# Approximate Controllability of Semilinear Control System Using Tikhonov Regularization

Authors: Ravinder Katta; N. Sukavanam
Journal: International Journal of Differential Equations
 (2017)

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## Abstract

For an approximately controllable semilinear system, the problem of computing control for a given target state is converted into an equivalent problem of solving operator equation which is ill-posed. We exhibit a sequence of regularized controls which steers the semilinear control system from an arbitrary initial statex0 to an ϵ neighbourhood of the target state xτ at time τ>0 under the assumption that the nonlinear function f is Lipschitz continuous. The convergence of the sequences of regularized controls and the corresponding mild solutions are shown under some assumptions on the system operators. It is also proved that the target state corresponding to the regularized control is close to the actual state to be attained.


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## Body

## 1. Introduction

Controllability is one of the qualitative properties of a control system that occupies an important place in control theory. Controllable systems have many applications in different branches of science and engineering (see [1–12] for an extensive review on controllability literature).LetV and U be Hilbert spaces called state and control spaces, respectively. Let Y=L2J,U and X=L2J,V be the function spaces. The inner product and the corresponding norm on a Hilbert space are denoted by ·,·, ·, respectively.Consider the semilinear control system(1)dxdt=Axt+But+ft,xt,xt0=x0,where A:DA⊆V→V is a densely defined closed linear operator which generates a C0 semigroup Tt, t≥0. B:U→V is a bounded linear operator and f:J×V→V is a nonlinear function where J=t0,τ⊆0,∞. If f≡0, then the resultant system is called the corresponding linear system which is denoted by (1)∗.Foru∈Y, the mild solution (see [13]) of (1) is given by(2)xt=Tt-t0x0+∫t0tTt-sBusds+∫t0tTt-sfs,xsds.The control system (1) is said to be exactly controllable if, for every x0 and xτ∈V, there exists u∈Y such that the mild solution x∈X verifies the condition xτ=xτ.The control system (1) is said to be approximately controllable if, for every ϵ>0 and for every x0 and xτ∈V, there exists u∈Y such that the corresponding mild solution x∈X satisfies(3)xτ-xτ≤ϵ.In [3], Naito proved the approximate controllability of semilinear system (1) under some assumptions which are given below.Theorem 1 (see [3]).
The semilinear control system (1) is approximately controllable under the following conditions:(i)
TheC0 semigroup Tt is compact ∀t>0.(ii)
The nonlinear functionft,x is Lipschitz continuous with respect to x; that is, ft,x1-ft,x2≤x1-x2∀x1, x2∈X, t∈J, where c>0 is Lipschitz constant.(iii)
ft,x≤M0, where M0 is a positive constant.(iv)
For everyp∈X, there exists a q∈RB¯ such that L1p=L1q, where RB is the range of the bounded linear operator B and L1:X→V is a bounded linear operator defined as (4)L1p=∫t0τTτ-spsds.Condition (iv) of Theorem1 implies that the corresponding linear system 1∗ is approximately controllable; for more details one can see the proof in [3].In this paper, we study the problem of computing control for an approximately controllable semilinear system for a given target state by converting it into an equivalent linear operator equation which is ill-posed. We find sequence of regularized controlsun,λ:n∈N,λ>0 using Tikhonov regularization and the mild solutions {xn,λ:n∈N,λ>0} corresponding to un,λ:n∈N,λ>0. Under some assumptions we prove the convergence of un,λ and xn,λ.The outline of the paper is as follows. In Section2, regularized control, its corresponding mild solutions, their convergence, and limitations due to the presence of nonlinearity are discussed. Section 3 is devoted to illustrating our theory through an example. Conclusions are made in Section 4.


