Thesis (Ph.D.) - Cairo University - Faculty of Science - Department of Mathematics

In this thesis, we continue the development of the theory of C0-semigroups of bounded linear operators from a Banach space X into itself on a semigroup time scale T. Also, we study the stabilizability of control dynamic equations of the form x{u2206}(t) = Ax(t) + Bu(t), t {u2208} T, x(x) = xx {u2208} D(A), and control Volterra integro-dynamic equations of the form x{u2206}(t) = Ax(t) + Z t 0 G(t, s)x(s){u2206}s + Bu(t), t {u2208} T, x(x) = xx {u2208} D(A), where G(t,s) is continuous in the variable t and rd-continuous in the variable s, A is the generator of a C0-semigroup and B {u2208} L(U, D(A)), the space of all bounded linear operators from a Banach space U ( the control space ) to the domain D(A) of A, by the feedback control u : T {u2192} U. Finally, we investigate many types of stability of abstract dynamic equations of the form x{u2206}(t) = F(t, x), x(x) = xx {u2208} X, t {u2208} Tx+ := [x, {u221E})T, where F : T{u00D7}X {u2192} X is rd-continuous in the {uFB01}rst argument with F(t,0) = 0, by using the Lyapunov{u2019}s second method. We construct a Lyapunov function to obtain new su{uFB03}cient conditions for stability of some of abstract dynamic equations on time scales.

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