Solvability of the Nonlocal Initial Value Problem and Application to Design of Controller for Heat-Equation with Delay (Journal of Mathematical Study, 2019)
In this paper, we study the solvability of a distribution-valued heat equationwith nonlocal initial condition. Under proper assumption on parameters we get the explicit solution of the distribution-valued heat equation. As an application, we furtherconsider the stabilization problem of heat equation with partial-delay in internal control. By the parameterization design of feedback controller, we show if the integralkernel functions are determined by the solution of the distribution heat equation withnonlocal initial value problem, then the closed-loop system can be transformed into asystem 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 thetarget system.
Keywords: Abstract heat equation; Solvability; nonlocal initial value condition; internal delayed control; integral-type feedback controller; exponential stability.