### On optimal control of systems with interface side conditions

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Boundary value problems for generalized linear differential equations and the corresponding controllability problems are dealt with. The adjoint problems are introduced in such a way that the usual duality theorems are valid. As a special case the interface boundary value problems are included. In contrast to the earlier papers by the author the right-hand side of the generalized differential equations as well as the solutions of this equation can be in general regulated functions (not necessarily...

The paper deals with the linear differential equation (0.1) ${\left(p{u}^{\text{'}}\right)}^{\text{'}}+{q}^{\text{'}}u={f}^{\text{'}\text{'}}$ with distributional coefficients and solutions from the space of regulated functions. Our aim is to get the basic existence and uniqueness results for the equation (0.1) and to generalize the known results due to F. V. Atkinson [At], J. Ligeza [Li1]-[Li3], R. Pfaff ([Pf1], [Pf2]), A. B. Mingarelli [Mi] as well as the results from the paper [Pe-Tv] concerning the equation (0.1).

n this paper we investigate systems of linear integral equations in the space ${\mathbb{G}}_{L}^{n}$ of $n$-vector valued functions which are regulated on the closed interval $[0,1]$ (i.e. such that can have only discontinuities of the first kind in $[0,1]$) and left-continuous in the corresponding open interval $(0,1).$ In particular, we are interested in systems of the form x(t) - A(t)x(0) - 01B(t,s)[d x(s)] = f(t), where $f\in {\mathbb{G}}_{L}^{n}$, the columns of the $n\times n$-matrix valued function $A$ belong to ${\mathbb{G}}_{L}^{n}$, the entries of $B(t,.)$ have a bounded variation on $[0,1]$ for any...

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