| Abstract: |
| A functional-analytic reformulation of the heat equation with finite memory in a bounded domain, subjected to boundary control actions, brings about an integro-differential equation in a Hilbert space where the memory pertains to both the state and control variables; the resulting unbounded control operator occurs in the convolution term, besides the free dynamics generator. In the case of smooth kernels, an approach to the study of certain sought-after control-theoretic properties of the solutions -- such as, e.g., controllability -- utilizes MacCamy's trick along with the theory of Volterra integral equations of the second kind, and then operator semigroup theory. In this talk we will describe current work toward obtaining a representation formula for the solutions, based on a ``state space representation'' approach, inspired by the one introduced by C. Dafermos to deal with evolution equations with infinite memory and pursued since the 1970s on. The distinct tasks and technical challenges which need to be handled will be discussed, particularly but not exclusively with an eye to the study of the associated optimal control problem with quadratic functionals.
(The talk is based on ongoing joint work with Paolo Acquistapace, Univ. di Pisa (Ret.).) |
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