Finite determining parameters feedback control for distributed nonlinear dissipative systems - a computational study

We present a computational study of a simple finite-dimensional feedback control algorithm for stabilizing solutions of infinite-dimensional dissipative evolution equations such as reaction-diffusion systems, the Navier-Stokes equations and the Kuramoto-Sivashinsky equation. This feedback control scheme takes advantage of the fact that such systems possess finite number of determining parameters or degrees of freedom, namely, finite number of determining Fourier modes, determining nodes, and determining interpolants and projections. In particular, the feedback control scheme uses finitely many of such observables and controllers that are acting on the coarse spatial scales. We demonstrate our numerical results for the stabilization of the unstable zero solution of the 1D Chafee-Infante equation and 1D Kuramoto-Sivashinksky equation. We give rigorous stability analysis for the feedback control algorithm and derive sufficient conditions relating the control parameters and model parameter values to attune to the control objective.


Introduction
Efficient algorithms for control and stabilization of flows for various transportreaction-diffusion systems have many important applications in several areas of science and engineering.Representative applications include stabilizing flame front propagation, catalytic rod, chemical vapor deposition -a process often used in the semiconductor industry to produce thin films for defense-related applications and microelectronics manufacturing, and nonlinear control of Czochralski crystal growth processes -a well established industrial process used for the production of silicon needed in the construction of wafers for fabrication of microelectronic chips and other micro devices, to name a few (see, e.g., [9] for a detailed description of set-up and models for these industrial and engineering applications).
For several of these applications, the model equations governing its dynamics are classified as dissipative dynamical systems that are known to have a finite number of degrees of freedom.Thus, various feedback control approaches are based on reduced order models.To name some of these works, we direct the reader to [1,2,20,23,37,52,53,56] and references therein.Despite many efforts, there are only a few rigorous analytical work in feedback control theory justifying these algorithms.There is also the conventional set-back on the real time implementation of these algorithms in industrial control systems.
In [4], the authors introduced a simple finite-dimensional feedback control algorithm for stabilizing solutions of infinite-dimensional dissipative evolution equations, such as reaction-diffusion systems, the Navier-Stokes equations and the Kuramoto-Sivashinsky equation.The feedback control scheme uses finitely many observables and controllers which is consistent and stems from the fact that such systems possess a finite number of determining parameters or degrees of freedom, for example, finite number of determining Fourier modes, determining nodes, and determining interpolants and projections.
We note that based on this new control algorithm a new continuous data assimilations algorithm that has applications for weather prediction was developed in [3], (see also [21,22] for abridged continuous data assimilation).With the assumption that the observational data measurements are free of noise, the authors in [3] provided sufficient conditions on the spatial resolution of the collected coarse mesh data and the relaxation parameter that guarantees that the approximating solution obtained from this algorithm converges to the unknown reference solution over time.Then, in [6], the performance of this linear feedback control algorithm applied to data assimilations where the observational data contains stochastic measurement errors was studied.The algorithm is applied to the 2D Navier-Stokes equations.The resulting equation in the algorithm is a Navier-Stokes-like equation with stochastic feedback term that attunes the large scales in the approximate solution to those of the reference solution corresponding to the measurements.They found resolution conditions on the observational data and the relaxation/nudging parameter in which the expected value of the difference between the approximate solution, recovered by proposed linear feedback control algorithm and the exact solution is bounded by a factor which depends on the Grashof (Reynolds) number multiplied by the variance of the noise, asymptoti-cally in time.We also note that the feedback controller proposed in [4] can be used for stabilization of the zero steady state solutions of a wide class of nonlinear dissipative wave equations, (see.eg.[42]).
In this paper, we implement the feedback control algorithm for a simple reaction-diffusion equation, the Chafee-Infante equation, which is the real version of the complex Ginzburg-Landau equation, and for the Kuramoto-Sivashinksy equation (KSE), a model for flame front propagation or flowing thin films on inclined surface.We give rigorous stability analysis for the feedback control algorithm for stabilizing the zero unstable steady state solution and derive the sufficient conditions relating the control parameters and model parameter values.The analysis for the more general case of stabilizing a nonzero reference solution of the KSE is provided in the appendix.
As an initial test case, we implement the control algorithm where the actuators are taken to be the first m Fourier modes and the control inputs prescribe the amplitude of the modes.Then, we implement the control algorithm using determining volume elements (local spatial averages) and as well as the determining volume elements based on nodal values.We also apply the algorithm for control of nonlinear parabolic PDE system to some recent feedback control case studies of the catalytic rod with or without uncertainty on the variables and give a comparison of results to existing control algorithms.For the catalytic rod with uncertainty, the control objective is the regulation of the temperature profile in the rod through the manipulation of the temperature of the cooling medium when the dimensionless heat of reaction β T is unknown and time-varying (see pp. 102-117 in [9], for more details).In their control process there is one available control actuator and one point measurement sensor placed at the center of the rod.It was found that in the presence of uncertainty, their proposed algorithm regulates the temperature profile at the desired steady state under a certain tolerance which depends on the uncertainty level in the system parameters.We present our results on a similar computational study using the new simplified method for control of instabilities in the presence of uncertainty in the model parameters.
We note that the newly proposed feedback control for controlling general dissipative evolution equations using finitely many determining parameters like determining modes, nodes, and volume elements, is without requiring the presence of separation in spatial scales, in particular, without assuming the existence of an inertial manifold (see, eg. [53] for using inertial manifold for feedback control).In addition, it is worth mentioning that one can use this approach to show that the long-time dynamics of the underlying dissipative evolution equation, such as the two-dimensional Navier-Stokes equations, can be imbedded in an infinite-dimensional dynamical system that is induced by an ordinary differential equations, named determining form, which is governed by a globally Lipschitz vector field (see, e.g.[24,25] and [38]).

