Linear Control Systems in Banach Spaces

Throughout the chapter, we consider the linear autonomous control system \(\displaystyle {} \begin {aligned}{} \dot {y}(t)&=Ay(t)+Bu(t)\\ y(0)&=y_0 \end {aligned} \) where the state \(y(t)\) belongs to a Banach space X, \(y_0\in X\) , the control \(u(t)\) belongs to a Banach space U, \(A:D(A)\rightarrow X\) is the generator of a \(C_0\) semigroup \((S(t))_{t\geqslant 0}\in \mathcal {G}(M,\omega )\) on X, and \(B\in L(U,X_{-1})\) . The space \(X_{-1}\) has been defined in the previous chapter.