Control and observation problems in Banach spaces. Optimal control and maximum principle. Applications to ordinary differential equations in \(\mathbb{R}^n\) (Q2187850)

The author presents an abstract operator model which applies in particular to controllability and observability of systems of ordinary differential equations \[ y'(t) = A(t)y(t) + B(t)u(t) \quad (0 \le t \le T) \qquad y(0) = 0\, , \quad y(T) = e \, . \] If $S(t, s)$ is the solution operator of the homogeneous equation then \[ A_T u = \int_0^T S(T, t) B(t)u(t) dt \] is an operator from the control space $U_T$ into $\mathbb{R}^n$, and the controllability problem reduces to the operator equation $A_Tu = e.$ Approximate controllability is then equivalent to observability of the adjoint system $A^*_T e^* = u^*,$ and there are further equivalences for stronger versions of controllability and observability. Optimality, in particular the maximum principle can be also accommodated within this abstract scheme. The author uses various tools of functional analysis in Banach spaces in particular results from his previous paper [Dokl. Math. 96, No. 2, 477--479 (2017; Zbl 1382.49020); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 476, No. 4, 377--380 (2017)]. See also the author's paper [Dokl. Math. 99, No. 2, 152--155 (2019; Zbl 1420.49008); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 485, No. 2, 153--157 (2019)].