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In this paper, we study the solvability of a distribution-valued heat equation with nonlocal initial condition. Under proper assumption on parameters we get the explicit solution of the distribution-valued heat equation. As an application, we further consider the stabilization problem of heat equation with partial-delay in internal control. By the parameterization design of feedback controller, we show if the integral kernel functions are determined by the solution of the distribution heat equation with nonlocal initial value problem, then the closed-loop system can be transformed into a system which is called the target system of the exponential stability under the bounded linear transformation. By selecting different distribution-valued kernel functions, we give the inverse transformation. Hence the closed-loop system is equivalent to the target system.
}, issn = {2617-8702}, doi = {https://doi.org/10.4208/jms.v52n2.19.02}, url = {http://global-sci.org/intro/article_detail/jms/13155.html} }In this paper, we study the solvability of a distribution-valued heat equation with nonlocal initial condition. Under proper assumption on parameters we get the explicit solution of the distribution-valued heat equation. As an application, we further consider the stabilization problem of heat equation with partial-delay in internal control. By the parameterization design of feedback controller, we show if the integral kernel functions are determined by the solution of the distribution heat equation with nonlocal initial value problem, then the closed-loop system can be transformed into a system which is called the target system of the exponential stability under the bounded linear transformation. By selecting different distribution-valued kernel functions, we give the inverse transformation. Hence the closed-loop system is equivalent to the target system.