# On Controllability for Heat Conduction with Delay

Michael Pokojovy*

We discuss controllability for abstract delay differential equations of the form $$\begin{split} x_{t}(t) &= A_{1} x(t) + A_{2} x(t - \tau) + B_{1} u(t) + B_{2} u(t - \tau) \text{ for } t \in [0, T], \\ x(0) &= u^{0}, \\ x(t) &= \varphi(t) \text{ for } t \in [-\tau, 0], \end{split} \notag$$ where $A_{1}$ is a linear operator with a maximal $L^{p}$-regularity on a Banach space $X$ and $A_{2}$ is no lower order perturbation of $A_{1}$. The theory will then be applied to a regularized delay heat equation.

Mathematics Subject Classification: 35K90

Keywords: partial differential equations; delay in higher order terms; controllability

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