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

In this thesis, we use an appropriate Lyapunov function and derive su{uFB03}cient conditions for the non-linear dynamic control Equations on a time scale T of the form x{u2206}(t) = A(t)x(t) + B(t)u(t) + f(t, x(t), u(t)), t {u2208} T to be uniformly stabilizable, uniformly exponentially stabilizable or h{u2212} stabiliz- able. Here, A : T {u2192} Rn{u00D7}n is an n {u00D7} n matrix valued function and B : T {u2192} Rn{u00D7}m is an n {u00D7} m matrix valued function, m {u2264} n. We {uFB01}nd a suitable control u, which leads to make the feedback non-linear dynamic equation is uniformly stabilizable, uniformly exponentially stabilizable or h{u2212} stabilizable. We also investigate su{uFB03}cient conditions for these types of stabilizability as well as |-uniform stabilizability when A, B : T {u2192} L(H), where H is a Hilbert space. The results of this thesis were prepared in the following articles [1] Alaa E. Hamza, Nada A. Laabi, Uniform Exponential Stabilizability of Dynamic Control Equations on Time Scales, Int. J. Contemp. Math. Sciences, Vol. 8, 2013, no. 10, 469 - 480. [2] Alaa E. Hamza, M. H. Abu-Risha and Nada A. Laabi, Lyapunov Stabilizability for Nonlinear Dynamic Control Systems on Time Scales, Applied Mathematical Sciences, Vol. 7, 2013, no. 71, 3497 - 3510.

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