On well-posedness of integro-differential equations in weighted \(L^ 2\)- spaces
The paper is devoted to the well-posedness of the initial value problem \[ {d \over dt} \left[ \int^ 0_{-r} g(s) x(t+s) ds \right] = \int^ 0_{-r} d \mu (s)x(t+s) + f(t), \quad t > 0, \tag{1} \] \[ x(s) = \varphi (s) \quad \text{for} \quad - r < s < 0, \tag{2} \] where \(0<r \leq + \infty\), \(g(\cdot) \in L^ 1 (-r,0)\) is nonnegative, nondecreasing and weakly singular at \(s = 0\), \(\mu (s) = \mu_ 0 (s) - \int^ s_ 0 a(m)dm\), with \(\mu_ 0 (\cdot)\) of bounded variation and \(a (\cdot)\) locally absolutely continuous on the interval \((-r,0)\). This work is motivated by the need to develop a framework for the analysis of numerical methods for designing control laws for aeroelastic systems. Using semigroup theory the authors prove that the weighted Hilbert space \(L^ 2_ g\) is a suitable well-posed state space model for such problems. The main result of the paper is Theorem 3.1. Suppose that \(\int^ 0_{-r} | d \mu_ 0 (s) |/g(s) < \infty\) holds. Then for any \(T>0\), \(\varphi \in L^ 2_ g\) and \(f \in L^ 2 (0,T)\), the problem (1)--(2) has a unique solution \(x(t)\), \(t \geq 0\), and there exists a constant \(K = K(T)\) such that \[ \bigl | x (t + \cdot) \bigr |^ 2_{L^ 2_ g} \leq K \biggl \{ | \varphi |^ 2_{L^ 2_ g} + \int^ t_ 0 \bigl | f(s) \bigr |^ 2 ds \biggr \}. \] Many results are also obtained for some special cases of the problem (1)--(2). Some numerical results based on this framework were given by the second author and \textit{J. Turi} [SIAM J. Numer. Anal. 28, 1698-1772 (1991; Zbl 0744.65103)].
scientific article; zbMATH DE number 716375