Applied Mathematics & Information Sciences

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This paper contributes to a design of stabilizing compensators for the stabilizable systems in the class. A strongly continuous quasi semigroup approach is implemented as a generalization of a strongly continuous semigroup for autonomous systems. Stability of the non-autonomous linear control system is identified by a uniformly exponential stability of a strongly continuous quasi semigroup on the state space. The results showed that in the infinite-dimensional state space, if the closed-loop non-autonomous linear control system was stabilizable and detectable, there existed an infinite-dimensional stabilizing compensators for the system. The assigned controller is given by u = Fxˆ where xˆ is the Luenberger observer. In any non-autonomous Riesz-spectral system, there exists a finite- dimensional compensator for the system. The construction of the compensator is based on the separation of the unstable eigenvalues of the corresponding Riesz-spectral operator. The numbers of the unstable eigenvalues are defined to be an order of the compensator. An example of the non-autonomous heat equation is given to assert the theoretical results.

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