Algebraic-delay differential systems: $C^0$-extendable submanifolds and linearization

Abstract:

Consider the abstract algebraic-delay differential system, \begin{eqnarray*} x’(t) &=& Ax(t)+F(x(t),a(t)), \\ a(t) &=& H(x_t,a_t) . \end{eqnarray*} Here $A$ is a linear operator on $D(A)\subseteq X$ satisfying the Hille-Yosida conditions, $x(t)\in \overline {D(A)}\subseteq X$, $a(t)\in {\mathbf {R}}^n$, and $X$ is a real Banach space. Let $C_0\subseteq \overline {D(A)}$ be closed and convex, and $K\subseteq \mathbf {R}^n$ be a compact set contained in the ball of radius $h>0$ centered at $0$. Under suitable Lipschitz conditions on the nonlinearities $F$ and $H$ and a subtangential condition, the system generates a continuous semiflow on a subset of the space of continuous functions $C([-h,0],C_0\times \mathbf {R}^n)$, which is induced by the algebraic constraint. The object of this paper is to find conditions under which this semiflow is also differentiable with respect to initial data. In the motivating example coming from modelling the dynamics of an age structured population, the nonlinearities $F$ and $H$ are not Fréchet differentiable on the sets $C_0\times K$ and $C([-h,0],C_0\times K)$, respectively. The main challenge of obtaining the differentiability of the semiflow is to determine the right type of differentiability and the right phase space. We develop a novel approach to address this problem which also shows how the spaces on which the derivatives of solution operators act reflect the model structure.