]>
(6.65) |
with appropriate initial conditions. Here denotes the displacement, is the temperature deviation from the reference temperature and , are positive constants depending on the material properties.
Substituting we can formally reduce the first two equations of (6.65) to the form
This suggests taking the Hilbert space
where denotes the standard Sobolev space of the first order, as the state space with the scalar product
In this space, the system equations can be rewritten in the abstract form
with
(6.66) |
where denotes the standard Sobolev space of the second order.
Proof. It is enough to prove that is compact. The form of can be determined by solving the system of equations
(6.67) |
with boundary conditions
(6.68) |
Here and , . The solution of (6.67), (6.68) is
(6.69) |
(6.70) |
(6.71) |
Furthermore, , and .
The operator
where , is of rank one, see [86, Theorem 6.1, p. 129], and therefore it is compact. The value of this operator at is the function
Define the operator
It is self–adjoint with the inverse
The operator has a system of eigenvectors, corresponding to eigenvalues , , which forms an orthonormal basis in ,
Since
the operator is by definition [86, p. 136] a HS–operator belonging to . The system
constitutes an orthonormal basis in . It is a system of eigenvectors of the extension of onto , which will be denoted also by ,
The operator ,
is a HS–operator and therefore it is compact. Indeed,
where denotes the constant function , . Hence, applying the Schwarz inequality we get
and is a HS–operator directly by the definition, see [86, p. 136]. Next, consider the composition of the bounded integral operator
with ,
By the result of [86, Theorem 6.4, p. 131] it is a HS–operator from and we have proved that all operators defined by the right–hand side of (6.69) are compact.
The next step is to prove that the injection operator from into , corresponding to the right–hand side of (6.70), is a HS–operator. Indeed,
and we recall once more the definition of a HS–operator given in [86, p. 136] to conclude that the injection operator is compact.
It remains to prove that the right–hand side of (6.71) is expressed also by compact operators. Observe that (6.71) can be written shortly as
We already know that is a HS–operator and therefore it suffices to prove that the composition operator is a HS–operator from . In virtue of [86, Theorem 6.4, p. 131] this requires proving that the first order differentiation operator from to is bounded. However, this easily follows from the definition, together with the fact that the norm of such an operator equals .
Now, all operators defining the right–hand sides of (6.69), (6.70), (6.71) are compact which completes the proof of compactness of . □
Lemma 6.5.2. The operator (6.66) generates a semigroup of contractions on .
Proof. For we have
Hence is dissipative. From the definition of an adjoint operator [86, p. 68] we find
and it is not difficult to see that
This implies that is also dissipative and thus, by [69, Corollary 4.4, p. 15], generates a –semigroup of contractions on . □
Proof. By the result of [63, Lemma 3.1, p. 497] we have
where
is the subspace on which acts isometrically. It follows from the proof of Lemma 6.5.2 that
It is an invariant subspace, on which the two first components of coincide with the unitary semigroup generated on by the skew–adjoint operator
Hence
where is a –periodic function to ensure
But satisfies the equation
and therefore, integrating both sides from to , we get
Since we have
This means that
Substituting one obtains , while for one gets . Hence
However this means that the function is –periodic for any , which is possible only if . This yields and thus
We have proved that
Since is a semigroup of contractions and is dense in , we can apply the results of [86, Problem 4.28, p. 81] to conclude that (6.72) holds. □
Lemma 6.5.3 has an important consequence when considering the lq problem (6.6) for the system (6.1) with given by (6.66) and with a compact operator . If is a compact operator and the solution of lq problem (6.6) exists then is compact too and, by the result of [81], is EXS. But if is EXS then is stabilizable, is detectable and the lq problem has a unique solution. Thus if is compact, the exponential stability of the semigroup is necessary and sufficient for solvability of the lq problem. In this context the result of Lin and Zheng [64] is extremely important. They have proved, using the Prüss–Huang–Weiss theorem [71], [50], [87], that is EXS. The proof is very technical.
Remark 6.5.1. A more fine structure of can be exhibited,
where is a normal discrete dissipative operator and is a skew–adjoint operator.
Since , the spectrum of , is equally spaced on we cannot apply any perturbation result to conclude that is similar to a normal operator, which is supposed to be true. If it would be the case then the proof of EXS of the semigroup were considerably simplified. For the time being we do not know whether there exists any simpler proof that is EXS.