Interpolant operators as feedback controllers
This section reviews the type of interpolant operators that we will use in our computational studies.
For ϕ ∈ H 1 ([0, L]) we define Consider a general linear map ) which is an interpolant operator that approximates the identity operator with error of order h.Specifically, it approximates the inclusion map i : holds for every φ ∈ H 1 ([0, L]), where c is a dimensionless constant.The last inequality is a version of the well-known Bramble-Hilbert inequality that usually appears in the context of finite elements [10].

Fourier modes
Consider a periodic function φ.The interpolant operator I h acting on φ is defined as the projection onto the first N Fourier modes; where the Fourier coefficients are given by a

Finite volume elements
The volume element operator is also an interpolant operator satisfying (2).Given ϕ, we define where is the characteristic function of the interval J k , for k = 1, . . ., N , serving as the actuator shape function, and where represents the amplitude for the given actuator.Here, the local averages, ϕ k , for k = 1, ..., N, are the observables, which also serve as the feedback controllers.

Interpolant operator based on nodal values
We also consider here where the observables are the values φ(x * k ), where . ., N .In this case the feedback control scheme is given by where χ is the characteristic function of the interval J k , for k = 1, . . ., N , serving as the actuator shape function.
3 Motivating example: Chafee-Infante equation We recall the Chafee-Infante reaction-diffusion equation (or the real version of the complex Ginzburg-Landau equation) on the interval [0, L], with no flux boundary condition for α > 0, large enough, and given initial condition u(x, 0) = u 0 (x).In [4] the authors designed a feedback control algorithm to stabilize an unstable steady state solution of (6a)-(6b) either by observing the values of the solutions at certain nodal points, local averages of the solutions in subintervals of [0, L], or by observing finitely many of their Fourier modes.In the case of the steady state v(x) ≡ 0, as a particular example, the number of feedback control needed to stabilize this zero solution is proposed (which was then rigorously justified in [4], see, also [5,36] and [57] for a similar analysis) to be proportional to the dimension of unstable manifold which is αL 2 π 2 ν .To shed some light on this proposed value, we linearize equation (6a) about the steady state solution v ≡ 0, to obtain One can assume a particular solution of the form v(x, t) = a k (t) cos( k x L π), with given initial condition v 0 (x) = A k cos( kx L π), where A k ∈ IR.Plugging in this ansatz, we obtain the equation for the time dependent coefficients as follows whose solution is simply One then observes that for α > 0, large enough, all the low wave numbers k 2 < α L 2 π 2 ν are unstable, and hence, one needs at least αL 2 π 2 ν number of parameters to stabilize v ≡ 0.

