On the maximal parameter range of global stability for a nonlocal thermostat model

Abstract

The global asymptotic stability of the unique steady state of a nonlinear scalar parabolic equation with a nonlocal boundary condition is studied. The equation describes the evolution of the temperature profile that is subject to a feedback control loop. It can be viewed as a model of a rudimentary thermostat, where a parameter controls the intensity of the heat flow in response to the magnitude of the deviation from the reference temperature at a boundary point. The system is known to undergo a Hopf bifurcation when the parameter exceeds a critical value. Results on the characterization of the maximal parameter range where the reference steady state is globally asymptotically stable are obtained by analyzing a closely related nonlinear Volterra integral equation. Its kernel is derived from the trace of a fundamental solution of a linear heat equation. A version of the Popov criterion is adapted and applied to the Volterra integral equation to obtain a sufficient condition for the asymptotic decay of its solutions.

Properties of the kernel of the Volterra integral equation

In order to avoid distracting the reader from the flow of ideas in the proof of our main results, we collect a number of elementary or well-known properties of the kernel of the Volterra integral equation (3.1) in this appendix.

Remarks A.1

(a) As already exploited in the proof of Proposition (3.2), a full spectral resolution of the operator $A_N$ can be computed in order to obtain a series representation for the (kernel of the) semigroup generated by $-A_N$, which reads

$$\begin{aligned} N(t,x):=e^{-tA_N}\delta _0=\sum _{k=0}^\infty c_k\cos (kx)e^{-tk^2} \end{aligned}$$

(A.1)

and is the fundamental solution for the parabolic homogeneous Neumann problem on the interval $[0,\pi ]$, where $c_0=\frac{1}{\pi }$ and $c_k=\frac{2}{\pi }$ for $k\ge 1$. Consequently, the integral kernel a satisfies $a=-N(t,\pi )$, i.e. it holds that

$$\begin{aligned} a(t)=-\frac{1}{\pi }+\frac{2}{\pi }\sum _{k=1}^\infty (-1)^{k+1}e^{-tk^2},\, t>0. \end{aligned}$$

(A.2)

Besides the convolution kernel a, the forcing term f can also be expressed in terms of the basis of eigenfunctions to give

$$\begin{aligned} f(t)=\sum _{k=0}^\infty \langle u_0,\varphi _k \rangle e^{-tk^2}\varphi _k(\pi )= {\bar{u}}_0+\sqrt{\frac{2}{\pi }}\sum _{k=1}^\infty (-1)^k \hat{u}_{0k}e^{-tk^2}, \end{aligned}$$

(A.3)

for ${\hat{u}}_{0k}=\langle u_0,\varphi _k \rangle$ and $k\in {\mathbb {N}}$.

(b) For $u_0\in {\text {H}}^1$ the forcing function is in ${\text {BUC}}^\infty \bigl ( (0,\infty )\bigr )$ and its n-th derivative satisfies

$$\begin{aligned} f^{(n)}\in {\text {L}}^p\bigl ((0,\infty )\bigr )\text { for }p\in [1,\infty ] \;,\;n\ge 1\;. \end{aligned}$$

This follows from (A.3), i.e. from the exponential convergence of $e^{-tA_N}u_0$ to the constant function ${\bar{u}}_0$ as $t \rightarrow \infty \,$.

(c) The fundamental solution (A.1) can also be obtained from the heat kernel on the whole real line:

$$\begin{aligned} H(t,x):=\frac{1}{\sqrt{4\pi t}}e^{-\frac{x^2}{4t}},\, x\in {\mathbb {R}},\, t>0. \end{aligned}$$

First one obtains the kernel $H_\pi$ for the $2\pi$-periodic heat equation by “periodization”

$$\begin{aligned} H_\pi (t,x):=\sum _{k\in {\mathbb {Z}}}H(t,x-2\pi k),\, x\in [0,2\pi ). \end{aligned}$$

