The Population Growth Rate Of The Control System Controllability Of Second Order Linear Parabolic System Observability Inequality The Existence Problem

There are two parts in this paper. In first part, we study the controllability of the species system with control as the increasing rate. Based on the dual method, by Carleman estimation and Kakutani fixed-point theorem, we prove the balanced distribution, approximate controllability of the system and give the result that the system can not be null controllable when the initial value satisfies some conditions. In the second part, we discuss the existence of observability inequality of the second-order parabolic system. By studying the controllability of the P-Laplace equation, we prove that the observability inequality fails when the smoothness of the coefficient of the second-order term is weak.