Chafee-Infante equations: Finite volume elements
The following scheme was proposed as a feedback control system for (6a)-(6b) to stabilize the steady state solution v ≡ 0: where I h is specified in section 2.
The results concerning global existence, uniqueness and stabilization for general family of finite-dimensional feedback control system that includes system (9a)-(9b) as a particular case, were established in [4].They showed that every solution u of (9a)-(9b) tends to zero, as t → ∞, under specific explicit assumptions on N , in terms of the physical parameters ν, α, L and µ.Their main results are stated in the next proposition and theorem.It is worth mentioning that similar results were first introduced and proved in [33] (see also [15] and [39]).
where h = L N .Moreover, where Using the proposition above, applying energy estimates, and giving specific explicit assumptions on N , in terms of the physical parameters ν, α, L and µ, the authors of [4] obtained the following theorem: Theorem 3.1 ([4]).Let N and µ be large enough such that µ ≥ ν 2π h 2 > α, where α > 0 and h = L N .Then u(t) L 2 tends to zero, as t → ∞, for every solution u(t) of (9a)-(9b).As mentioned in [4], the assumptions of Theorem 3.1, in particular, that N > 4π 2 ν , is consistent with the fact that the dimension of the unstable manifold about v ≡ 0 is of order of L 2 α ν (see for e.g., [5,36,57]).It was also noted there that one can use the same idea to stabilize any other given solution, v(x, t), of (6a)-(6b) by using a slightly modified feedback control in the right-hand side of (9a)-(9b) of the form . Note that in [44], it was shown that the dissipative system (6a)-(6b) has only two determining close enough nodes.Whether one can design a feedback control scheme that stabilizes the zero stationary solution using only two controllers at these nodes is an open problem of an ongoing research.

Discretization scheme, set-up
We solve the Chafee-Infante equation with or without feedback control, (6a)-(6b) or (9a)-(9b), respectively, with initial data u 0 = A cos(3x), with A = 1, supplemented with homogeneous Neumann boundary conditions using finite differences in space and explicit time stepping.We note that when A 1, the simulation exhibits similar behavior.We solve it on the time interval [0, 1], and spatial interval [0, 1] and choose the step-size ∆t and spatial discretization width ∆x so that the ratio r = ν∆t/(∆x) 2 < 1/2 for stability of the numerical scheme.We used α = 100 and ν = 1.

Without feedback control
The Chafee-Infante equations in the case where there is no feedback control (µ = 0) has an unstable trivial steady state.Starting with some initial condition close to zero, the solution grows in time until it reaches another steady solution characterized by constant function equal to its maximum value proportional to the √ α as expected.

