[Paper Review] Construction of Exact Control for a One-Dimensional Heat Equation with Delay

D. Ya. Khusainov, Michael Pokojovy+1 more|arXiv (Cornell University)|Jan 22, 2014

Stability and Controllability of Differential Equations9 references3 citations

TL;DR

This paper establishes exact controllability for a one-dimensional heat equation with delay in both lower and highest-order terms using an explicit representation of the control function. By transforming the equation via a phase shift and employing the delayed exponential function, the authors derive a Fourier series solution and construct a control that steers the system to a desired final state under sufficient Sobolev regularity of the data, ensuring classical regularity of both solution and control.

ABSTRACT

We prove an exact controllability result for a one-dimensional heat equation with delay in both lower and highest order terms and nonhomogeneous Dirichlet boundary conditions. Moreover, we give an explicit representation of the control function steering the system into a given final state. Under certain decay properties for corresponding Fourier coefficients which can be interpreted as a sufficiently high Sobolev regularity of the data, both control function and the solution are proved to be regular in the classical sense both with respect to time and space variables.

Motivation & Objective

• To establish exact controllability for a one-dimensional heat equation with discrete delay in both lower and highest-order terms.
• To provide an explicit representation of the control function that steers the system to a given final state.
• To ensure classical regularity of both the solution and control under sufficient Sobolev regularity of the initial, boundary, and forcing data.
• To extend analytical tools for distributed systems with delay by leveraging the delayed exponential function and Fourier series decomposition.

Proposed method

• Transform the original equation using a phase shift $ v(x,t) = e^{ u x} u(x,t) $ to eliminate first-order terms, reducing the problem to a form with delayed second-order terms.
• Introduce the delayed exponential function $ \exp_{\tau}(D,t) $ to handle the time-delayed dynamics in the solution representation.
• Represent the solution as a Fourier series in $ \sin(\pi n x / l) $, with coefficients derived from integrals involving the delayed exponential and initial/boundary data.
• Derive an integral equation for the control by matching the final state condition, leading to a solvable equation in the Fourier coefficients of the control.
• Solve the integral equation by assuming a control form involving $ e^{-L_n(T-t)} $, leading to an explicit formula for the control coefficients.
• Use the identity $ \int_{-\tau}^{T-\tau} \exp_{\tau}(D_n,t) dt = \frac{1}{D_n}(\exp_{\tau}(D_n,T) - 1) $ to invert the integral equation and obtain the control coefficients.

Experimental results

Research questions

• RQ1 Can exact controllability be achieved for a one-dimensional heat equation with delay in both lower and highest-order terms?
• RQ2 Is it possible to construct an explicit representation of the control function that steers the system to a desired final state?
• RQ3 Under what regularity conditions on the data does the control and solution remain classical in time and space?
• RQ4 How can the delayed exponential function be used to solve parabolic equations with distributed delays?
• RQ5 What is the structure of the solution and control in terms of Fourier series when delays are present in both spatial derivatives and the state?

Key findings

• The paper proves exact controllability for the one-dimensional heat equation with delay in both lower and highest-order terms under sufficient Sobolev regularity of the data.
• An explicit formula for the control function is derived in the form $ U(x,t) = \sum_{n=1}^\infty U_n(t) \sin(\pi n x / l) $, with $ U_n(t) = e^{-L_n(T-t)} \frac{R_n(T) D_n}{\exp_{\tau}(D_n,T) - 1} $.
• The solution and control are shown to be classical (smooth) in both time and space variables when the Fourier coefficients of the data decay sufficiently fast, corresponding to high Sobolev regularity.
• The control construction relies on transforming the equation to eliminate first-order terms and using the delayed exponential function to represent the solution of the delayed PDE.
• The key identity $ \int_{-\tau}^{T-\tau} \exp_{\tau}(D_n,t) dt = \frac{1}{D_n}(\exp_{\tau}(D_n,T) - 1) $ enables the inversion of the integral equation for the control.
• The final state condition leads to $ R_n(T) = \Psi_n - s_{1n}(T) - s_{2n}(T) - m_n(\mu_1,\mu_2)(T) $, which determines the required control amplitude for each Fourier mode.