Preliminaries

This chapter serves as a simple introduction on the necessary mathematics basis for later chapters. It first introduces some basic results in functional analysis. This is the minimal knowledge in understanding the system control theory of partial differential equations, where the state spaces are infinite dimensional, contrast to the finite-dimensional systems where the state space is the Euclidean space R_n or C_n. This chapter introduces briefly some basic facts on C₀ -semigroup theory without proofs. To study systems described by the partial differential equations, the theory of the Sobolev spaces is also necessary in the sense that the solution to a partial differential equation is usually the weak solution. This is in sharp contrast to finite-dimensional systems, where the derivative is always the classical derivative. This chapter only lists some very basic results of the Sobolev space for the convenience of citations in later chapters.