Kuramoto-Sivashinsky equation: Overview
We illustrate the application of the proposed feedback control algorithm to the stabilization of the zero solution of another nonlinear dissipative PDE known as the 1D KSE.The KSE equation was originally derived and introduced as a turbulence model for magnetized plasmas [46,13], chemical reaction-diffusion processes [45], and flame front propagation [55].KSE also models the motion of a thin viscous fluid or thin film flowing down an inclined wall [8].In one space dimension, it is written as subject to the periodic boundary conditions, and initial condition: where u(x, t), for example, describes the height of the film fluctuations, and the parameters γ and ν are given positive constants.Equation ( 12) can be nondimensionalized by substituting u → γu/ L, t → t L2 /γ, x → Lx, and ν → L2 γν, with L = L 2π .In this case one gets the same equation as before with the modification γ = 1 and L = 2π.
To give some justification for the choice of the number of feedback controllers (for our method the actuators and sensors are in the same locations) it is necessary to know the number of unstable modes of the steady state v(x) ≡ 0 for a given instability parameter value ν.In order to obtain a heuristic value for the number of unstable modes, one can perform a simple analysis by linearizing equation ( 12) about v ≡ 0, subject to periodic boundary conditions, to obtain the linear equation Assuming a particular solution of the form v(x, t) = a k (t) e ikx yields the equation ȧk for the time dependent coefficients.We solve equation (15) with initial condition v 0 (x) = A k e ikx , with A k ∈ IR, which yields This shows that all the low wave numbers k < 1 √ ν are unstable.Thus, one needs at least 1 √ ν number of parameters to stabilize v ≡ 0. Since the smaller value of ν, for which the solution u(x, t) = 0 is about to become unstable is ν = 1, implies that the nonlinear system (12), is locally unstable when ν < 1.

KSE without feedback control 4.1.1 Discretization scheme, set-up
To show some proof-of-concept corresponding to our heuristic calculations of the number of unstable modes, we present our simulations for the case where the ν > 1 and the case where the ν < 1.We choose the initial condition 1e −10 * cos x (1 + sin x) for both cases.Observe that for ν = 1.1 > 1, our linear stability analysis shows exponential decay to the linearly stable steady state zero solution.Figure 3a is consistent with this result.The final profile of the film height is given in Figure 3b.
For the case where ν = 4/15 < 1, our numerical simulation is illustrated in Figures 4a and 4b.In the context of a thin film flowing in an inclined surface, Figures 4a and 4b illustrate the unwanted wavy fluctuations that develop in time.The film height initially starts close to zero and then at around time t = 32 a pattern starts to evolve.At time t = 80, the solution is eventually attracted to a stable traveling wave.We show only up to some particular final time to display a clear transition between patterns or structures.A common goal (see, e.g.[1], [2], [47]) is to identify a control strategy to suppress the undesirable occurrences of these wavy patterns by using actuators/sensors that control the thin film's thickness, for example, by adjusting dynamically in time a valve that serves as a source or sink at specific location in the inclined plane where the liquid film is flowing.One also needs to design a control algorithm that is likely to achieve the stabilization of the film height to the zero solution in real time making the implementation tractable for industrial control system problems.

KSE with feedback control
To stabilize the steady state solution v = 0 of (12), we implement the proposed feedback control algorithm in [4], applied to KSE, which is given as follows: subject to the periodic boundary conditions, and initial condition u(x, 0) = u 0 (x), with 2π 0 u(x, 0) dx = 0, where the interpolant operator I h acting on u can be defined as one of the general interpolants listed in section 2 satisfying certain properties.Note that the interpolant polynomial here is shifted by its average in the whole domain [0, 2π] to maintain the invariance of the zero average for the controlled equation, (see e.g., equation (36) in the numerical results section below).We begin by establishing global well-posedness and stability results for the proposed feedback control algorithm.

