Perturbation method for first- and complete second-order differential equations (Q495740)

The authors consider in a Banach space \(X\) the first-order problem \[ u^{\prime}(t) - Au(t) = f(t)z, \;\;0\leq t\leq \tau; \quad u(0)=u_0, \] with the additional information \(\Phi[u(t)] = g(t), \;\;0\leq t\leq \tau\), where \(z\in X\), \(A:D(A)\subset X \rightarrow X\) is the generator of a \(C_0\)-semigroup of linear operators, \(u_0\in D(A)\), \(g\in C([0,\tau]; \, \mathbb{R})\), \(\Phi \in X^{\ast}\), and \((u,f)\) is the unknown to be determined. The method used in this paper to solve this inverse problem is to reduce it to a direct Cauchy problem for an evolution equation associated with a perturbation \(A+B\) which is also a generator. Both the hyperbolic and parabolic cases are solved. Moreover, a similar second-order problem is analyzed together with a related identification problem. Some applications are also discussed.