Then by introducing the Riemann theta function

$$\begin{aligned} \theta _1(\tau ,z):=\sum _{k\in {\mathbb {Z}}}e^{\pi i k^2 \tau }\,e^{2\pi i kz} \,,\, \tau , z \in {\mathbb {C}}\,,\, {\text {Im}}(\tau )>0 \end{aligned}$$

and by setting

$$\begin{aligned} \theta (t,x):=\theta _1\left( \frac{it}{\pi },\frac{x}{2\pi }\right) =\sum _{k\in {\mathbb {Z}}}e^{-k^2 t}\,e^{ikx}\,\,, t>0,\, x\in {\mathbb {R}}\,, \end{aligned}$$

it is directly verified with (A.1) that

$$\begin{aligned} N(t,x)=\frac{1}{\pi }\,\theta (t,x)\,,\, t>0,\, x\in [0,\pi ]\,. \end{aligned}$$

(A.4)

It is a classical result (see e.g. [19]) that the periodic heat kernel $H_\pi$ can be represented by $\theta \,$, i.e.

$$\begin{aligned} H_{\pi }(t,x)=\frac{1}{2\pi }\,\theta (t,x)\,,\, t>0\,,\, x\in [0,2\pi ]\,. \end{aligned}$$

As a consequence of (A.4), the Neumann fundamental solution N can be written in terms of the periodic heat kernel

$$\begin{aligned} N(t,x)=2H_{\pi }(t,x)=H_\pi (t,x)+H_\pi (t,2\pi -x)\,,\, t>0,\, x\in [0,\pi ]\,. \end{aligned}$$

(A.5)

The fundamental solution for the Neumann problem can thus be constructed from the heat kernel on ${\mathbb {R}}\,$ by “periodization” and “reflection”.

(d) For a and for the “shifted kernel” $a_s(t):=a(t)+\frac{1}{\pi }$ it holds that

$$\begin{aligned} \lim _{t\rightarrow \infty } a(t)=-\frac{1}{\pi },\, \lim _{t\rightarrow \infty } a_s(t)=0, \end{aligned}$$

and for $n \ge 1$

$$\begin{aligned} \lim _{t\searrow 0} a(t)=\lim _{t\searrow 0} a_s^{(n)}(t)=0\,. \end{aligned}$$

Also note that, by definition,

$$\begin{aligned} a^{(n)}=a_s^{(n)}, \end{aligned}$$

for $n\ge 1$. The limits for $t\rightarrow \infty$ are obtained directly from (A.2). To determine the one-sided limits, we observe that

$$\begin{aligned} -a(t)=N(t,\pi )=2H_{\pi }(t,\pi ) \end{aligned}$$

and therefore the limits for ${t\searrow 0}$ follow from the well-known properties of the heat kernel on the unit circle. In particular from the concentration of the kernel’s mass at $x=0$ as ${t\searrow 0}\,$, more precisely

$$\begin{aligned} \lim _{t\searrow 0}\, H_{\pi }(t,x) = \lim _{t\searrow 0}\, H^{(n)}_{\pi }(t,x)=0 \,, n\ge 1 \,,\, x\in (0,2\pi )\;. \end{aligned}$$

In the sequel, we will not need the one-sided limits for the derivatives $a^{(n)}\,$. For the sake of being more self-contained, we also provide an alternative, more direct argument for $\lim _{t\searrow 0} a(t)=0\,$. The Jacobi theta function

$$\begin{aligned} \vartheta _4(z,q):=\sum _{k=-\infty }^{\infty }(-1)^k q^{k^2}e^{2kiz}=1+2\sum _{k=1}^{\infty }(-1)^k q^{k^2}\hbox {cos}(2kz) \end{aligned}$$

is particularly simple for $z=0$

$$\begin{aligned} \vartheta _4(q):=\vartheta _4(0,q)=1+2\sum _{k=1}^{\infty }(-1)^k q^{k^2}. \end{aligned}$$

It is known ([2], Chapter 3 ) that $\lim _{q\nearrow 1} \vartheta _4(q) = 0$ and therefore

$$\begin{aligned} \lim _{q\nearrow 1} \sum _{k=1}^{\infty }(-1)^k q^{k^2} =-\frac{1}{2}\,. \end{aligned}$$

(A.6)

Therefore by setting $q=e^{-t}$, we obtain $\lim _{t\searrow 0} \, a(t)=0$ from

$$\begin{aligned} \lim _{t\searrow 0} \, \frac{2}{\pi }\sum _{k=1}^{\infty }(-1)^{k+1} \, e^{-tk^2} = \frac{1}{\pi } \end{aligned}$$

and (A.2).

