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Observability of systems with time-lag. (English. Russian original) Zbl 0644.34016

Differ. Equations 23, No. 4, 393-400 (1987); translation from Differ. Uravn. 23, No. 4, 598-608 (1987).
The author considers the quasidifferential equation (1) \(q^ n_ px=0\) where \(q^ 0_ Px=p_{00}x\), \(q^ k_ Px=p_{kk}\frac{d}{dt}q_ P^{k-1}x+\sum^{k-1}_{j=0}p_{kj}q^ j_ Px\) \((k=1,...,n)\), x:(\(\alpha\),\(\beta)\to {\mathbb{R}}\), \(\alpha\),\(\beta\in {\mathbb{R}}\), \(P=(p_{ik})^ n_ 0\), \(p_{ik}=0\) if \(i<k\). Here x has absolutely continuous quasi-derivatives \(q^ k_ P\) for \(k=0,...,n-1\). The functions \(p_{ik}: (\alpha,\beta)\to {\mathbb{R}}\) \((i>k)\) are Lebesgue measurable and the functions \(1/p_{ii}\) \((i=0,...,n)\), \(p_{ik}/p_{ii}\) \((i=1,...,n\); \(k=0,...,i-1)\) are locally integrable. A point \(t_ 0\in (\alpha,\beta)\) is called a P-zero of x of multiplicity \(k\geq 1\) if \((q_ P^{i-1}x)(t)=0\) for \(i=1,...,k\) and \((q^ k_ Px)(t_ 0)\neq 0\). The equation (1) is called non-oscillating in the interval \(I\subset (\alpha,\beta)\) if the sum of the multiplicities of the P-zeros in I is at most \(n-1\). The equation (1) is transformed into a first order system \(\dot z=Az\). Multipoint boundary conditions of the form \(c'_{v_ ii}z(t)=\gamma_{v_ ii}\) \((v_ i=1,...,k_ i;i=1,...m)\) are considered. Under some additional conditions the author proves that this boundary value problem has a unique solution if and only if the equation (1) is non-oscillating. The general result is applied to second order equations. For simple twopoint boundary conditions concrete non-oscillating-conditions are discussed.
Reviewer: M.Möller

MSC:

34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
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