Local Exact Controllability to the Trajectories of Burgers–Fisher Equation

This paper is addressed to study local exact controllability to the trajectories of the Burgers–Fisher (BF) equation. By using the global Carleman estimate for the second-order parabolic operator, we establish the observable inequality and obtain the exact controllability to the trajectories of the linear system. Then, by local inverse theory, we consider the controllability result for the Burgers–Fisher equation.


Introduction
Burgers-Fisher-type equations describe the interaction between reaction mechanism, convection effect and diffusion process. Due to this, these equations have a wide range of applications in plasma physics, fluid physics, capillarygravity waves, nonlinear optics, chemical physics and population dynamics [1][2][3].
Burgers-Fisher-type equation is as follows: where a, b and d are the nonnegative numbers. As we know, the approximation numerical methods for the Burgers-type equations have been developed by many researchers, including the moving mesh PDE method [4], the Adomian decomposition method [5,6], the direct discontinuous Galerkin method [7], and the B-spline quasiinterpolation method [8].
Due to the quadratic nonlinearity of the Burgers-Fisher equation, the nonlinear phenomena results in very complex and unfavorable behaviors (e.g., blowing up, shock waves, and chaos). Control by using internal or external actuation has been expected as an effective method to reduce or totally avoid those undesired phenomena.
We consider the controlled system described by the Burgers-Fisher equation: y t − y xx + αyy x − βy(1 − y) � uχ ω , y(t, 0) � 0 � y(t, 1), y x (t, 0) � 0 � y x (t, 1), where u(t, x) is an internal control and ω is a nonempty open interval of [0, 1]. Our problem is to guide the solution of (2) to a given trajectory. More accurately, for any given time T > 0 and a suitable space X, Y(seen below), if with any initial value y 0 ∈ X and the given y ∈ Y satisfying y t − y xx + αyy x − βy(1 − y) � 0, y(t, 0) � 0 � y(t, 1), en, there exists a control u(t, x) such that the solution y of (2) satisfies y(0, x) � y 0 and can touch y(T, x) � y(T, x).
Letting q � y − y, we get a new controlled system: 1), It is easy to find that the exact controllability to the trajectories of system (2) is equivalent to the null controllability of system (4).
For the convenience of narration, we firstly introduce some notations as a preliminary: (i) L y y � y t − y xx + (2βy + αy x − β)y + αyy x is a linear operator.
(iv) Let To end this introductory, let us mention how this work is organized. Sections 2 and 3, respectively, establish the well posedness of the linear Burgers-Fisher equation and the nonlinear one. In Section 4, we establish the Carleman estimate for the second parabolic operator (similar estimates can be found in [20][21][22]). Section 5 is contributed to the null controllability of system (4).

Well Posedness of Linear BF Equation
then y is called a weak solution of the following system: 2 Mathematical Problems in Engineering e main result for the well posedness of linear system (8) is as follows: Proposition 1. Let a, b ∈ L 2 (0, T; L ∞ (0, 1)), if y 0 ∈ L 2 (0, 1) and f ∈ L 1 (0, T; L 2 ), then there exists a unique solution of (8): y ∈ C([0, T]; L 2 ) ∩ L 2 (0, T; H 1 ) and there is a constant C independent of y 0 , f, a, b, and T, such that Proposition 1 will be proved in the following four sections step-by-step. Firstly, we establish a Galerkin approximate solution for the linear BF equation; secondly, we prove the existence of this solution through a series of mathematical estimates; finally, the uniqueness of the solution is proved.

Let
Ay � y xx , Obviously, A is the second-order operator defined on L 2 (0, 1). We can find an orthogonal basis φ k ∞ k�1 of L 2 (0, 1) as the eigenfunctions corresponding to the eigenvalues λ k of operator A. Let where C k m (t)(k � 1, 2, . . . , m) is the solution of the following ordinary differential equation According to the classical theory of the ordinary differential equation, equation (13) has a unique solution in the interval [0, T m ].

Energy
Estimates. Next, we prove that the solution y m mentioned above is bounded when T m ⟶ T.
Multiplying the first formula of equation (13) by C k m (t) on both sides and taking summation about k from 1 to m, the following is obtained: y mt − y mxx , y m + ay m + by mx , y m � f m , y m . (14) at is to say Noticing that erefore Mathematical Problems in Engineering 3 According to Gronwall's inequality, we have Integrating both sides of (17) about t on [0, T] and combining with (18), we can get Combining (18) and (19), we have Let us denote H 1 * (0, 1) as the dual space of H 1 (0, 1) and H 1 * for convenience. Similarly, L ∞ (0, 1), L 2 (0, 1), and so on; also, omit (0, 1) for convenience.

Noticing that
According to the above two estimates and (20), we can obtain which implies y mt m ≥ 0 is bounded in L 1 (0, T; H 1 * ).