(e) The kernel a (extended by 0 for $t\le 0$) satisfies

$$\begin{aligned} a\in {\text {BC}}^\infty ({\mathbb {R}},{\mathbb {R}}) \end{aligned}$$

and its n-th derivatives satisfy

$$\begin{aligned} a^{(n)}\in {\text {L}}^p({\mathbb {R}}) \,, p\in [1,\infty ]\,,\, n\ge 1\;. \end{aligned}$$

This follows from the kernel’s representation (A.2).

(f) Similarly the shifted kernel $a_s=a+\frac{1}{\pi }\,$, extended smoothly to $t=0$ by setting $a_s(0)=\frac{1}{\pi }\,$, satisfies

$$\begin{aligned} a_s\in {\text {BC}}^\infty ([0,\infty ),{\mathbb {R}})\cap {\text {L}}^p\bigl ((0,\infty )\bigr ) \;\;,\; p\in [1,\infty ]\;. \end{aligned}$$

Again, this follows from (A.2). Note that the extension of $a_s$ to negative arguments by zero creates a discontinuity at $t=0\,$. This will be relevant when computing its Fourier transform.

(g) The series representation of the Fourier transforms of $a_s$ and $a_s':=a_s^{(1)}$ will be needed in the sequel to recover the Popov stability criterion from the Volterra integral equation. They are given by

$$\begin{aligned} {\hat{a}}_s(\omega )=\frac{2}{\pi }\sum _{k=1}^\infty (-1)^{k+1} \frac{k^2-i \omega }{k^4+\omega ^2} \end{aligned}$$

(A.7)

and by

$$\begin{aligned} \widehat{a_s'}(\omega )=-\frac{1}{\pi }+\frac{2}{\pi }\sum _{k=1}^\infty (-1)^{k+1} i \omega \frac{k^2-i \omega }{k^4+\omega ^2}, \end{aligned}$$

(A.8)

respectively. Note that, here, we define $a_s(t)$ also for negative arguments by setting its value to zero for $t<0\,$. As a consequence of this extension, $a_s$ is discontinuous at $t=0\,$, which impacts the Fourier transform of its distributional derivative $a_s'\,$.

To determine the Fourier transforms, we use (A.2) and interchange the order of summation and integration, which is justified by the uniform convergence in (A.2). To compute ${\hat{a}}_s$ observe that

$$\begin{aligned} {\hat{a}}_s(\omega )=\int \nolimits _{{\mathbb {R}}}e^{-i\omega t}a_{s}(t)\hbox {d}t=\int \nolimits _{0}^{\infty }e^{-i\omega t}a_{s}(t)\hbox {d}t \end{aligned}$$

and therefore

$$\begin{aligned} {\hat{a}} _s(\omega )= & {} \frac{2}{\pi }\sum _{k=1}^\infty (-1)^{k+1}\int \nolimits _{0}^{\infty } e^{-i\omega t} e^{-tk^2}\,\hbox {d}t=\frac{2}{\pi }\sum _{k=1}^\infty (-1)^{k+1} \int \nolimits _{0}^{\infty } e^{-(i\omega +k^2)t} \,\hbox {d}t\\= & {} \frac{2}{\pi }\sum _{k=1}^\infty (-1)^{k+1}\frac{1}{k^2+i\omega }\,. \end{aligned}$$

