Identification of the initial temperature from the given temperature data at the left end of a rod

In this study, we consider the following optimal control problem:

Choose a control v(x) ∈ L₂(0, l) and a corresponding u such that the pair (v, u) minimizes the functional

Jα<!-- a -->v=∫<!-- ? -->0Tu0,t;v−<!-- - -->yt2dt+α<!-- a -->vxL20,l2 $$\begin{array}{} \displaystyle J_{\alpha }\left(v\right)=\int^T_0{{\left[u\left(0,t;v\right)-y\left(t\right)\right]}^2dt}+\alpha {\left\|v\left(x\right)\right\|}^2_{L_2\left(0,l\right)} \end{array}$$(1)

subject to the parabolic problem;

ut−<!-- - -->k(x)uxx=fx,t,x,t∈<!-- ? -->Ω<!-- O -->:=0,l×<!-- ? -->0,Tux,0=vx,x∈<!-- ? -->0,lux0,t=0,ul,t=0,t∈<!-- ? -->0,T $$\begin{array}{} \displaystyle u_t-{\left(k(x)u_x\right)}_x=f\left(x,t\right),\, \, \, \, \left(x,t\right)\in \Omega :=\left(0,l\right)\times \left(0,T\right] \\\displaystyle u\left(x,0\right)=v\left(x\right),\, \, \, x\in \left(0,l\right) \\\displaystyle u_x\left(0,t\right)=0,\, \, \, u\left(l,t\right)=0\, ,\, \, \, \, t\in \left(0,T\right] \end{array}$$(2)

where y ∈ L₂(0, T) is a given target function and f is a known function.

L₂(0, l) is the Banach space consisting of all measurable functions on (0, l) having the norm

vL20,l=∫<!-- ? -->0lvx2dx1/2. $$\begin{array}{} \displaystyle {\left\|v\right\|}_{L_2\left(0,l\right)}={\left(\int^l_0{{\left(v\left(x\right)\right)}^2dx}\right)}^{1/2}. \end{array}$$

With the choice of the functional in (1), we mention the observation of u(0, t; v) in L₂(0, T) for the control v(x) ∈ L₂(0, l). In (1), α > 0 is the parameter of regularization and it can be found by the Tikhonov regularization method [14].

Let admissible set V_ad be closed and convex subset of the space L₂(0, l). Let’s denote by u(x, t; v) the solution of the parabolic problem (2), corresponding to the given element v ∈ V_ad.

Many papers have already been published to study the control problems for the parabolic equations. Bushuyev [1] has controlled the function f(x) in the parabolic problem u_t + Au = σ(x, t)f(x) with Dirichlet boundary conditions. Munch and Periago [2] have studied the optimal distribution of the support of the internal null control of minimal L₂–norm for the 1-D heat equation. In [3], the numerical approximation of an optimal control problem for a linear heat equation have been presented by Munch and Periago. Yu [4] has established the equivalence of minimal time and minimal norm control problems for the semi-linear heat equations. Zheng, Guo and Ali [5] have investigated the stability of the optimization problem for a multidimensional heat equation. Zheng and Yin [6] have studied the optimal time for the time optimal control problem governed by an internally controlled semi-linear heat equation. In [7, 8, 9], the inverse problems with different controls and cost functions have been investigated.

Lions [10] has examined the optimal control problem of the initial condition for the parabolic systems from the measured temperature at the final time. Hasanov and Mukanova [11] have investigated the problem of determining of the initial condition f(x) in the heat equation u_t = (k(x)u_x)_x with boundary conditions u(0, t) = 0 and u_x(l, t) = 0 from the measured final data u_T(x) = u(x, T).

In this study, we formulate the control problem whose solution implies the minimization of the distance measured in a suitable norm between the solution of the problem at left end and a given target. The aim of this work is to determine the optimal function v^∗ in closed and convex set V_ad ∈ L₂(0, l) such that

Jα<!-- a -->v∗<!-- * --><<!-- ? -->Jα<!-- a -->v,∀<!-- ? -->v∈<!-- ? -->Vad. $$\begin{array}{} \displaystyle J_{\alpha }\left(v^*\right) \lt J_{\alpha }\left(v\right),\, \, \, \forall v\in V_{ad}. \end{array}$$

The plan of the paper is as follows: In section 2, we give the existence of the unique optimal solution of the problem (1)-(2) by getting the weak lower semi-continuity of the cost functional J_α. In section 3, we find the gradient of the cost functional via adjoint problem approach and constitute the minimizing sequence for the functional (1). Finally, we obtain the approximate optimal control function for a numerical example.

