On Assignment of the Upper Bohl Exponent for Linear Time-Invariant Control Systems in a Hilbert Space by State Feedback

: We consider a linear continuous-time control system with time-invariant linear bounded operator coefﬁcients in a Hilbert space. The controller in the system has the form of linear state feedback with a time-varying linear bounded gain operator function. We study the problem of arbitrary assignment for the upper Bohl exponent by state feedback control. We prove that if the open-loop system is exactly controllable then one can shift the upper Bohl exponent of the closed-loop system by any pregiven number with respect to the upper Bohl exponent of the free system. This implies arbitrary assignability of the upper Bohl exponent by linear state feedback. Finally, an illustrative example is presented. arbitrary assignment of the upper Bohl exponent of linear feedback with a time-varying linear bounded gain operator function. We exact controllability of the open-loop system sufﬁcient for arbitrary assignability of the upper Bohl exponent of the closed-loop system. We plan to extend these results to systems bounded operator A but generating a C 0 -continuous semigroup. We plan to prove similar for systems with dynamic output feedback. Further development of these results their extension to systems with periodic coefﬁcients and with arbitrary time-varying non-periodic


Introduction
Consider a linear control system: Here x ∈ X and u ∈ U are the state and control vectors respectively, X and U are some finite-dimensional or infinite-dimensional Banach spaces. Suppose that the controller in system (1) has the form of linear static state feedback u(t) = U(t)x(t). The closed-loop system has the form: Now we consider the elements of the gain operator U(t) as controlling parameters. The problems of control over the asymptotic behavior of solutions to systems (2) by means of elements of gain operator U(t) (in particular, the problem of stabilization for system (2)) belong to the classical problems of control theory. First results relate to stationary systems in finite-dimensional spaces. It was proved for complex [1] and real [2] finite-dimensional (X = R n , U = R m ) time-invariant (A(t) ≡ A, B(t) ≡ B) systems that the condition of complete controllability of system (1) is necessary and sufficient for the arbitrary assignment of the eigenvalue spectrum λ 1 , . . . , λ n of the closed-loop system (2) by means of time-invariant (U(t) ≡ U) feedback. This implies, in particular, stabilizability of (2) by means of U(t) ≡ U. First results for time-varying periodic systems in finite-dimensional spaces were obtained in [3]: It was proved that the complete controllability of system (1) is necessary and sufficient for the arbitrary assignment of the characteristic multipliers ρ 1 , . . . , ρ n of the closed-loop system (2) by means of periodic feedback. For time-varying non-periodic systems in finite-dimensional spaces, first results on stabilization were obtained in [4][5][6]. A transformation reducing system (2) to a canonical (block)-Frobenius form was used, which allows one to solve the eigenvalue assignment problem. However, rather restrictive conditions on the smoothness and boundedness of the coefficients of system (1) are required there. These conditions were weakened in [7] to the condition of uniform complete controllability in the sense of Kalman [8], and, on the basis of this property, sufficient conditions for exponential stabilization of system (2) were obtained. The proof of exponential stability is carried out using the second Lyapunov method (the Lyapunov function method).
In the framework of the first Lyapunov method of studying systems of differential equations in finite-dimensional spaces, a natural generalization of the concept of eigenvalue spectrum for non-stationary systems is the spectrum of Lyapunov exponents (see [9][10][11]). In addition to Lyapunov exponents, other Lyapunov invariants are known (that is, characteristics that do not change under the Lyapunov transformation, see [12]), which characterize the asymptotic behavior of solutions to a linear system of differential equations, for example, the Bohl exponents, the central (Vinograd) exponents, the exponential (Izobov) exponents, etc. In a series of studies [13][14][15][16][17], the results on arbitrary assignability of Lyapunov exponents and other Lyapunov invariants for system (2) in finite-dimensional spaces were proved, based on the property of uniform complete controllability in the sense of Kalman. In recent studies [18][19][20][21][22][23], these results have been partially extended to discrete-time systems. In finite-dimensional spaces, the Lyapunov exponents, the Bohl exponents, and other Lyapunov invariants were studied, for example in [24][25][26] for continuous-time systems and in [27][28][29][30][31][32][33] for discrete-time systems.
In this paper, we studied the problem of arbitrary assignment of the upper Bohl exponent for continuous-time systems in an infinite-dimensional Hilbert space. The brief outline of the paper is as follows. In Section 2, some notations, definitions, and preliminary results are given and the concepts used throughout the paper are defined, as well as some basic theories, methods, and techniques. In Section 3, we analyze the problem of arbitrary assignment of the upper Bohl exponent by means of linear state feedback with a time-varying linear bounded gain operator function for linear time-invariant control system in a Hilbert space with bounded operator coefficients and prove that the property of exact controllability of the open-loop system is sufficient for arbitrary assignability of the upper Bohl exponent of the closed-loop system. Section 4 provides an illustrative example that emphasizes the theory. In Section 5, we revise the results obtained in the paper and also showcase future developments of the theory.