Existence, uniqueness and stability results using finite parameters feedback control
We recall that the existence and uniqueness of solution, as well as the existence of finite-dimensional global attractor to the system (12) were first established in [49] (see also [57]) for odd initial data.The long-time boundedness results was later improved and extended to any mean-zero initial data in [14,35].Improved estimates were later established in [7] and [34].
In this section we establish the global existence and uniqueness for the general feedback algorithm stated in (16); and that the I h feedback control is stabilizing the steady state solution v ≡ 0 of ( 12).This will be accomplished by choosing µ large enough and then choosing h small enough, under the assumptions Our estimates here are formal, but these steps can be justified by the Galerkin approximation procedure and then passing to the limit while using the relevant compactness theorems.We will omit the rigorous details of this standard procedure and provide only the formal a-priori estimates (see, e.g., [57]).
Let us now establish the aformentioned formal a-priori bounds for the solution which are essential for guaranteeing global existence and uniqueness.System ( 16) can be rewritten as Taking the L 2 -inner product of ( 18) with u, integrating by parts, and using the periodic boundary conditions, the cubic nonlinear term disappears and we obtain 1 2 and applying the Cauchy-Schwarz, we get 1 2 .
Using Young's inequality, we reach 1 2 Let us recall the fact that for every φ ∈ H 1 per (0, L), with L 0 φ dx = 0, (see Prop. 5 (ii) page 299 of [3]) and thanks to Poincaré inequality, one has for some constant c.Hereafter, we abuse the notation for an arbitrary constant c which may change from line to line.
Using the interpolation inequality φ x 2 L 2 ≤ φ L 2 φ xx L 2 and the approximation inequality (21) one has 1 2 By Young's inequality we have and substituting ( 23) to ( 22) we obtain 1 2 for every t ∈ [0, T ).Thanks to the assumption ( 17), we have that By dropping the u xx 2 L 2 term on the left-hand side of ( 24) and applying Gronwall's inequality we obtain In particular, by assumption (17) we find that u ∈ L ∞ ([0, T ], L 2 ), for all T > 0. Also notice that from (24) and ( 17) one can conclude that for every τ > 0 Using (26) and ( 27) one can easily show the continuous dependence of the solutions of ( 16) on the initial data and the uniqueness, using similar energy estimates, provided the assumptions ( 21) and ( 17) hold.
In conclusion, from the above, and in particular thanks to (26) and ( 27), we have the following theorem: In particular, we established (see (26)) that under that above assumption the feedback control interpolant operator I h is stabilizing the steady state solution v ≡ 0 of (12) in L 2 .
Remark 5.1.In the KSE (12), the term u xx is responsible for internal destabilizing of the system, as it has been demonstrated in studying (14), and the term u xxxx is responsible for dissipating energy and stabilizes the system.In order to show that the KSE is dissipative, one takes advantage of the nonlinear term uu x which transfers energy from low modes (produced by the term u xx ) to high modes which is then dissipated by the term u xxxx .In other words, the nonlinear term is responsible for preventing the solution to growing indefinitely to infinity.On the other hand, the feedback control term is a different mechanism which stabilizes the null state.The feedback control term in (18) is responsible for stabilizing the low unstable modes (large spatial scales); together with u xxxx it neutralizes the instability generated by the term u xx .Specifically, the dissipation term u xxxx dissipates the spillover and instabilities of the small scales that I h (u) might produce while stabilizing the large spatial scales.

H 1 − stability
Here we conclude that under the same assumption (17), the feedback control interpolant operator I h is stabilizing the steady state solution v ≡ 0 of (12) in the H 1 norm.
We start by taking the inner product of (18) with −u xx , writing I h (u) as in (19), and integrating by parts, to get 1 2 Applying Cauchy-Schwartz and Hölder's inequality, using the fact that in one dimension, H 1 → L ∞ , (and we have the available Agmon's inequality φ 2 L ∞ ≤ φ L 2 φ x L 2 ), and then applying Young's inequality we obtain 1 2 Combining like terms we obtain 1 2 and also recall that we have (21).The right-hand side of equation ( 30) becomes (Here we used the Young's inequality a • b ≤ 3 4 a 4/3 + 1 4 b 4 ).Thus we have 1 2 . It follows that the coefficient of u x L 2 , which we denote as We drop the positive term ν 2 u xxx 2 L 2 in (32), recalling that Q < 0 and the fact that lim t→∞ u(t) 2  L 2 = 0, we obtain from Gronwall's inequality that lim sup t→∞ u x (t) 2 L 2 = 0.

