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

Vasilii Zaitsev,Marina Zhuravleva

doi:10.3390/math8060992

Abstract

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.

Highlights

IntroductionIn 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
Consider a linear control system:ẋ (t) = A(t) x (t) + B(t)u(t), t ∈ R. (1)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
We studied the problem of arbitrary assignment of the upper Bohl exponent for continuous-time systems in an infinite-dimensional Hilbert space

Summary

Introduction

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. We studied the problem of arbitrary assignment of the upper Bohl exponent for continuous-time systems in an infinite-dimensional Hilbert space. 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. 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), κ( A + BU ) = μ. The corresponding definition in finite-dimensional spaces was given in [13] (see [48]) for the upper (and lower) central (and Bohl) exponents

Main Results

Example

Conclusions