Weak Chaos and Controllability Conditions in Discrete Systems

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 righthand 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

( + 1, )= ( ) ( , )+ ( ) ( )+ ℎ ( )+ + ( 1( , ), … , −1( , ), +1( , ), … , ( , )), , = ̅1̅̅,̅̅ with boundary conditions ( 1 ) (A.3), (∙, )= , where ( ), ( )∶ → - are a set of bounded operators, from the Banach space B into itself.

Assume that

It means that:

||| ||| = ∈ || ( )|| < + ∞, |||ℎ ||| = ∈ ||ℎ ( )|| < +∞,

: ∞( , )→ are the linear and bounded operators which translates bounded solutions of ( 1 ) into the Banach space BS, αi are the elements of Banach space BS. We find conditions of the existence of bounded solutions ( 1 ), ( 2 ) which turn (ε = 0) in one of the bounded solutions of generating boundary-value problem 0( + 1)= ( ) 0( )+ ( ) ( )+ ℎ ( ), ∈ , = ̅1̅̅,̅̅ ( 3 )

0(∙)= . ( 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:

( + 1)= ( ) ( ). ( 5 ) It should be noted that an arbitrary solution of a homogeneous system can be represented as: ( )= Φ ( , ) ( ), ≥ , where:

( − 1) ( − 2)… ( ), > Φ ( , )= { , = .

It is clear, that

Also, we denote

Φ ( , 0 )= ( − 1) ( − 2)… (0).

( ):= Φ ( , 0)and (0)= .

Traditionally [12], the mappings Φ ( , )are called the evolutionary operators of the problem ( 5 ). Suppose that the system ( 5 ) is exponentially dichotomous [4, 12] on the semiaxes + and − with projectors and in the space B respectively, which means that there are projectors { { ( 2 = )and ( 2 = ), 1,2 ≥ 1, 0 < 1(,2) < 1

− || ( ) −1( )|| ≤ 1 ( 1( ))

, ≥ || ( )( − ) −1( )|| ≤ 1 ( 1( )) − , ≥ ,

− || ( ) −1( )|| ≤ 2 ( 2( ))

, ≥ − || ( )( − ) −1( )|| ≤ 2 ( 2( )) , ≥ , constants such that for arbitrary , ∈ + (dichotomy on +). for arbitrary , ∈ − (dichotomy on −).

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]). projectors and respectively, and the operators

Lemma 1. Suppose that a homogeneous system is dichotomous on the semi-axes + and − with = − ( − ): → are generalized invertible [5]. The solutions of the equation ( 1 ) bounded on the entire axis exist if and only if the following conditions are satisfied: If the conditions ( 6 ) hold, the set of bounded solutions has the following view:

∑+=∞−∞ ( + 1)(ℎ ( )+ ( ) ( ))= 0. 0 ( , )= ( ) ( ) + ( [ℎ + ])( ), ∈

where are generalized inverse Green’s operators [12, 13] on with the following properties: ( [ℎ + ])( )= ( ) ( )+ ℎ ( ), ∈ , ( 6 ) ( 7 ) ( 8 ) ( 9 ) ( 10 ) where ( )( )≔ ( + 1)− ( ) ( ): ∞( , )→ ∞( , ), ( + 1)=

−1( + 1), −are generalized inverse to the operators , projectors ( )= − − and = − − (see [3], [13]),

Remark 2. It should also be noted that if bounded solutions are united together at zero as follows: (0 + )= (0 − )+ , which project space B on the kernels ( )of the operators and the subspaces ≔ ⊝ ( ) respectively ( = ⨁ ( )).

Remark 1. We have the following estimates for the norm of the solutions: || 0( , )|| ≤ 1( 1) || ( ) || + 1( 1) || −|| ( 1 1− 11 + 1 −2 22)(|||ℎ ||| + ||| ||| ||| |||)+ 1 (1+ 1−( 1) ) 1− 1

(|||ℎ ||| + ||| ||| ||| |||), ≥ 0, ||| ||| ||| |||)+ 2 (1+ 2−( 2)− +1)

1− 2 From these inequalities follows the estimates || 0( , )|| ≤ 2( 2)− || ( ) || + 2( 2)− || −|| ( 1 1− 11 + 1 −2 22)(|||ℎ ||| + (|||ℎ ||| + ||| ||| ||| |||), ≤ 0. ||| 0||| ≤ || ( ) || + || −|| ( 1

1− 11 + 1 −2 22)(|||ℎ ||| + ||| ||| ||| |||)+ + 11+− Λ21 (|||ℎ ||| + ||| ||| ||| |||), where = { 1, 2}, 1 = { 1, 2}, 2 = { 1, 2}. where 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: where = (∙) ( ( )): → . If the operators are generalized invertible [6], then equation ( 11 ) is solvable [6] if and only if the following conditions are satisfied:

= − ( [ℎ + ])(∙),

( − ( [ℎ + ])(∙))= 0, where = − − ( ( 11 ) have the following forms

Thus, we obtain the following theorem.

= ( )⨁ ). Under conditions ( 12 ) the sets of solutions of the system = −( − ( [ℎ + ])(∙))+ ( ) ̅, ̅ .

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: ( , ̅)= ( ) ( ) ( ) ̅+ ( ̅[ℎ + , ])( ), where ( ̅[ℎ + , ])( )are generalized Green’s operators in the form

( ̅[ℎ + , ])( )= ( [ℎ + ])( )+ + ( ) ( )

−( − ( [ℎ + ])(∙)).

Remark 3. It should be noted that the operators in ( 2 ) can be for example represent two-point or multi-point boundary-value problems:

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 ̅ from the Banach space B (see ( 13 )) satisfy the following operator system for the generating elements: ( 11 ) ( 12 ) ( 13 ) (14) (15) where , ∈ , ( )∈ ( , conditions at the infinity: (∙)= 1 →−∞ ( )+ 2 →+∞ ( ).

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 )) ), = ̅1̅̅,̅̅ are linear and bounded operators. Another example is in the forms: the condition (15).

∑

=1 0(∙)= , .

In such way we obtain boundary-value problem with components of system 0 which can connect by +∞

2.3. Controllability conditions

= ∑+=∞−∞ ( + 1) ( ), = − ∑+=∞−∞ ( + 1)ℎ ( ),

= 1 ( [ ])(∙), = 1( − ( [ℎ ])(∙)).

We can rewrite the systems of operator equations (16), (17) in the form of matrix operator equation = , 0: → ( × 1) , ∈ , ∈ ( × 1) , where 0 = ( [ 11] 0 0

0 [ 22]

… 0 , = ( …2), If the operator 0 is generalized invertible [6], then we can obtain the following theorem. Theorem 2. (Reachability conditions). Under condition

( 0∗) = 0, the linear generating boundary-value problem ( 3 ), ( 4 ) is controllable. The sets of controls can be represented in the following form: = 0− + ( 0) , for any = ( 11, 12, 21, 22, … , 1, 2)∈ ( × 1) .

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 ) || 0( , ̅)|| ≤ ( 1) (|| ( ) ( ) ̅|| + || ||)+ (|||ℎ |||)+ (||| |||||| |||), ≥ 0, || 0( , ̅)|| ≤ ( 2)− (|| ( ) ( ) ̅|| + || ||)+ (|||ℎ |||)+ (||| |||||| |||), ≤ 0, with corresponding constants and functions .

Remark 5. Presented theorem gives us conditions of the input-to-state stability and chaoticity (see also [1],[11]). [ 1 1] = [ 22] .

… ([ ]) (17) (18) (19) (20)

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

3. Acknowledgements

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

4. References

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