It is assumed that

f∈<!-- ? -->L2Ω<!-- O -->,kx∈<!-- ? -->L∞<!-- 8 -->0,l,0<<!-- ? -->k1≤<!-- = -->kx≤<!-- = -->k2,∀<!-- ? -->x∈<!-- ? -->0,l. $$\begin{array}{} \displaystyle f\in L_2\left(\Omega \right),\, \, k\left(x\right)\in L_{\infty }\left(0,l\right),\, \, {0 \lt k}_1\le k\left(x\right)\le k_2,\, \, \forall x\in \left(0,l\right). \end{array}$$(3)

We can define the generalized solution u(x, t) of the problem (2) as a element of V̂^1,0(Ω) satisfying the following identity

∫<!-- ? -->0T∫<!-- ? -->0l−<!-- - -->uη<!-- ? -->t+kuxη<!-- ? -->xdxdt=∫<!-- ? -->0T∫<!-- ? -->0lfx,tη<!-- ? -->x,tdxdt+∫<!-- ? -->0lv(x)η<!-- ? -->(x,0)dx $$\begin{array}{} \displaystyle \int^T_0{\int^l_0{\left(-u{\eta }_t+ku_x{\eta }_x\right)dxdt}}=\int^T_0{\int^l_0{f\left(x,t\right)\eta \left(x,t\right)dxdt}}+\int^l_0{v(x)\eta (x,0)dx} \end{array}$$(4)

for all η ∈ Ĥ^1,1(Ω) and η(x, T) = 0 (see [12]). Here H^1,1(Ω) is the Sobolev space of functions with the norm [12]

uH1,1Ω<!-- O -->=∫<!-- ? -->Ω<!-- O -->u2+ux2+ut2dxdt1/2 $$\begin{array}{} \displaystyle {\left\|u\right\|}_{H^{1,1}\left(\Omega \right)}={\left(\int_{\Omega }{\left[u^2+u^2_x+u^2_t\right]dxdt}\right)}^{{1}/{2}} \end{array}$$

and Ĥ^1,1(Ω) := {η ∈ H^1,1(Ω) : η (l, t) = 0, ∀ t ∈ (0, T]}.

Here V^1,0(Ω) = C([0, T];L₂(0, l)) ∩ L₂((0, T); H¹(0, l)) and V̂^1,0(Ω) is a subspace of V^1,0(Ω) whose elements equal to zero for x = l.

It is known that for every v(x) ∈ L₂(0, l), the boundary value problem (2) admits a unique generalized solution u ∈ V^1,0(Ω) (see [7], [12]).

We give the solvability of the optimal control problem (1)-(2). Let’s give the increment △v to v such that v + △v ∈ V_ad and show the solution of (2) corresponding v + △v by u_△ = u(x, t; v + △v) . Then the function △u = u_△ – u will be the solution of the following difference problem:

△<!-- ? -->ut−<!-- - -->kx△<!-- ? -->uxx=0,x,t∈<!-- ? -->Ω<!-- O -->:=0,l×<!-- ? -->0,T△<!-- ? -->ux,0=△<!-- ? -->vx,x∈<!-- ? -->0,l△<!-- ? -->ux0,t=0,△<!-- ? -->ul,t=0,t∈<!-- ? -->0,T. $$\begin{array}{} \displaystyle {\triangle u}_t-\ {\left(k\left(x\right){\triangle u}_x\right)}_x=0,\, \, \, \, \left(x,t\right)\in \Omega :=\left(0,l\right)\times \left(0,T\right] \\ \triangle u\left(x,0\right)=\triangle v\left(x\right),\, \, \, x\in \left(0,l\right) \\ \triangle u_x\left(0,t\right)=0,\, \, \triangle u\left(l,t\right)=0,\, \, \, \, t\in \left(0,T\right]. \end{array}$$(5)

Inequalities of the type given by (11) are termed variational inequalities.

Let us introduce the Lagrangian L(u, v, ψ) given by

Lu,v,ψ<!-- ? -->=Jα<!-- a -->v+ψ<!-- ? -->,ut−<!-- - -->kxuxx−<!-- - -->fx,tL2Ω<!-- O --> $$\begin{array}{} \displaystyle L\left(u,v,\psi \right)=J_{\alpha }\left(v\right)+{\left\langle \psi ,u_t-{\left(k\left(x\right)u_x\right)}_x-f\left(x,t\right)\right\rangle }_{L_2\left(\Omega \right)} \end{array}$$(12)

where the functional J_α(v) is defined by (1) and the function ψ(x, t) is the Lagrange multiplier.

Using the δL = 0 stationarity condition, we have the following adjoint problem:

ψ<!-- ? -->t+k(x)ψ<!-- ? -->xx=0,x,t∈<!-- ? -->Ω<!-- O -->:=0,l×<!-- ? -->0,Tψ<!-- ? -->x,T=0,x∈<!-- ? -->0,lk0ψ<!-- ? -->x0,t=2u0,t;v−<!-- - -->yt,ψ<!-- ? -->l,t=0,t∈<!-- ? -->0,T $$\begin{array}{} \displaystyle {\psi }_t+{\left(k(x){\psi }_x\right)}_x=0,\, \, \, \, \left(x,t\right)\in \Omega :=\left(0,l\right)\times \left(0,T\right] \\ \psi \left(x,T\right)=0,\, \, \, \, x\in \left(0,l\right) \\ k\left(0\right){\psi }_x\left(0,t\right)=2\left[u\left(0,t;v\right)-y\left(t\right)\right],\, \, \, \psi \left(l,t\right)=0,\, \, \, t\in \left(0,T\right] \end{array}$$(13)

Now, we investigate the variation of the functional J_α(v). The difference functional