To find the Fourier transform of $a_{s}'$, one can proceed similarly and we evaluate the occurring integral as follows

$$\begin{aligned} {\widehat{a'_s}}(\omega )&=\int \nolimits _{0}^{\infty }e^{-i\omega t}a_{s}'(t)dt= \lim _{\varepsilon \rightarrow 0}\Big \{\frac{2}{\pi }\sum _{k=1}^\infty (-1)^{k+2} \int \nolimits _{\varepsilon }^{\infty } k^2 e^{-k^2t} e^{-i\omega t} \,\hbox {d}t\Big \}\\&=\lim _{\varepsilon \rightarrow 0}\Big \{\frac{2}{\pi }\sum _{k=1}^\infty (-1)^{k+1} e^{-k^2t} e^{-i\omega t}\Big |_\varepsilon ^\infty \Big \}-i\omega \frac{2}{\pi } \sum _{k=1}^\infty \int \nolimits _0^\infty (-1)^{k}e^{-k^2t} e^{-i\omega t}\, \hbox {d}t. \end{aligned}$$

By using (A.6), we find the result

$$\begin{aligned} {\widehat{a'_s}}(\omega )=-\frac{1}{\pi }+\frac{2}{\pi }\sum _{k=1}^\infty (-1)^{k+1} i\omega \frac{(k^2-i\omega )}{k^4+\omega ^2}. \end{aligned}$$

(h) The Laplace transform ${\mathcal {L}}(a)$ of a is given by

$$\begin{aligned} {\mathcal {L}} (a)(s)=-\frac{1}{\pi s}-\frac{2}{\pi }\sum _{k=1}^\infty (-1)^k\frac{1}{s+k^2} \,,\,s\in \{ z\in {\mathbb {C}} \, |\, {\text {Re}}(z)>0\}\;. \end{aligned}$$

The Laplace transform ${\mathcal {L}}(a)$ of the kernel also has the closed form representation

$$\begin{aligned} {\mathcal {L}} (a)(s)=-\frac{1}{\sqrt{s}\sinh (\pi \sqrt{s})}\,. \end{aligned}$$

The series representation of the Laplace transform is obtained from (A.2) by elementary integrations. Its explicit representation is obtained in [13] (formula (10)) in a different context and is derived again in Sect. 4.3. A more elementary direct computation using a partial fraction expansion is also given in, e.g. [4].

(i) We note that $G_+(s):=-{\mathcal {L}}(a)$ can be expressed in terms of the Fourier transform of $a_s$ and that of $a_s'\,$. In fact, for $\omega \in {\mathbb {R}}$, it holds that

$$\begin{aligned} {\text {Re}}\bigl ( G_+(i \omega )\bigr )=-{\text {Re}} \bigl ( {\hat{a}}_s(\omega )\bigr ) \end{aligned}$$

and

$$\begin{aligned} \omega {\text {Im}}\bigl ( G_+(i \omega )\bigr )= {\text {Re}}\bigl ( \widehat{a'_s}(\omega )\bigr ) . \end{aligned}$$

The inequality

$$\begin{aligned} {\text {Re}}\bigl ( {\hat{a}}_s(\omega )\bigr )+ q{\text {Re}}\bigl ( \widehat{a'_s}(\omega )\bigr )-\frac{1}{\beta }<0 \,,\omega \in {\mathbb {R}}\,, \end{aligned}$$

(A.9)

is then equivalent to

$$\begin{aligned} {\text {Re}}\bigl ( G_+(i \omega )\bigr )-q \omega {\text {Im}}\bigl ( G_+(i \omega )\bigr )>-\frac{1}{\beta } \,,\omega \in {\mathbb {R}}\,. \end{aligned}$$

(A.10)

We will use this relationship between ${\mathcal {L}}(a)$ and ${\hat{a}}_s$ and $\widehat{a'_s}$ to verify that the stability condition (A.9) obtained from the analysis of the integral equation (3.1) is precisely the Popov stability criterion (A.10) applied to the transfer function $G_+\,$. We refer, e.g. to [17][Chapter 7, Equation (7.19)] for a statement and a discussion of the Popov criterion (A.10) in the context of transfer functions associated with finite-dimensional nonlinear feedback systems.