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

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.

### 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
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