Volume 52, Issue 2
Solvability of the Nonlocal Initial Value Problem and Application to Design of Controller for Heat-Equation with Delay

J. Math. Study, 52 (2019), pp. 127-159.

Published online: 2019-05

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• Abstract

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.

93D15, 93C20, 35B35

2016210036@tju.edu.cn (Xiao-Pei Liu)

gqxu@tju.edu.cn (Gen-Qi Xu)

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@Article{JMS-52-127, author = {Liu , Xiao-Pei and Xu , Gen-Qi}, title = {Solvability of the Nonlocal Initial Value Problem and Application to Design of Controller for Heat-Equation with Delay}, journal = {Journal of Mathematical Study}, year = {2019}, volume = {52}, number = {2}, pages = {127--159}, abstract = {

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} }
TY - JOUR T1 - Solvability of the Nonlocal Initial Value Problem and Application to Design of Controller for Heat-Equation with Delay AU - Liu , Xiao-Pei AU - Xu , Gen-Qi JO - Journal of Mathematical Study VL - 2 SP - 127 EP - 159 PY - 2019 DA - 2019/05 SN - 52 DO - http://doi.org/10.4208/jms.v52n2.19.02 UR - https://global-sci.org/intro/article_detail/jms/13155.html KW - Abstract heat equation, Solvability, nonlocal initial value condition, internal delayed control, integral-type feedback controller, exponential stability. AB -

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.

Xiao-Pei Liu & Gen-Qi Xu. (2019). Solvability of the Nonlocal Initial Value Problem and Application to Design of Controller for Heat-Equation with Delay. Journal of Mathematical Study. 52 (2). 127-159. doi:10.4208/jms.v52n2.19.02
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