Some properties of the attainable set for the abstract control problem with application to controllability
For linear differential control systems described by the following evolution equation: \({dx\over dt} = Ax(t) + Bu(t)\), \(x(0) = x_0\), \(y(t) = Cx(t)\) the attainable set \[ K(t) := \{x \in X: x = \int^t_0 S(t - \tau) Bu (\tau) d\tau,\quad u(\cdot) \in L_2([0,t]), U)\} \] and the unobservable set \[ N(t) = \{x \in X : CS(\tau) x = 0,\quad \forall \tau \in [0,t]\} \] where \(A\) is a linear operator generating a strongly continuous semigroup \(\{S(t)\}_{t \geq 0}\) on a Banach space \(X\), \(B: U \to X\) and \(C : X \to Y\) are linear bounded operators and \(Y\), \(U\) are Banach spaces, are investigated. Conditions of independence of \(t\) for the closure \(\overline {K(t)}\) and \(N(t)\) have been established. Those conditions are useful for investigations of approximate controllability and observability of control systems under consideration.
scientific article; zbMATH DE number 761130
scientific article