Generalized differential equations in the space of regulated functions (boundary value problems and controllability). (English) Zbl 0744.34021

Tvrdý, Milan

Math. Bohem. 116, No. 3, 225-244 (1991).

Extensions of some known results for systems of the form \[ x(t)-x(0)- \int^ t_ 0[dA(s)]x(s)=f(t)-f(0),\qquad t\in[0,1], M_ x(0)+\int^ t_ 0 K(s)[dx(s)]=r \] are given when \(f\) is regulated and not necessarily of bounded variation (i.e. the limits \(f(t+)=\lim_{\tau\to t+}f(\tau)\), \(f(s-)=\lim_{\tau\to s-}f(\tau)\) exist for all \(t\in[0,1]\)). In particular, existence and uniqueness theorems are proved and an evolution operator is found. This is done by defining the operator \[ {\mathcal A}x={x(t)-x(0)-\int^ t_ 0[dA(s)]x(s)\choose Mx(0)+\int^ t_ 0K(s)[dx(s)]}. \] The dual operator \({\mathcal A}^*\) is determined and this leads to some controllability results.

Reviewer: S.P.Banks (Sheffield)

MSC:

34B10	Nonlocal and multipoint boundary value problems for ordinary differential equations
34H05	Control problems involving ordinary differential equations
26A45	Functions of bounded variation, generalizations

Keywords:

generalized differential equations; adjoint operators; existence and uniqueness theorems; controllability