Controllability and optimal controllability for operator equations of the first kind in (B)-spaces: Examples for ODE in \(\mathbb{R}^n\) (Q2313249)

The abstract control system in this paper consists of two Banach spaces $U$ and $E$ and a linear closed operator $A$ mapping a dense domain $D(A) \subseteq U$ into $E$. The control problem is that of finding solutions $u_e \in D(A)$ of the operator equation \[ Au_e = e \tag{1} \] for a given $e \in E$. Assuming solutions exist, the solution $\bar u_e$ is called optimal if \[ \|\bar u_e\|_U = \inf_{Au_e = e} \|u_e\|_U. \tag{2} \] The equation (1) is controllable if solutions exist for every right side $e$, that is, if $AD(A) = E$. If $\overline{AD(A)} = E$ the equation is approximately controllable. Controllability can be reduced to properties of the adjoint $A^*$; for instance, the system is approximately controllable if and only if $A^*u^* = 0 \ \Longrightarrow \ u^* = 0$, a property called observability. A stronger version of observability is equivalent to controllability. The author studies (1) and (2) using these results and other functional analysis tools. The results are applied to control problems described by ordinary differential systems \[ y'(t) = A(t)y(t) + B(t)u(t) \quad (0 \le t \le T), \qquad y(0) = 0, \quad y(T) = e. \]