A Variable Structural Control for a Hybrid Hyperbolic Dynamic System

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.

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, then discuss the spectral properties and semigroup generation of the system operator.

Spectral Analysis and Semigroup Generation
We start this section with considering the system (1.1) in the underlying Hilbert space H = L 2 (0, 1) 2 . Define the (2.1) Then the system (2.1) can be written an an evolution equation in H: Proof. Given (f, g, b) ∈ X, we solve that is, Let's denote by M (x, y, λ) the fundamental matrix of the system d dx On the other hand, we see from the boundary condition in (2.1) that where It can be eventually seen from (2.5) that R(λ, A) is compact for any λ ∈ ρ(A).
Theorem 2.2. The operator A defined by (2.1) generates a C 0 -semigroup T (t) on H.
Proof. We need only to prove the assertion for the case C ≡ 0 because is a bounded operator by assumption (H2), and bounded perturbations do not affect C 0 -semigroup generations. For the sake of simplicity, we assume that H is real.
The idea is to define an equivalent norm on H by properly choosing some positive weighting functions It is easily verified that H * , the dual space of H, consisting of all elements where q denotes the conjugate number of p, which satisfies 1 p + 1 q = 1.

For any
We estimate I i separately. It is clear from the expression of I 3 that e ij v j (0), we see that Because λ i (0) > 0 and µ j (0) < 0 from (H1) , we can always find g j (0) > 0 and f i (0) > 0 such that holds, which implies that I 2 ≤ 0.
We now estimate I 4 by means of the inequalities (|a| + |b|) p ≤ 2 p (|a| p + |b| p ) and |a| 1 p |b| 1 q ≤ |a| p + |b| q which hold for any real a and b, we have with α i and β j denoting the obvious constants. Subsequently, it can be seen that If we choose f i (1) > 0, g j (1) > 0 such that for any 1 ≤ i ≤ N and N + 1 ≤ j ≤ n, then The estimations of I i above show that there exists a constant M such that Now we choose a weighting functions f i (x) and g i (x) such that they satisfy (2.10) and (2.11), and hence define a norm in H according to (2.3). Because A − M is dissipative and A has the properties stated in the Lemma 2.1, we can assert from [9] and [11] that A generates a C 0 -semigroup on H, and the Theorem 2.2 is established now.

A Variable Structural Control
Let's establish and discuss a structural control problem for the hybrid hyperbolic system (2.2) In the rest part of this paper, we are going to show that the actual sliding mode W (t) will approach uniformly to the ideal sliding modeW (t) under certain conditions. and I − P are commutative because A and P are commutative. We see that LetÃ denote the infinitesimal generator of T 2 (t). Since the limit on the left exists, we can assert that x ∈ D(Ã) and In the boundary layer T 1 (t) ≤ δ, let's introduce the equivalent control as follows Hence, the solution of (3.7) can be expressed as follows: (3.8) and therefore, the solution of (3.4) can be written as Subtracting (3.9) into (3.8) yields Since P A = AP , we see that P T 1 (t) = P T 1 (t). It should be emphasized that (I − P )P = 0 and T 2 (t) = (I − P )T 1 (t), and consequently, Thus, The proof of the theorem is complete.
We see from the Theorem 3.2 that the solution of the beam system can be approximated by ideal sliding mode in any accuracy.

Conclusion
In the present paper, 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. An evolution equation corresponding to the beam system is established in an appropriate Hilbert space. A spectral analysis and semigroup generation of the system operator for the system are studied. Finally, a variable structural control is proposed, and a significant result that the solution of the system can be approximated by the ideal variable structural model under the control is obtained.