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On an inverse problem for a second order differential equation in Banach space. (Russian) Zbl 0692.34064

This paper is on problems that can be thought of as control problems or as boundary value problems for two types of differential equations in Banach spaces. One of the equations in question is the second order linear abstract hyperbolic equation \((1)\quad u''(t)=Au(t)+\Phi (t)p+F(t),\) where A is the infinitesimal generator of a cosine function in a Banach space H, the function \(\Phi\) (t) (resp. F(t)) is real-valued (resp. H-valued) and continuous in \(0\leq t\leq T\) and \(p\in H\). As is well known, (1) has a unique (mild) solution that satisfies the initial conditions \(u(0)=u_ 0\), \(u'(0)=u_ 1\), where \(u_ 0,u_ 1\in H\). The author considers the following problem: given \(u_ 2\in H\), determine p in such a way that the solution u(t) satisfies the “final condition” (2) \(u(T)=u_ 2\). The other equation is the abstract elliptic equation \((3)\quad u''(t)=Au(t)+\Phi (t)p,\) where A is strongly positive (the resolvent set contains (-\(\infty,0)\) and \(\| (A+\lambda I)^{-1}\| \leq M/\lambda\) for \(\lambda <0)\). Under suitable conditions on \(\Phi\) (t), the equation (3) has a unique solution u(t) satisfying \(u(0)=u_ 0\), \(u(T)=u_ 2\), where \(u_ 0,u_ 2\in H\). The problem this time is to determine p in such a way that the solution u(t) satisfies in addition \(u'(0)=u_ 1\). For each problem, the author gives conditions that guarantee a solution. The results are applied to the Poisson equations of elasticity theory.
Reviewer: H.O.Fattorini

MSC:

34G10 Linear differential equations in abstract spaces
34A55 Inverse problems involving ordinary differential equations
35Q99 Partial differential equations of mathematical physics and other areas of application
93B05 Controllability
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