Strong and Weak Stabilizability: Lyapunov Type Approaches
CALIFORNIA UNIV LOS ANGELES SCHOOL OF ENGINEERING AND APPLIED SCIENCE
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This theses addresses the problem of determining stabilizing controls for distributed parameter systems. The focus is on controls which provide strong or weak stabilization to the system. One approach to the stabilization of finite dimensional systems and exponential stabilization of infinite dimensional systems has been the use of Lyapunov type functionals. This is one technique which is developed and extended here, to provide new conditions for strong or weak stability. A new functional is presented, and if this functional is strictly positive, a certain semigroup will be strongly stable. This functional suggests an inequality relation which, if satisfied guarantees the weak stability of uniformly bounded semigroups. The relationship between contraction semigroups is examined on a Hilbert space and shift semigroups on a related Hilbert space. In particular, strongly stable semigroups are found to be equivalent in a certain sense to a backward shift semigroup. This provides an alternative view point for strong stability. Since stable semigroups are uniformly bounded and since this condition is important in verifying stability we examine this phenomena. Some new observations are presented to illustrate conditions under which perturbations of uniformly bounded semigroups remain uniformly bounded.
- Operations Research