Numerical results: Stabilizing KSE by implementing two types of interpolant operators I h
In this section we present the numerical results for the new feedback control algorithm (16).We illustrate through matlab computer simulations, the application of the proposed new method for stabilizing the unsteady zero solution of the Kuramoto-Sivashinksy equations.As a simple initial test case, we consider the example where the actuators are taken to be the first m modes, where m depends on the number of unstable modes and the control inputs prescribe the amplitude of the modes.We also simulate the control algorithm using determining volume elements (local spatial averages) and as well as the determining nodal values.The stabilization of 1D KSE has been addressed in several earlier works, for example in [1], [2], and [47], in which the common starting point is reduced-order system, that can accurately describe the dynamics of the KSE.Then, from this resulting reduced-order system, the feedback controller can readily be designed and synthesized by taking advantage of the reduced-order techniques, called the approximate inertial manifold (AIM) and the proper orthogonal decomposition (POD) methods.Several other works also addressed the issue of selecting the optimal actuator/sensor placement so that the desired control energy budget are achieved with minimal cost, (see for example [48] and references therein).
We utilize a simple modification of the Exponential Time Differencing fourthorder Runge-Kutta (ETDRK4 ) method used to solve the KSE with feedback control.This exponential time differencing scheme was originally derived by Cox and Matthews in [18] and was later modified by Kassam and Trefethen in [43], treating the problem of numerical instability in the original scheme.This overcomes a stiff type problem via the exponential time differencing, a method which uses the idea similar to the method of integrating factor.The implementation of ETDRK4 for the KSE equation was presented as a simple example in [43].We have adapted a similar code for a fixed computational domain, incorporated the parameter ν as the system parameter, and incorporated the feedback control term appropriately.

Case 1: Controlled KSE with finite modes 6.1.1 Projection onto Fourier modes as an interpolant operator
For a motivating simple example, we implement the proposed feedback control system in [4] for the KSE to stabilize the steady state solution v = 0 of (12), which is given as follows: subject to the periodic boundary conditions, with L = 2π, and initial condition u(x, 0) = u 0 (x), with L 0 u(x, 0) dx = 0, where the interpolant operator I h acting on u is defined as in (3) Here we do not assume any symmetry on the initial data except that its spatial average is zero.Designing the feedback control based on the first m, L−periodic Fourier modes, although not practical, is a common practice for testing grounds of the control method (for example, see [1,2] and references therein).Here we present a similar initial test, that is, where feedback control is given simply by (3).We are able to implement this by modifying the definition of the operator L in [43] to accommodate the feedback control term −µI h and design a routine for the feedback control scheme that depends on the number of unstable modes and the type of interpolation operator I h .
In Figure 5a and 5b we illustrate the solution to controlled problem using ET-DRK4.He we start with the initial condition u 0 (x) = cos(x).* (1 + sin(x)).The figure in 5a illustrates the open-loop profile showing instability of the u(x, t) = 0 steady state solution for 0 < t < 40 for ν = 4/15 < 1, then the feedback control with µ = 20 is turned on for t > 40 which exponentially stabilizes the system.Figure 5b shows the top view profile of u(x, t).

Case 2: Stabilizing KSE by implementing finite volume elements
Using the notation of section 3.1 we consider the proposed feedback control system in [4] to stabilize the steady state solution v = 0 as follows: subject to the periodic boundary conditions, with L = 2π, and initial condition u(x, 0) = u 0 (x), where the interpolant operator acting on u, I h (u), is defined as a slight modification of (3) as follows, where We implement this proposed control algorithm by modifying the main timestepping loop via the 4th order Runge-Kutta in section 4 of [43] to accommodate the feedback control term −µI h and designing a function for the sensor/actuator placements.

Control turned on at t = t c = 0
We denote by t c the time when the feedback control is turned on.Figure 6 illustrates the solution to controlled problem when the control is turned on at t = 0.He we used the initial condition The closed-loop profile shows exponential stabilization of the u(x, t) = 0 steady state solution.The number of controls is N C = 4, which is proportional to the number of unstable modes.