Notations, Definitions, and Preliminary Results
Let X be a Banach space, X * be dual to X. By L(X 1 , X 2 ) we denote a Banach space of linear bounded operators A : X 1 → X 2 . If A ∈ L(X 1 , X 2 ), then A * ∈ L(X * 2 , X * 1 ) is its adjoint operator. By I : X → X denote the identity operator. Consider a linear system of differential equations: We suppose that the following conditions hold: (a) A(t) ∈ L(X, X) for any t ∈ R; By a solution of system (3) we will understand, by definition, a solution of the integral equation: where Due to conditions imposed on A(·), a solution (4) of (3) is a continuous, piecewise continuously differentiable function and satisfies (3) almost everywhere ( [45], Ch. III, Sect. 1.1, 1.2).
Proof. By using Lemma 1, we obtain: Let us consider another linear system of differential equations: Suppose that the operator function C(t) also satisfies conditions (a), (b), (c), i.e., C(t) ∈ L(X, X) ∀t ∈ R , C(·) is piecewise continuous, and sup t∈R C(t) = c < +∞. By Θ(t, τ) denote the evolution operator of system (11). Because of conditions imposed on C(·), we have the inequality: where L(t) is a bounded linear operator function with a bounded inverse: The following criterion holds (see [45], Ch. IV, Sect. 2, Lemma 2.1, (a)). (3) and (11) are kinematically similar on R if and only if there exists an operator function R t → L(t) ∈ L(X, X) satisfying (13) and such that the evolution operators of the systems are connected by the relation:
Let us state sufficient conditions for kinematical similarity of systems (3) and (11) on R analogous to the corresponding conditions in a finite-dimensional space (see, e.g., [46]).

Lemma 5.
Suppose that the operator functions A(t) and C(t) satisfy conditions (a), (b), and (c), and there exists a sequence Then systems (3) and (11) are kinematically similar on R.
Proof. By using the group property (B) of evolution operators, we obtain for all j > i: By (C), (15) holds for any i, j ∈ Z. Let us construct the operator function: By (15), we have L(t i ) = I, i ∈ Z. Next, by (16), we have: Hence, (14) is fulfilled. Let us prove that (13) is satisfied. Let t ∈ R be an arbitrary number. Then, since We have: Then, taking (6) and (12) into account, we obtain: Hence, (13) holds. By Lemma 3, the lemma is proved.
Suppose that the controller in system (17) has the form of the linear state feedback: where U(t) ∈ L(X, U) ∀t ∈ R, U(·) is piecewise continuous, and sup t∈R U(t) < +∞. We say that the gain operator function U(·) satisfying these conditions is admissible. The closed-loop system has the form: By Φ U (t, τ) we denote the evolution operator of system (19).
Definition 4. We say that system (17) admits a λ-transformation if there exists a constant σ > 0 such that, for any λ ∈ R, there exists an admissible gain operator function U(·) ensuring that the evolution operator Φ U (t, τ) of system (19) satisfies the relation: for all k ∈ Z.
This definition was given in [13] for systems (17) in finite-dimensional spaces (see also [48]). It is related to the definition of a λ-transformation of system (3). (20) that, for the evolution operator Φ U (t, s) of system (19), the relation

Remark 1. It follows from
Theorem 1. Suppose that system (17) admits a λ-transformation. Then, for any λ ∈ R, there exists an admissible gain operator function U(·) such that the closed-loop system (19) and system (7) are kinematically similar on R.

