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.


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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