Case 3: Controlled KSE with interpolant operator based on nodal values
As we have mentioned earlier, designing the feedback control based on the first m, 2π-periodic Fourier modes, or based on determining volume elements may not be as practical to implement in industrial setting.This is because you would require a proportional amount of sensors or controllers as the number of grid points used in the computer simulations.Here we present a similar application of the control algorithm but where the feedback control uses (5).The amount of physical sensors or controllers one needs is proportional to the number of unstable modes for given parameter values.We are able to implement this simply by modifying  the feedback control routine that uses the value of the function at the middle of each subintervals instead of taking the values at every discretized spatial points and then averaging them as in the previous example with the determining volume elements.
with I h (u) defined as in (5).Using similar initial condition, u 0 = 1e −10 cos x(1 + sin x), the same number of controllers N C = 4 and relaxation parameter µ = 20 which is turned on at t c = 40, we can see in Figure 7, that the film height starts to destabilize around t = 32 and then once feedback control is turned on at time t c = 40 it stabilizes exponentially to zero again.We recall that the simulation time is truncated to give a clear picture of the stabilization process.We summarize our numerical experiments in Table 2.

Predictive control of catalytic rod with or without uncertainty variables
We illustrate the application of the proposed new method for control of nonlinear parabolic PDE system to some recent feedback control case studies of the catalytic rod.We consider the catalytic rod example introduced in [9].Consider a long thin rod in a reactor where pure species A is fed into the system and a zeroth-order exothermic catalytic reaction of the form A → B takes place on the rod.The reaction is assumed to be exothermic which then allows the cooling medium that is in contact with the rod for decreasing the temperature.
Under the assumptions of constant density and heat capacity of the rod, constant conductivity of the rod and constant temperature at both ends of the rod and unlimited supply of species A in the furnace, the model that describes the evolution of the dimensionless temperature u(x, t) in the reactor as described in [9] is written as follows subject to homogeneous Dirichlet boundary conditions: and initial condition: where β T denotes a dimensionless heat of reaction, γ denotes a dimensionless activation energy, β U denotes a dimensionless heat transfer coefficient, and q(t) the manipulated input (supplied by the cooling medium), with b(x) the actuator distribution shape function which was taken to be b(x) = 2 π sin(x) in [9] chosen in order to supply maximum cooling in the middle of the rod.

Case 1: Uncontrolled catalytic rod
For the typical values of the model parameters, the steady state solution u(x, t) = 0, when there is no control available, is unstable.Starting with an initial data with small perturbation near zero, the temperature evolves to another stable steady state where the temperature profile has a hot-spot in the middle.We run the simulation with the initial condition u 0 (x) = 1e −3 sin(2x) on the spatial interval [0, π] and time interval [0,6].We obtain the following results illustrated in Figure 8.The axes are in units of ∆x = π/20 and ∆t = 6/1000.

Case 2: Controlled catalytic rod
We now investigate the new feedback control scheme for the catalytic rod problem which takes the following form: For the given parameters for the catalytic rod problem in (42), we observe one unstable mode and so we supply our control algorithm with N C = 1.We put one actuator in the middle of the rod at x = π/2.The interpolant operator is defined as I h (u) = ū χ Under this feedback control scheme, using the same initial condition, we observed stabilization of the trivial steady state solution as shown in Figure 9.

Case 3: Application to catalytic rod with uncertainty
For the catalytic rod with uncertainty, the control objective is the regulation of the temperature profile in the rod through the manipulation of the temperature of the cooling medium in the presence of time-varying uncertainty in the heat of the reaction β T (page 102 [9] for more details).The author in [9] proposed a procedure for the synthesis of the robust controllers that achieve arbitrary degree of asymptotic attenuation of the effect of the uncertain variables on the output based on the construction of higher dimensional approximation of the state slowvariables subsystems stemming from the concept of inertial manifold.Here we apply a similar study using the new feedback control algorithm.
For simplicity, we used β T = β T +θ(t), where β T = 50 and θ(t) = sin(0.524t).The location of the actuator is at x = π/2.In the presence of the uncertainty in some model parameters values, the control algorithm is able to compensate for the uncertainty.The result on the bounds for u depends on the size of the error in the measurements.Our results are shown in Figure 110 for the case where the initial condition has amplitude of A = 1e −10 .Although the stabilizing effect is not exponential, we see the eventual stabilization to zero.

