛)𝑃𝑖 𝑃𝑁(𝐷𝑖) π‘‰π‘–βˆ’ (𝛼𝑖 βˆ’ 𝑙𝑖 (𝐺𝑖 [β„Žπ‘– + 𝐡𝑖 𝑒𝑖 ])(βˆ™)).                          (14)
   Remark 3. It should be noted that the operators 𝑙𝑖 in (2) can be for example represent two-point
or multi-point boundary-value problems:
                                                                         𝑛

                     𝑙𝑖 π‘₯𝑖 (βˆ™) = 𝐴𝑖1 π‘₯𝑖 (π‘š ) βˆ’ 𝐴𝑖2 π‘₯𝑖 (0),   𝑙𝑖 π‘₯𝑖 (βˆ™) = βˆ‘ 𝐢𝑖 (𝑗 )π‘₯𝑖 (π‘šπ‘— ),
                                                                       𝑗=1
where π‘š, π‘šπ‘— ∈ 𝑍, 𝐢𝑖 (𝑗 ) ∈ 𝐿(𝐡, 𝐡𝑆), 𝑗 = Μ…Μ…Μ…Μ…Μ…
                                         1, 𝑛 are linear and bounded operators. Another example is
conditions at the infinity:
                              𝑙𝑖 π‘₯𝑖 (βˆ™) = 𝐴𝑖1 π‘™π‘–π‘š π‘₯𝑖 (𝑛) + 𝐴𝑖2 π‘™π‘–π‘š π‘₯𝑖 (𝑛).
                                               π‘›β†’βˆ’βˆž               𝑛 β†’+∞
    Moreover, boundary-value problems (3), (4) in the represented form are the systems of independent
boundary-value problems. We can consider more general linear boundary conditions (instead of (4))
in the forms:
                                βˆ‘π‘π‘˜=1 π‘™π‘˜ π‘₯π‘˜0 (βˆ™) = 𝜌, 𝜌 πœ– 𝐡𝑆.                                    (15)
                                                                              0
In such way we obtain boundary-value problem with components of system π‘₯𝑖 which can connect by
the condition (15).

2.2. Bounded solutions. Nonlinear case

     Using theorem 1 we can obtain necessary condition of the existence of the sets of bounded solutions
for the nonlinear boundary-value problem (1), (2). Suppose that generating boundary-value problem
(3), (4) is solvable. It means that the conditions (6), (12) are satisfied. In this case we can easily obtain
the following theorem.
   Theorem 2. (Necessary condition). Suppose that conditions (6), (12) (the sets of solutions (3), (4)
has the form (13)) are satisfied and nonlinear problem (1), (2) has bounded solutions. Then 𝑐̅𝑖 from the
Banach space B (see (13)) satisfy the following operator system for the generating elements:



                                                                                                          99
                                +∞

𝐹𝑖1 (𝑐̅1 , 𝑐̅2 , … , 𝑐̅𝑝 ) ≔                                        0 (
                                βˆ‘ 𝐻𝑖 (π‘˜ + 1)𝑅𝑖 (π‘₯10 (π‘˜, 𝑐̅1 ), … , π‘₯π‘–βˆ’1               0 (
                                                                        π‘˜, 𝑐̅𝑖 βˆ’1 ), π‘₯𝑖+1 π‘˜, 𝑐̅𝑖+1 ), … , π‘₯𝑝0 (π‘˜, 𝑐̅𝑝 ) ) = 0
                               π‘˜= βˆ’βˆž
 𝐹𝑖2 (𝑐̅1 , 𝑐̅2 , … , 𝑐̅𝑝 ) ≔ 𝑃𝑉 𝑖 (𝑙𝑖 (𝐺𝑖 (𝑅𝑖 (π‘₯10 (βˆ™, 𝑐̅1 ), … , π‘₯π‘–βˆ’1
                                                                    0 (               0 (
                                                                        βˆ™, 𝑐̅𝑖 βˆ’1 ), π‘₯𝑖+1 βˆ™, 𝑐̅𝑖+1 ), … , π‘₯𝑝0 (βˆ™, 𝑐̅𝑝 ) ))(βˆ™)) = 0.
                                    𝐡1



