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Degenerate first order identification problems in Banach spaces. (English) Zbl 1115.34012

Favini, Angelo (ed.) et al., Differential equations. Inverse and direct problems. Papers of the meeting, Cortona, Italy, June 21–25, 2004. Boca Raton, FL: CRC Press (ISBN 1-58488-604-8/hbk). Lecture Notes in Pure and Applied Mathematics 251, 1-15 (2006).
Let \(X\) be a Banach space, \(\tau>0\) be fixed and \(u_0,z\in D(A)\), where \(A\) is the generator of an analytic semigroup in \(X\); let \(\Phi\in X^*\) and \(g\in C^1([0,\tau],\mathbb R)\). The first result of the paper gives conditions under which the problem \[ u'(t)+Au(t)=f(t)z\;(0\leq t\leq\tau), \]
\[ u(0)=u_0, \;\Phi[u(t)]=g(t)\;(0\leq t\leq\tau) \] has a unique solution \((u,f)\), \(u\) and \(f\) belonging to certain Hölder classes of functions. Then, the following, possibly degenerate problem is considered. Let \(X\) be a reflexive space, \(L\), \(M\) be two closed linear operators in \(X\) with \(D(L)\subset D(M)\) and \(L\) being invertible, \(\Phi\in X^*\) and \(g\in C^{1+\theta}([0,\tau],\mathbb R)\), for some \(\theta\in(0,1)\). Conditions are given assuring that there is a unique solution \((u,f)\) of the problem \[ \frac d{dt}((Mu)(t))+Lu(t)=f(t)z\;(0\leq t\leq\tau), \]
\[ (Mu)(0)=Mu_0,\;\Phi[Mu(t)]=g(t)\;(0\leq t\leq\tau). \] Finally, these abstract results are applied to some concrete identification problems for partial differential equations.
For the entire collection see [Zbl 1098.34002].

MSC:

34A55 Inverse problems involving ordinary differential equations
34G10 Linear differential equations in abstract spaces
47D06 One-parameter semigroups and linear evolution equations
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