Case 4: Nodal-point observational measurements
For our last numerical test, we repeat Case 2 but with the feedback control scheme is given by with I h (u) = u(π/2)χ [0,π] .So the actuator and sensor are both located in the middle of the rod at x = π/2.We have observed similar behavior as in the case of the finite volume case.For this reason we do not present the figures.We summarize our numerical experiments in Table 3.In this section we will establish the global existence, uniqueness and stability results for the general feedback algorithm stated as where I h is a feedback control interpolant operator that is stabilizing the solution v = u * of the ( 12).This will be accomplished by choosing µ large enough and then choosing h small enough, under the assumptions (54), below, and that Under the conditions stated in ( 46) and ( 54) we derive formal a-priori bounds for the difference u−u * that are essential for guaranteeing the decay of u−u * L 2 to zero.These a-priori estimates, together with the global existence of u * , form the key elements for showing the global existence of the solution u of (45).Notably, these formal steps can be justified by the Galerkin approximation procedure.Uniqueness is obtained using similar energy type estimates.
To show convergence of u to the reference solution u * , we consider the difference w = u − u * .The evolution equation for w, which is obtained by subtracting ( 12) and ( 45), is Multiplying by w, integrating by parts with respect to x using periodic boundary conditions (the cubic nonlinear term disappears) and using (19), we obtain A straightforward calculations, using the Poincaré, Cauchy-Schwarz, Young and Agmon Hereafter, we abuse the notation for an arbitrary constant c, which may change from line to line.By assumption (46) that cµh 4 ≤ ν we have Thus, where the last inequality is due to the assumption (46), i.e., µ > 4/ν.In conclusion we have 1 2 and by Gronwall's inequality it follows that Since u * is a solution of KSE, it suffices to show that for large enough t we have to get an upper estimate, that decays like e − µ 8 t , for w(t) 2 L 2 .This can be obtained by requiring additional assumption on µ in terms of the size of the absorbing ball for KSE ( [49], [57]).We then proceed noting that by Sobolev imbedding theorem in one dimension we have u * From the mathematical theory of 1D KSE (see, e.g.[7,14,34,49,50,57]) we have that lim sup where R 2 is a number which depends on the ν and L. Therefore lim sup Thus, if we assume that We obtain the following: The stability in H 1 can be obtained by a slight modification of the previous analysis.Here we conclude that under the assumption ( 46) and ( 54), the feedback control interpolant operator I h is stabilizing the steady state solution v ≡ u * of (12) in the H 1 norm.Taking the inner product of (47) with −w xx , writing I h (u) as in (19) and integrating by parts, yields with where for Q 5 , we used the approximation identity (21) and the assumption (46) that ν ≥ µ ch 4 .Collecting like terms we have 1 2 By assumption (46), that µ > 4 ν , we have Let > 0 be a given arbitrarily small number.Thanks to Theorem 8.1 there exists a T 0 ( ) > 0, large enough, such that for all t ≥ T 0 we have

Figure 1 : 4 Figure 2 :
Figure 1: (a) Open-loop profile showing instability of the u(x, t) = 0 steady state solution.The solution increases with bound proportional to √ α.(b) Top-view.

Figure 5 :
Figure 5: (a) Open-loop profile showing instability of the u(x, t) = 0 steady state solution for 0 < t < 40 for ν = 4/15 < 1, then the feedback control with µ = 20 is turned on for t > 40 which exponentially stabilizes the system.(b) Top view profile of u(x, t).

Figure 7 :
Figure 7: (a) With u 0 = 1e −10 cos x(1 + sin x), the film height starts to destabilize around t = 32 and then once feedback control is turned on at t c = 40, the solution stabilizes to zero again.(b) A top view of the controlled profile.

5 Figure 8 :
Figure 8: (a) Open-loop profile showing instability of the u(x, t) = steady state solution.(b) Top-view of u(x, t).
, where ū is the spatial average of u(x, t) on the interval [0, π].