## 2. Regularized Control

Definition 2 (well-posed problem).
LetU and V be normed linear spaces and L:U→V be a linear operator. The equation(5)Lu=vis said to be well-posed if the following holds: (i)
For everyv∈V, there exists a unique u∈U such that Lu=v.(ii)
L-1 is a bounded operator. Equivalently, for every v∈V and for every ϵ>0, there exists a δ>0 with the following properties: If v~∈V with v~-v≤δ and if u,u~∈U are such that Lu=v and Lu~=v~, then u-u~≤ϵ.Definition 3 (ill-posed problem).
Equation (5) is said to beill-posed if L violates one of the conditions for well-posedness.Theorem 4 (Tikhonov regularization, see [14]).
LetU and V be Hilbert spaces and L:U→V be a bounded linear operator. Then for each v∈V and λ>0, there exists a unique uλv∈U which minimizes the map(6)u⟼Lu-v2+λu2,u∈U.Moreover, for each λ>0, the map(7)Rλ:v⟼uλv,v∈V,is a bounded linear operator from V to U and Rλv≔L∗L+λI-1L∗v, where L∗ is the unique adjoint of the bounded linear operator L.Theorem 5 (see [14]).
Forλ>0, the solution uλ of the operator equation(8)L∗L+λIuλ=L∗vminimizes the function u→Lu-v2+λu2,u∈U, and Luλ-v→0asλ→0.Definition 6.
Forv∈V and λ>0, the element uλ∈U as in Theorems 4 and 5 is called the Tikhonov regularized solution of Lu=v.Lemma 7 (14).
LetU,V be Hilbert spaces and L∈BU,V. Then for λ>0,(9)L∗L+λI-1=LL∗+λI-1≤1λ,L∗L+λI-1L∗=L∗LL∗+λI-1≤12λ.For more details on ill-posed problems and regularization methods one can refer to [14–20].LetL:Y→V be a linear operator defined as (10)Lu=∫t0τTτ-sBusds.Assumption 8.
(i) System (1) is approximately controllable.
(ii)P=[M2bcτ-t0)3/2+2Mcλτ-t0/2λ<1, where λ>0 is a regularization parameter (to be chosen appropriately) and M,b are given by (11)Tt≤M,B≤b.In our analysis, we assume that the control system (1) satisfies Assumption 8. We obtain a sequence of controls and corresponding mild solutions for semilinear system (1) iteratively and also prove that this sequence of controls steers the semilinear control system from an initial state x0 to an ϵ neighbourhood of the final state xτ at time τ>0.Consider(12)xnt=Tt-t0x0+∫t0tTt-sBunsds+∫t0tTt-sfs,xn-1sds,where xnτ=x1, for all n=0,1,2,…, and unt is a control function such that xnτ=x1. We start with an initial (guess) mild solution x0t. To find unt such that xnτ=x1, we need to solve(13)Lun=vn,where (14)Lun=∫t0τTτ-sBunsds,vn=x1-Tτ-t0x0-∫t0τTτ-sfs,xn-1sds,n=0,1,2,….Since (13) is ill-posed in the sense of Hadamard [21], any small perturbations in vn can lead to large deviations in the solution. Hence, in practice it is not advisable to solve (13) directly to obtain un; one has to look for stable approximations un,λ, λ>0, such