AN OPTIMAL CONTROL FOR A HYBRID HYPERBOLIC DYNAMIC SYSTEM

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. In this paper, we are concerned with a hybrid hyperbolic dynamic system formulated by partial differential equations with initial and boundary conditions. An optimal energy control of the system is investigated. First, the system is transformed to an abstract evolution system in an appropriate Hilbert space, and then semigroup generation of the system operator is discussed. Finally, an optimal energy control problem is proposed and it is shown that an optimal energy control can be obtained by a finite dimensional approximation.

In this paper, our goal is to investigate an optimal energy control of the system.First, we transfer the system to an abstract Cauchy problem in an appropriate Hilbert space, and then discuss the semigroup generation of the system operator.Finally, we propose an optimal energy control problem and show that the optimal energy control exists and it can be obtained by a finite dimensional approximation.

SEMIGROUP GENERATION OF THE SYSTEM
Consider the system (1.1) in the underlying Hilbert space H = L 2 (0, 1) 2 .Define the Then the system (2.1) can be written an an evolution equation in H : (2) Lemma 2.1 The operator A definde by (2.2) has compact resolvent and hence σ (A ) consists only isolated eigenvalues.
Proof.Given ( f , g, b) ∈ X, we solve Denote by M(x, y, λ ) the fundamental matrix of the system (4) dy On the other hand, we see from the boundary condition in (2.4) v(1) dy where 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 namely, define the norm on H as 3) It is easily verified that H * , the dual space of H , consisting of all elements (u * , v * , d * ) with where q denotes the conjugate number of p, which satisfies 1 p + 1 q = 1.For any (u, v, d) ∈ D(A), (u, v, d) = 0 and any (u * , v * , d * ) ∈ F((u, v, d)) ⊂ H , where F denotes the duality set.A direct calculation shows that We estimate I i separately.It is clear from the expression of I 3 that e i j 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| q which hold for any real a and b, we have with α i and β j denoting the obvious constants.Finally, 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 (2.6) and (2.7) and then 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 conclude from the standard argument in [6] that A generates a C 0 -semigroup on H

AN OPTIMAL ENERGY CONTROL
In this section, we will discuss an optimal control problem of the hyperbolic system (2.2): where both state space H and control space U are Hilbert spaces, the state function ) is a control of the system.
In this section, we shall discuss a specific optimal control, that is, the minimum energy control of the system (3.1).We know that the minimum energy control in an abstract space is, in general, the minimum norm control.So, from mathematics point of view, the existence and uniqueness of the optimal control are essential.If these are true, then how to obtain the optimal control is a significant problem.The main content of this paper is to solve these essential and significant issue.
From the theory of operator semigroup, we see that for every control element u(W (•), •) ∈ L 2 ([0, T ], U ), the system (3.1) has an unique mild solution let ϕ(•) be an arbitrary element in C([0, T ]; H ), and define the admissible control set of the system (3.1) as follows where ε is any positive number.
It can be seen from (3.3) that U ad is not empty and contains infinitely many elements related to ϕ and ε.The minimum energy control problem is actually to find the element u, satisfying where u 0 is said to be a minimum energy control element.Proof.Convexity.For any u 1 , u 2 ∈ U ad and a real number λ , 0 < λ < 1, it is easy to see from and hence Closedness.Suppose {u n } ⊂ U ad , and lim n→∞ u n − u * = 0.It can be shown that u * ∈ U ad .
In fact, from the definition of U ad we see that Thus, u * ∈ U ad , and U ad is a closed set.The proof is complete.Theorem 3.2 There exists an unique minimum energy control element in the admissible control set U ad for the system (3.1)Proof.Since L 2 ([0, T ], U ) is a Hilbert space, it is naturally a strict convex Banach Space.
From the preceding Lemma, we have seen that U ad is a closed convex set in L 2 ([0, T ], U ), it follows from [2] that there is an unique element u 0 ∈ U ad such that According to the definition (3.4), u 0 is just the desired minimum energy control element of the system (3.1).The proof is complete.
Finally, we shall show that the minimum energy control element can be approached.It is obvious that {u n } is a bounded sequence in L 2 ([0, T ]; U ), and so there is a subsequence {u n k } of {u n } such that {u n k } weakly converges to an element u in L 2 ([0, T ]; U ) (see [3]).
Since U ad is a closed convex set in L (3.10)

Lemma 3 . 1
The admissible control set U ad defined by (3.3) is a closed convex set in Hilbert space L 2 ([0, T ]; U ).

Theorem 3 . 3
Suppose that u 0 is the minimum energy control element of the system (1.1), then there exists a sequence {u n } of U ad such that {u n } converges strongly tou 0 in L 2 ([0, T ]; U ),namely, lim n→∞ u n − u 0 = 0 Proof.Let {u n } be a minimized sequence in the admissible control set U ad , then it follows that u n+1 ≤ u n , n = 1, 2, • • • (3.8) and lim n→∞ u n = in f { u : u ∈ U ad } (3.9) 2 ([0, T ]; U ) (see Lemma 3.1), we see from Mazur's Theorem that U ad is a weakly closed set in L 2 ([0, T ]; U ), thus ũ ∈ U ad .Combining (3.2) and employing the properties of limits of weakly convergent sequence on norm yieldin f { u : u ∈ U ad } ≤ ũ ≤ lim k→∞ u n k = lim n k →∞ u n k = lim n→∞ u n = inf{ u ; u ∈ U ad }.
semigroup in Hilbert space H , there is a constant M > 0 such that sup 0≤t≤T S(t) ≤ M. On the other hand, since W (s) is differentiable on [0, T ], it is continuous on [0, T ], and hence {W (s) : s