Abstract
Abstract: In this paper, we are concerned with a hybrid hyperbolic dynamic system formulated by partial differential equations with initial and boundary conditions. First, the system is transformed to an abstract evolution system in an appropriate Hilbert space, and spectral analysis and semigroup generation of the system operator is discussed. Subsequently, a variable structural control problem is proposed and investigated, and an equivalent control method is introduced and applied to the system. Finally, a significant result that the state of the system can be approximated by the ideal variable structural mode under control in any accuracy is derived and examined.
Highlights
A great attention has been paid to the dynamics and control of flexible beam
We are concerned with the following general hyperbolic dynamic system with static boundary condition in one space variable in normal form studied in [1]-[3]:
A variable structural control problem for a hybrid hyperbolic dynamic system dominated by partial differential equations subject to the boundary shear force feedback is investigated
Summary
A great attention has been paid to the dynamics and control of flexible beam (see [1]-[8]). Where (H1) K(x) = diag{λ1(x), λ2(x), · · · , λm(x), μ1(x), μ2(x), · · · , μk(x)} is a diagonal n × n, (n = m + k), matrix with real entries λj(x), μj(x) ∈ C1[0, 1], λj(x) > 0, μi(x) < 0, ∀ x ∈ [0, 1], i = 1, 2, · · · , k, j = 1, 2, · · · , m. (H2) C(x) = diag{c1(x), c2(x), · · · , cn(x)} is an n × n diagonal matrix with continuous entries in x ∈ [0, 1];. In order to investigate the variable structural control problem for the system, first, let’s transfer the system to an abstract Cauchy problem in an appropriate Hilbert space, discuss the spectral properties and semigroup generation https://rajpub.com/index.php/jam of the system operator
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