that Lun,λ-vn→0 as λ→0. For this we shall use the Tikhonov regularization for obtaining the control function un,λ which is given below:(15)un,λ=L∗LL∗+λI-1vn,n=0,1,2,….Convergence of{un,λ}andxn,λ. We have the sequence of regularized controls un,λ and the sequence of corresponding mild solutions xn,λ for each n∈N, λ>0. The inner product and the corresponding norm on the function space L2J,U are given below.Foru,w∈L2J,U, (16)u,wL2J,U=∫Jus,wsUds,wL2J,U=∫JwsU2ds.Theorem 9.
Under Assumption8 and for fixed λ>c, the sequences un,λ, xn,λ are convergent with respect to n in L2J,U, L2J,V, respectively.Proof.
AsL2(J,U), L2J,V are complete spaces, it is sufficient to prove un,λ and xn,λ are Cauchy sequences in L2J,U, L2J,V, respectively.
We haveun,λ=L∗LL∗+λI-1vn,(17)un+1,λ-un,λL2J,U=L∗LL∗+λI-1vn+1-vnL2J,U≤12λvn+1-vnV≤12λ∫t0τTτ-sfs,xn,λs-fs,xn-1,λsdsV≤12λMc∫t0τxn,λs-xn-1,λsVds≤12λMcτ-t0xn,λ-xn-1,λL2J,V≤12λMcτ-t0Pxn-1,λ-xn-2,λL2J,V≤12λMcτ-t0P2xn-2,λ-xn-3,λL2J,V⋮≤12λMcτ-t0Pn-1x1,λ-x0L2J,V=12λMcτ-t0Pn-1u1,λ-u0,λL2J,U,where P=[M2bcτ-t0)3/2+2Mcλτ-t0/2λ.
By Assumption8 of (ii), P<1; hence for large value of n, the sequence un,λ is Cauchy. Therefore un,λ converges.
Similarly, we have(18)xn,λt=Tt-t0x0+∫t0tTt-sBun,λsds+∫t0tTt-sfs,xn-1,λsds,xn+1,λt-xn,λtV=∫t0tTt-sBun+1,λs-Bun,λsds+∫t0tTt-sfs,xn,λs-fs,xn-1,λsdsV≤M∫t0tBun+1,λs-Bun,λsVds+M∫t0tfs,xn,λs-fs,xn-1,λsVds≤Mb∫t0tun+1,λs-un,λsUds+Mc∫t0txn,λs-xn-1,λsVds≤Mb∫t0τun+1,λs-un,λsUds+Mc∫t0τxn,λs-xn-1,λsVds≤Mbτ-t0un+1,λ-un,λL2J,U+Mcτ-t0xn,λ-xn-1,λL2J,V≤MbMcτ-t02λxn,λ-xn-1,λL2J,V+Mcτ-t0xn,λ-xn-1,λL2J,V≤M2bcτ-t03/2+2Mcλτ-t02λxn,λ-xn-1,λL2J,V.Thus(19)xn+1,λt-xn,λtV≤M2bcτ-t03/2+2Mcλτ-t02λxn,λ-xn-1,λL2J,V.We have(20)xn+1,λ-xn,λL2J,V2=∫Jxn+1,λt-xn,λtV2dt.From (19) and (20), we get(21)xn+1,λ-xn,λL2J,V≤Pxn,λ-xn-1,λL2J,V≤P2xn-1,λ-xn-2,λL2J,V≤P3xn-2,λ-xn-3,λL2J,V⋮≤Pnx1,λ-x0L2J,V.Since P<1, for large value of n, the sequence xn,λ is also Cauchy; hence it converges.
This completes the proof.Remark 10.
In practice, to obtain better approximation to the sequence of controls,λ (regularization parameter) can be chosen such that P<1; that is, (22)M2bcτ-t03/2+2Mcλτ-t02λ<1⟹M2bcτ-t03/2+2Mcλτ-t0<2λ⟹2λMcτ-t0-1<-M2bcτ-t03/2⟹λ>M4b2c2τ-t032-2Mcτ-t02=Qsay.If Mcτ-t0≪1 then Q is very small. Then we get better approximation.
In many practical semilinear control systems, the nonlinear part is a perturbation, in the sense that the Lipschitz constant is sufficiently small so that the system is approximately controllable. In particular, the regularization parameterλ>c, where c is very small. Then λ can also be chosen sufficiently small. Hence we get a regularized control close to the exact solution.


