PROCESS-CONTROLLABILITY OF SEMILINEAR EVOLUTION EQUATIONS AND APPLICATIONS

This article focuses on the controllability for a class of semilinear evolution equations and applications to some specific differential equations. Without assuming compactness or equicontinuity of the related semigroups, the existence of mild solutions of semilinear equations in Hilbert space is demonstrated via the noncompact measure tool and the fixed point trick. In order to study the controllability of semilinear equations, new concepts are proposed, namely, exact controllability along any A-bounded Lipschitz continuous curve, and approximate controllability along any continuous curve. Furthermore, two new approximation methods, ``bisection method"" and ``equisection method,"" are introduced. The exact controllability along any A-bounded Lipschitz continuous curve and the approximate controllability along any continuous curve in the sense of the graph norm of semilinear evolution equations are obtained under the asymptotic condition on the norm of the controllability Gramian inverse operator near the zero point. In fact, our conclusions show that this is not only a result control but also a process control. Finally, the results presented in this paper are employed in resistance, inductance, voltage source type electrical circuit systems, one-dimensional nonhomogeneous transport systems, as well as differential equations with delay which have important effects on economic systems.