Definition 5.
We say that the upper Bohl exponent of system (17) is arbitrarily assignable by linear state feedback (18) if for any µ ∈ R there exists an admissible gain operator function U(·) such that, for the closed-loop system (19), The corresponding definition in finite-dimensional spaces was given in [13] (see also [48]) for the upper (and lower) central (and Bohl) exponents.

Main Results
Consider a time-invariant control system (17): Here x ∈ X, u ∈ U; X and U are separable Hilbert spaces; A ∈ L(X, X), B ∈ L(U, X); a := A , b := B . For Hilbert spaces H 1 , H 2 , we suppose that, if F ∈ L(H 1 , H 2 ), then F * ∈ L(H 2 , H 1 ), i.e., we identify H * i with H i . By ·, · denote the scalar product (in the corresponding space). If F * = F ∈ L(X, X), then the inequality F ≥ αI means, by definition, that Fx, x ≥ α x 2 for all x ∈ X.

Example
Let X = U = 2 where 2 is the is the space of all sequences x = (x 1 , x 2 , . . . , x n , . . .) with the . The space 2 is a separable Hilbert space ( [51], § 56). Consider a linear control system: where Considering elements of 2 as column-vectors with an infinite number of coordinates, one can identify the operators A and B with the following matrices with an infinite number of rows and columns: We will use the following denotations for the matrices of the form (59): It follows that system (55), (56), (57) is exactly controllable on [0, ϑ] for any ϑ > 0. Let us take ϑ = π. Let us show that the upper Bohl exponent of system (55), (56), (57) is arbitrarily assignable by linear state feedback (18). Consider the system:ẏ (t) = Fy(t), t ∈ R, y ∈ R 2 .

Remark 2.
The advantage of the developed method is that it allows us to establish the exact asymptotics (i.e., exact equality κ(A + BU) = µ) for the closed-loop system, in contrast to, e.g., [35], from which one can only obtain the inequality Λ(A + BU) ≤ κ for the upper Lyapunov exponent Λ of the closed-loop system. The problem of exact assignment of the upper Bohl exponent for a system in infinite-dimensional space in the presented formulation has not been previously investigated. Moreover, the developed method allows us to assign exact values for other asymptotic invariants of the closed-loop system (central exponents, exponential exponents etc.). A disadvantage is that the analytical expressions for the controller (and for solutions of the closed-loop system) can be complicated, in contrast to the stabilization problem [35]. This method can be applied to any system with the property of exact controllability. The choice of matrices in the example in a rather simple form was made for illustrative purposes because in this case the analytical expressions for the controller and for the solutions of the closed-loop system is not very complicated.

Conclusions
For a linear time-invariant control system in a Hilbert space with bounded operator coefficients, we examined the problem of arbitrary assignment of the upper Bohl exponent by means of linear state feedback with a time-varying linear bounded gain operator function. We have proved that the property of exact controllability of the open-loop system is sufficient for arbitrary assignability of the upper Bohl exponent of the closed-loop system. We plan to extend these results to systems without necessarily bounded operator A but generating a C 0 -continuous semigroup. We plan to prove similar results for systems with dynamic output feedback. Further development of these results may be their extension to systems with periodic coefficients and with arbitrary time-varying non-periodic coefficients, to systems in general Banach spaces, or to systems with discrete time. We expect to apply the results to specific systems, for example, to systems with delays, considering them as abstract systems of differential equations in an infinite-dimensional space.

Conflicts of Interest:
The authors declare no conflict of interest.