## 3. Application for an Approximately Controllable System

In this section, we illustrate the theory for an approximately controllable semilinear system. Letuλ≔limn→∞un,λ be the regularized control. Let xλ be the mild solution corresponding to uλ.Then from Theorem4 we see that(23)xλτ-xτ⟶0asλ⟶0which shows that the target state corresponding to the regularized control xλτ is close to the actual state xτ to be attained.Example 11.
Consider the semilinear heat equation given by the partial differential equation(24)∂z∂tx,t=∂2z∂x2x,t+ux,t+ft,zx,t,x∈0,l,(25)z0,t=0=zl,t,zx,0=g0x,g0∈L20,l,where zx,t represents the temperature at position x at time t, g0x is the initial temperature profile, and ux,t is the heat input (control) along the rod and f:J×V→V is a nonlinear function which is Lipschitz continuous.
We have(26)J≔0,τ,U=V=L20,l,U=L2J,U,V=V.Define the operator A by (27)Ah=d2hdx2,h∈DA,where (28)DA≔h∈L20,l:h′′∈L20,l,h0=0=hl.Let B=I, the identity operator on L20,l. By using the notations z~t≔z·,t,u~t≔u·,t,f~t,z~t≔ft,z·,t, (24) takes the form of a control system defined on L20,l which is given below:(29)dz~tdt=Az~t+u~t+f~t,z~t,t∈0,τ,(30)z~0=g0.The C0 semigroup generated by the operator A [22] is(31)Ttg=∑j=1∞e-j2π2/l2tg,ϕjϕj,whereϕjx=2lsin⁡jπxl.For u~∈U, the mild solution of (29) is given by (32)z~t=Ttg0+∫0tTt-su~sds+∫0tTt-sf~s,z~sds.Let L0:U→V be the operator defined by (33)L0u~=∫0τTτ-su~sds.Then we have(34)L0∗vs≔∑j=1∞e-j2π2/l2τ-sv,ϕjϕj,L0L0∗v=∑j=1∞σj2v,ϕjϕj,whereσj2≔1-e-2j2π2/l2τ2j2π2l2.Since the semigroup (31) is compact, L is a compact operator; consequently the control system (24) is approximately controllable. The control system (24) satisfies Assumption 8. Hence, the regularized control of system (24) for a given target state gτ (desired temperature profile) is obtained as follows:(35)z~n,λt=Ttg0+∫0tTt-su~n,λsds+∫0tTt-sf~s,z~n-1,λsds,where z~n,λτ=gτ, for all n=0,1,2,…, and u~n,λt is a control function such that z~n,λτ=gτ.(36)L0L0∗vn=∑j=1∞σj2vn,ϕjϕj,L0L0∗+λIvn=∑j=1∞σj2+λvn,ϕjϕj,L0∗ϕjs=e-j2π2/l2τ-sϕj.We have(37)u~n,λ=L0∗L0L0∗+λI-1vn.Thus, using (36) in (37) we get(38)u~n,λs=∑j=1∞1σj2+λvn,ϕjL0∗ϕjs=∑j=1∞e-j2π2τ-s/l2σj2+λvn,ϕjϕj,n∈N.
Error involved in the regularization procedure is given by(39)vn-L0u~n,λ=∑j=1∞λσj2+λvn,ϕjϕj,(40)vn-L0u~n,λ2=∑j=1∞λ2σj2+λ2vn,ϕj2,n∈N.From (40), it is clear that vn-L0u~n,λ→0 as λ→0.Problem 12.
Consider (24) and (25) with ft,zx,t=c∫01zx,t2dx, and g0x=sin⁡(πx), t∈J≔0,2, l=1, x∈0,1; that is, (41)∂z∂tx,t=∂2z∂x2x,t+ux,t+c∫01zx,t2dx,(42)z0,t=0=z1,t,zx,0=sin⁡πx.Here we have the system constants: M=1, τ=2, t0=0, and c is the Lipschitz constant.In order to obtain better approximation to the regularized control, the regularization parameterλ can be chosen in such a way that λ>8c2/(2-4c)2,c≠1/2. Then the semilinear control system (41) satisfies Assumption 8. Hence, the convergence of the sequences of regularized controls un,λ and the corresponding mild solutions xn,λ follows from Theorem 9.


## 4. Conclusions

In the mathematical control theory literature, Tikhonov regularization is not given much attention to the problems related to approximately controllable system. We use the Tikhonov regularization method and exhibited a sequence of regularized controls and their corresponding mild solutions. The convergence of the sequences under some assumptions has also been established. The results are illustrated with an example. However, the case whereB≠I should be considered for future work as the theory will change substantially.



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