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Linear differential equations with measures as coefficients and control theory. (English) Zbl 0813.34058

Prace Naukowe Uniwersytetu Śląskiego w Katowicach 1413. Katowice: Wydawnictwo Uniwersytetu Śląskiego (ISBN 83-226-0557-9/pbk). 134 p. (1994).
The book is a research monograph. Studied are differential equations of the form (1) \(\dot x(t) = A(t)x(t) + f(t)\), \(x(t_ 0) = x_ 0 \in \mathbb{R}^ n\) on an interval \((a,b)\), \(-\infty \leq a < b \leq \infty\), where \(A(t)\) is an \(n \times n\) matrix with measures (here Lebesgue-Stieltjes measure generated by a function of locally bounded variation) as entries and \(f(t)\) is some \(n\)-vector consisting of locally integrable functions or measures. The solution space of these equations is defined as discontinuous functions of locally bounded variation which are continuous from the right. In this set-up, fundamental properties are derived: Existence of a solution, fundamental matrix of the homogeneous equation, uniqueness of the solution under additional assumptions, Variation-of- Constants. For scalar systems, \(n=1\), of the form (1), monotonicity and oscillatority of the solution is studied. An analogon to the Bellman- Gronwall inequality is derived as well as some results on the inverse problem, on periodic – and even/odd systems (1), respectively their solutions, stability and stabilization by state feedback, controllability, attainable sets, minimum-energy and minimum-norm control, existence of optimal control, maximum principle, differential games.

MSC:

34G10 Linear differential equations in abstract spaces
34C25 Periodic solutions to ordinary differential equations
34D05 Asymptotic properties of solutions to ordinary differential equations
93B03 Attainable sets, reachability
93B05 Controllability
93D15 Stabilization of systems by feedback
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