## On well-posedness of integro-differential equations in weighted $$L^ 2$$- spaces.(English)Zbl 0813.34064

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 J. Turi [SIAM J. Numer. Anal. 28, 1698-1772 (1991; Zbl 0744.65103)].

### MSC:

 34K30 Functional-differential equations in abstract spaces 45J05 Integro-ordinary differential equations 34G10 Linear differential equations in abstract spaces 47D06 One-parameter semigroups and linear evolution equations 34K40 Neutral functional-differential equations

Zbl 0744.65103