Hyperbolic systems with finite time stabilization
Novosibirsk State University, Sobolev Institute of Mathematics, 630090, Novosibirsk, prosp. Akad. of Koptyug, 4
We consider initial-boundary value problems in the semistrip for first order linear hyperbolic decoupled autonomous systems
where is the unknown function and
Here are diagonal matrices the first of elements of the matrix are positive and another are negative; The constant matrix is the matrix of the reflection boundary conditions for (1), where
It is known  that the problem (1) is well-posed in and it is superstable by some , i.е. all solutions to this problem are reduced to zero faster than exponents in any degree . We investigate the spectrum and the resolvent of the operator and prove the following theorems.
Тheorem 1. The problem (1) is superstable (1) the problem has finite time stabilization to zero of all solutions (this time does not depend on ).
Тheorem 2. The problem (1) is superstable (1) the spectrum of is empty.
Тheorem 3. The problem (1) is superstable for any matrices the matrix of the absolute values of elements of is nilpotent.
The paper  is devoted to perturbed to (1) hyperbolic problems of the type
where is the matrix, is the known function. We prove that under small perturbations the problem (2) is exponentially stable in and it is smoothing from to . We show the use of these results to analyze the stability of mathematical models of chemical kinetics and control theory.
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