2.3. Controllability conditions

   Consider the case when the controllability sequences 𝑒𝑖 (𝑛) = 𝑒𝑖 are fixed for any n. Then,
conditions of reachability take the form of solvability of the following systems of operator equations:
                                                𝑄𝑖 𝑒𝑖 = 𝑔𝑖 ,                                       (16)
                                                           𝑅𝑖 𝑒𝑖 = 𝑀𝑖 ,                                                       (17)
where the corresponding operators 𝑄𝑖 , 𝑅𝑖 and elements 𝑔𝑖 , 𝑀𝑖 have the following form
                                    𝑄𝑖 = βˆ‘+∞
                                          π‘˜= βˆ’βˆž 𝐻𝑖 (π‘˜ + 1)𝐡𝑖 (π‘˜ ),              𝑔𝑖 = βˆ’ βˆ‘+∞
                                                                                        π‘˜= βˆ’βˆž 𝐻𝑖 (π‘˜ + 1)β„Žπ‘– (π‘˜ ),
                                     𝑅𝑖 = 𝑃𝑉 𝑖 𝑙𝑖 (𝐺𝑖 [𝐡𝑖 ])(βˆ™),                  𝑀𝑖 = 𝑃𝑉 𝑖 (𝛼𝑖 βˆ’ 𝑙𝑖 (𝐺𝑖 [β„Žπ‘– ])(βˆ™)).
                                                  𝐡1                                         𝐡1

We can rewrite the systems of operator equations (16), (17) in the form of matrix operator equation
                                 𝑒 = 𝐺, 𝐡0 : 𝐡𝑝 β†’ (𝐡 Γ— 𝐡1 )𝑝 , 𝑒 ∈ 𝐡𝑝 , 𝐺 ∈ (𝐡 Γ— 𝐡1 )𝑝 ,                                       (18)
where
                 𝑄
                [ 1]            0            …         0                                                         𝑔1
                 𝑅1
                                                                                                                [𝑀 ]
                                𝑄                                                  𝑒1                             1
                  0            [ 2]          …         0                                                         𝑔2
                                𝑅2                                                 𝑒2
𝐡0 =                                                            ,             𝑒 = ( ),                      𝐺 = [𝑀2 ] .
                                                                                   …                             …
                                    …                                              𝑒𝑝                            𝑔𝑝
                                                    𝑄𝑝                                                         ([𝑀𝑝 ])
                   0            0            …     [ ]
        (                                           𝑅𝑝      )
    If the operator 𝐡0 is generalized invertible [6], then we can obtain the following theorem.
    Theorem 2. (Reachability conditions). Under condition
                                                                    𝑃𝑁(𝐡0βˆ— ) 𝐺 = 0,                                           (19)
   the linear generating boundary-value problem (3), (4) is controllable. The sets of controls can be
                                 represented in the following form:
            𝑒 = 𝐡0βˆ’ 𝐺 + 𝑃𝑁(𝐡0) 𝑣,                   for any 𝑣 = (𝑣11 , 𝑣12 , 𝑣21 , 𝑣22 , … , 𝑣𝑝1 , 𝑣𝑝2 ) ∈ (𝐡 Γ— 𝐡1 )𝑝 .       (20)
   Remark 4. Substituting solutions (20) in (13) we obtain the set of bounded solutions with
corresponding set of controls u.
    Theorem 3. (Input-to-state stability conditions (see [1]-[3]). It is easy to show that under conditions
of the existence of bounded solutions from the inequalities (8), (9) we have the following estimates for
any bounded solution of (3), (4)
                                    𝑛
 ||π‘₯𝑖0 (𝑛, 𝑐̅𝑖 )|| ≀ 𝑀𝑖 (πœ†π‘–1 ) (||𝑃𝑁(𝐷𝑖) 𝑃𝑁(𝑉𝑖) 𝑐̅𝑖 || + ||𝛼𝑖 ||) + 𝛾𝑖 (|||β„Žπ‘– |||) + 𝛾𝑖 (|||𝐡𝑖 ||| |||𝑒𝑖 |||) , 𝑛 β‰₯ 0,
                                 βˆ’π‘›
||π‘₯𝑖0 (𝑛, 𝑐̅𝑖 )|| ≀ 𝑀𝑖 (πœ†π‘–2 )            (||𝑃𝑁(𝐷𝑖 ) 𝑃𝑁(𝑉𝑖) 𝑐̅𝑖 || + ||𝛼𝑖 ||) + 𝛾𝑖 (|||β„Žπ‘– |||) + 𝛾𝑖 (|||𝐡𝑖 ||| |||𝑒𝑖 |||) , 𝑛 ≀ 0,
with corresponding constants 𝑀𝑖 and functions 𝛾𝑖 .
   Remark 5. Presented theorem gives us conditions of the input-to-state stability and chaoticity (see
also [1],[11]).