△J_α(v) = J_α(v + △v) – J_α(v) is such as

△<!-- ? -->Jα<!-- a -->v=∫<!-- ? -->0T2u0,t;v−<!-- - -->2yt△<!-- ? -->u0,tdt+∫<!-- ? -->0T[△<!-- ? -->u0,t]2dt+α<!-- a -->∫<!-- ? -->0l(2v+△<!-- ? -->v)△<!-- ? -->vdx $$\begin{array}{} \triangle J_{\alpha }\left(v\right)=\int^T_0{\left[2u\left(0,t;v\right)-2y\left(t\right)\right]\triangle u\left(0,t\right)dt} \\ \qquad\qquad~ +~ \int^T_0{{[\triangle u\left(0,t\right)]}^2}dt+\alpha \int^l_0{(2v+\triangle v)\triangle vdx} \end{array}$$(14)

Using the identity between the difference problem and the adjoint problem the equation (14) can be rewritten as follows:

△<!-- ? -->Jα<!-- a -->v=∫<!-- ? -->0l−<!-- - -->ψ<!-- ? -->x,0+2α<!-- a -->v△<!-- ? -->vdx+∫<!-- ? -->0T[△<!-- ? -->u0,t]2dt+α<!-- a -->∫<!-- ? -->0l(△<!-- ? -->v)2dx. $$\begin{array}{} \displaystyle \triangle J_{\alpha }\left(v\right)=\int^l_0{\left\{-\psi \left(x,0\right)+2\alpha v\right\}\triangle vdx}+ \int^T_0{{[\triangle u\left(0,t\right)]}^2}dt+\alpha \int^l_0{{(\triangle v)}^2dx}{\rm \, }. \end{array}$$(15)

The Lemma 1 implies that the second integral in (15) is bounded by term o(vL20,l2 $\begin{array}{} \displaystyle {\left\|v\right\|}^2_{L_2\left(0,l\right)} \end{array}$). So Frechet differential at v ∈ V_ad of the cost functional J_α(v) can be defined as follows:

Jα<!-- a -->′v=−<!-- - -->ψ<!-- ? -->x,0+2α<!-- a -->v. $$\begin{array}{} \displaystyle J'_{\alpha }\left(v\right)=-\psi \left(x,0\right)+2\alpha v. \end{array}$$

We use the conjugate gradient method that is known to be very successful in linear optimization problems in order to compute a numerical approximation of the optimal control. According to this method the minimizing sequence is set by

vk+1=vk−<!-- - -->β<!-- ? -->kJα<!-- a -->′vk,k=0,1,2,⋯<!-- ? --> $$\begin{array}{} \displaystyle v_{k+1}=v_k-{\beta }_kJ'_{\alpha }\left(v_k\right),\, \, \, \, k=0,1,2,\cdots \end{array}$$(16)

where v₀ ∈ V_ad is a given initial iteration and β_k is a small enough relaxation parameter and assures that

Jα<!-- a -->vk+1<<!-- ? -->Jα<!-- a -->vk. $$\begin{array}{} \displaystyle J_{\alpha }\left(v_{k+1}\right) \lt J_{\alpha }\left(v_k\right). \end{array}$$

Concerning the choice of the parameter β_k, there are several possibilities and these can be found in any optimization books.

From the following strongly convex functional definition:

Jα<!-- a -->λ<!-- ? -->v1+1−<!-- - -->λ<!-- ? -->v2≤<!-- = -->λ<!-- ? -->Jα<!-- a -->v1+1−<!-- - -->λ<!-- ? -->Jα<!-- a -->v2−<!-- - -->χ<!-- ? -->λ<!-- ? -->1−<!-- - -->λ<!-- ? -->v1−<!-- - -->v2L20,l2 $$\begin{array}{} \displaystyle J_{\alpha }\left({\lambda v}_1+\left(1-\lambda \right)v_2\right)\le {\lambda J}_{\alpha }\left(v_1\right)+\left(1-\lambda \right)J_{\alpha }\left(v_2\right)-\chi \lambda \left(1-\lambda \right){\left\|v_1-v_2\right\|}^2_{L_2\left(0,l\right)} \end{array}$$

we can see that the cost functional (1) is strongly convex with the constant χ = α.

Using the strongly convexity of the cost functional, we can write the following inequality

vk−<!-- - -->v∗<!-- * -->2≤<!-- = -->Jα<!-- a -->vk−<!-- - -->Jα<!-- a -->v∗<!-- * -->,k=0,1,2,⋯<!-- ? --> $$\begin{array}{} \displaystyle {\left\|v_k-v^*\right\|}^2\le \left(J_{\alpha }\left(v_k\right)-J_{\alpha }\left(v^*\right)\right),\, \, k=0,1,2,\cdots \end{array}$$

This inequality show that the minimizing sequence (16) converges to optimal solution v^∗.

In this paper, we prove that the initial temperature can be controlled from the target u(0, t) in the heat problem. The Frechet differential of the cost functional considered can be obtained by using the adjoint problem. The minimizing sequence (16) converges to the optimal solution of the optimal control problem considered. We give a numerical example which shows that theoretical results are verified.