Functional differential equations of retarded and neutral type: Analytic solutions and piecewise continuous controls

Abstract

t E [0, T] (rather than quasi-piecewise analyticity, as obtained in Banks and Jacobs [3] for the non-neutral case, i.e., when Aij = 0, j = l,..., 1). This is done under the assumption that both the coefficients, the initial function and the initial derivative are analytic. (The choice of the interval [0, T] rather than any finite interval [to, t,] is done merely for convenience and for simplicity in notation.) Analyticity of the solution might fail, in general, but we find a condition on the initial data, which we call compatibility, and which guarantees an unbroken analyticity of the solution. A key step in our consideration is the determination by the structure of the equation of a finite subset d of [0, T] which contains all points which might ruin the analyticity of the solutions, This observation proves useful throughout. ‘In particular it enables us to drop the assumption made by Banks and Jacobs that the lags r, r, ,..., rI are commensurate. It also implies that solutions of the equation