                                                                                                                               100
   Remark 6. It should be noted that in more general case the linear and bounded operators 𝑙𝑖 can
translate bounded solutions of (1) into different spaces (instead of BS we can use 𝐡𝑆𝑖 , where 𝐡𝑆𝑖 are
Banash spaces and 𝐡𝑆𝑖 β‰  𝐡𝑆𝑗 ).

3. Acknowledgements

   The authors would like to thank the conference organizers for the opportunity to make report at the
conference DSMSI 2023.

4. References
[1] S. Dashkovskiy, M. Kosmykov. Input-to-state stability of interconnected hybrid systems,
    Automatica 49 (2013) 1068-1074. //doi.org/10.1016/j.automatica.2013.01.045.
[2] Sontag E.D. Mathematical control theory. Deterministic finite dimensional systems, 2nd ed.,
    Springer, New York, (1991).
[3] Gohberg, I. and Krupnik, N. One-dimensional linear singular integral equations, Birkhauser, Basel,
    1992. //doi.10.1007/978-3-0348-8647-5.
[4] K. Palmer. Exponential Dichotomies, the Shadowing Lemma and Transversal Homoclinic Points,
    Dynamics Reported 1 (1988).
[5] Diblik, J. Bounded solutions to systems of fractional discrete equations, Advances in Nonlinear
    Analysis 11 (2022) 1614--1630.
[6] Duan, L. and Huang, L. and Guo, Z. Stability and almost periodicity for delayed high-order
    Hopfield neural networks with discontinuous activations, Nonlinear dynamics 77 (2014) 1469β€”
    1484. // doi.10.1007/s11071-014-1392-3.
[7] Yanguang, Li. Homoclinic tubes in discrete nonlinear SchrΓΆdinger equation under Hamiltonian
    perturbations, Nonlinear dynamics 31 (2003) 393-434. // doi.10.1023/A:1023268714897.
[8] Jin, Y. and Zhu, X. and Liu, Y. Bifurcations of twisted heteroclinic loop with resonant eigenvalues,
    Nonlinear dynamics, 92 (2018) 557-573. // doi.10.1007/s11071-018-4075-7.
[9] Zhang, W. and Li, S.B. Resonant chaotic motions of a buckled rectangular thin plate with
    parametrically and externally excitations, Nonlinear dynamics, 62 (2010) 673β€”686.
    //doi.10.1007/s11071-010-9753-z.
[10] Li, S.B. and Shen, C. and Zhang, W. Homoclinic bifurcations and chaotic dynamics for a
     piecewise linear system under a periodic excitation and a viscous damping, Nonlinear dynamics,
     79 (2015) 2395–-2406. //doi.10.1007/s11071-014-1820-4.
[11] Slyn’ko V., Tunc C. Stability of abstract linear switched impulsive differential equations,
      Automatica 107 (2019), 433-441.
[12] O. Pokutnyi. Weak chaos in discrete systems. arxiv:2310.08220 (2023).
[13] Boichuk, A. and Samoilenko, A., Generalized Inverse Operators and Fredholm Boundary-Value
      Problems, 2nd ed., De Gruter, Berlin, 2016. //doi.10.1515/9783110378443.




                                                                                                    101