Existence of Weak Solutions.
We are in position to use the energy estimates to gain weak solutions. According to (20) and (23), combined with the Lions-Aubin theorem, y m ⟶ y, weakly in L 2 0, T; H 1 , y mt ⟶ y t , weakly in L 2 0, T; H 1 * , y mt ⟶ y t , strongly in L 2 0, T; H 1 .
As it is known Taking the limit on both sides in (26) and combining with (24) and (28), the following can be obtained: i.e., ∀t ∈ [0, T], we get Mathematical Problems in Engineering 5 which shows that y satisfies (7).

Uniqueness of Weak Solutions.
If y 1 and y 2 are two solutions of (8), letting y � y 1 − y 2 , then y satisfies y t − y xx + ay + by x � 0, When (20) is used for (31) and according to the convergence of solution, we obtain that is
Similar to formula (9), we have We have For any R, we note It is easy to see that the appropriate r, R > 0 can be selected to satisfy then Π B(0,R) ⊂ B(0, R). Now we are in the position to prove that Π is a contractive mapping.

Carleman Estimate
e main result in this section is given as follows: ere exist constants C 0 and C > 0 such that for λ > C 0 (T + T 2 ), ∀y ∈ D(A), we have In order to prove eorem 2, we need the following conclusion.

Proposition 2.
Let y t − y xx � f and u � θy � e η y, then where Substituting the above formulas into y t − y xx � f, we achieve (49) and (50). e proof of eorem 2 will be completed in the following four steps: Step 1: we will get the following estimate: Let (I 1 u, I 2 u) 2 � I i,j (i � 1, 2, 3; j � 1, 2), where I i,j denotes the L 2 inner product between the ith term of I 1 u and the jth term of I 2 u. For all I i,j , integrating by part and using the boundary conditions, we have

Mathematical Problems in Engineering
Adding the above equations all together, we have where According to the definition of η(x, t), we note that where |R 0 | ≤ λ 3 μ 3 φ 3 and |R 1 | ≤ Cλμφ.
Step 3: the term Q ω 0 λμφu 2 x dx dt can be absorbed in such that the following estimate can be obtained: In the following discussion, we consider two nonempty open intervals ω 1 , ω, satisfying: ω 0 ⊂ ω 1 , ω 1 ⊂ ω and select a nonnegative function χ ∈ C ∞ 0 (ω 1 ), which satisfies χ ≡ 1 in ω 0 . us According to the definition of Substituting (67) into (66) and taking enough small ε and sufficient large λ, it is easy to see that (65) holds.
Step 4: let us prove the Carleman estimate (48). Similar to formula (67), according to the definition of Combining (67) with (68), we have According to (65) and (68), we can get By substituting u with θy, we obtain Taking a suitable μ 0 , when μ ≥ μ 0 , the estimate can be reduced to which implies (48).

Controllability to the Trajectory
In this section, the controllability of linear system (73) is obtained by the duality of observability-controllability, and the result for the nonlinear system is obtained by means of a local inverse theorem, where the idea can be referred to [13]. Firstly, the conclusion for the linear system is as follows.

Mathematical Problems in Engineering
, has a solution y ∈ Y, satisfying y(·, T) � 0. Furthermore, there exists C > 0 such that Proposition 3 will be proved in three steps. According to the dual theory, in order to get the controllability of system (73), we need an observable inequality of dual operator (76) (for more details, refer [14]).
Step 1: we firstly prove the following estimate: Consider the dual operator: Similar to (48), the following estimate can be shown: According to (77), we have By adding the left and right sides of the two formulas separately, we have Step 2: we can obtain the following observable inequality: Let ξ � e λmϕ ϕ (1/2) and ξ ′ � e λmϕ ϕ ′ ϕ − (1/2) (λmϕ + (1/2)), then ξ ′ ≤ Ce λmϕ ϕ (3/2) � C e λMϕ L y * q| 2 + e λMϕ ϕ 3 Step 3: we will prove Proposition 3. Note the right side of (81) as which defines a norm on the space: Noticing that Q is a complete space of Q with respect to the above norm, Q is obviously a Hilbert space, on which the inner product is defined as Recalling the first formula in (73) Multiply both sides by q and integrating over Q that is Integrating by parts, we have that is e left of (90) defines a linear function: L: Q ⟶ R: with erefore, L is continuous. en, there exists a unique p on Q, such that L(q) � 〈p, q〉, ∀q ∈ Q. (93) Let y � e 2λMϕ L * y p and u � − e 2λMϕ ϕ 3 p, then (94) can also be written as follows: Combining (90) and (95), we obtain which implies y(x, T) � 0. Furthermore, in the right side of (94), letting q � p and taking L * y p � e − 2λMϕ y, that is In order to obtain ‖y‖ Y ≤ C(‖f‖ F + ‖u‖ U + ‖y 0 (x)‖ L 2 (0,1) ), we construct a new system.
e proof of Proposition 3 is completed. Next, we consider the controllability of the nonlinear system. We have the following result.