Of Evolution Equations, Stability Problems

In this paper we study the exponential stabilization of nonlinear evolutionary sys-tems with linear feedback controls. First, we consider the nonlinear heat system, and with the assumption that s> d/2 we can obtain the exponential stabilization of the sys-tem. Secondly, under the same condition as heat system's we also get the same result for nonlinear Schrodinger system. Furthermore, if the Schrodinger system is defocus-ing and the degree of nonlinearity is not too high, then using the energy decreasing property of the system we can obtain some better results, such as globally exponen-tial stabilization. We restrict our case to the cubic defocusing system because of the smoothness of the nonlinearity. However, our result can also be extended to other sim-ilar systems provided that some restrictions on regularity need to be made depending on the smoothness of the nonlinearity.