Abstract
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.
1. 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, capillary-gravity waves, nonlinear optics, chemical physics and population dynamics [1–3].
Burgers–Fisher-type equation is as follows:where 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 quasi-interpolation 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.
Although the controllability of infinite dimensional systems has been studied extensively, e.g., null controllability [9, 10], local controllability to trajectories [11–14], approximate controllability [15, 16], exact controllability [17, 18], and boundary controllability [19], control of the Burgers–Fisher equation is still in its infancy and remains open. In this paper, we will deal with the local exact controllability to the trajectories of the Burgers–Fisher equation.
We consider the controlled system described by the Burgers–Fisher equation:where 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 and a suitable space (seen below), if with any initial value and the given satisfying
Then, there exists a control such that the solution of (2) satisfies and can touch .
Letting , we get a new controlled system:
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) is a linear operator.(ii)Let , which satisfies
For given positive constants and , we construct two weight functions:(iii)For simplicity, we abbreviate as either or , where .(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 [20–22]). Section 5 is contributed to the null controllability of system (4).
2. Well Posedness of Linear BF Equation
Definition 1. If for any , there exists , such thatthen is called a weak solution of the following system:The main result for the well posedness of linear system (8) is as follows:
Proposition 1. Let , if and , then there exists a unique solution of (8): and there is a constant independent of , 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.
2.1. Galerkin Approximate Solutions
We consider the system
Let
Obviously, is the second-order operator defined on . We can find an orthogonal basis of as the eigenfunctions corresponding to the eigenvalues of operator .
Letwhere 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 .
2.2. Energy Estimates
Next, we prove that the solution mentioned above is bounded when .
Multiplying the first formula of equation (13) by on both sides and taking summation about from 1 to , the following is obtained:
That is to say
Noticing that
Therefore
According to Gronwall’s inequality, we have
Integrating both sides of (17) about on and combining with (18), we can get
Combining (18) and (19), we have
Let us denote as the dual space of and for convenience. Similarly, , , and so on; also, omit for convenience.
Noticing that
Similarly
According to the above two estimates and (20), we can obtainwhich implies is bounded in .
2.3. 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, there exists such that
According to the first formula in (13), for , we have
Multiplying (25) with and integrating with respect to on , we obtainwhere is the characteristic function on .
As it is known
Therefore
Taking the limit on both sides in (26) and combining with (24) and (28), the following can be obtained:i.e., , we getwhich shows that satisfies (7).
2.4. Uniqueness of Weak Solutions
If and are two solutions of (8), letting , then satisfies
When (20) is used for (31) and according to the convergence of solution, we obtain
Thusthat is
3. Well Posedness of Nonlinear BF Equation
Theorem 1. For , , and , if there exists a positive number , such thatthen the nonlinear systemhas a unique solution: .
Proof. Inspired by [12]. Let , satisfying (35).
We define a mapping:where is the solution of the following system:Comparing (36) and (38), we know that is the solution of (36) if and only if is a fixed point of the mapping .
Similar to formula (9), we haveWe haveFor any , we noteIt is easy to see that the appropriate can be selected to satisfythen .
Now we are in the position to prove that is a contractive mapping.
Taking is the solution of the following system:Similar to (9) and choosing appropriate , we can obtainNoticing thatWe haveThus, the mapping is contractive when . By the Banach fixed-point theorem, has a unique fixed point, i.e., , which is a solution of (36).
4. Carleman Estimate
The main result in this section is given as follows:
Theorem 2. There exist constants such that for , , we have
In order to prove Theorem 2, we need the following conclusion.
Proposition 2. Let and , thenwhere
Proof. Substituting the above formulas into , we achieve (49) and (50).
The proof of Theorem 2 will be completed in the following four steps: Step 1: we will get the following estimate: Let , where denotes the inner product between the term of and the term of . For all , integrating by part and using the boundary conditions, we have Adding the above equations all together, we have where According to the definition of , we note that where . Noticing that , we have Combining (54), (56) and (57), we obtain (52). Step 2: we prove the following estimate: In fact, from (52) we get Noticing that , we obtain In the formula above, if we take , then holds for , that is From (59) and (62), we can infer that Combining (62) with (63), (58) is easy to be achieved. Step 3: the term can be absorbed in such that the following estimate can be obtained: In the following discussion, we consider two nonempty open intervals , satisfying: and select a nonnegative function , which satisfies in . Thus According to the definition of , then Substituting (67) into (66) and taking enough small and sufficient large , 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 , we have Combining (67) with (68), we have According to (65) and (68), we can get By substituting with , we obtain Taking a suitable , when , the estimate can be reduced to which implies (48).
5. 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.
Proposition 3. and , then there exists a control , such thathas a solution , satisfying . Furthermore, there exists 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 Similarly By adding the left and right sides of the two formulas separately, we have Step 2: we can obtain the following observable inequality: Let and , then Step 3: we will prove Proposition 3. Note the right side of (81) as which defines a norm on the space: Noticing that is a complete space of with respect to the above norm, is obviously a Hilbert space, on which the inner product is defined as
Recalling the first formula in (73)
Multiply both sides by and integrating over that is
Integrating by parts, we havethat is
The left of (90) defines a linear function: :with
Therefore, is continuous. Then, there exists a unique on , such that
Thus
Let , then (94) can also be written as follows:
Combining (90) and (95), we obtainwhich implies .
Furthermore, in the right side of (94), letting and takingwe have
Thereforethat is
In order to obtain , we construct a new system.
Let , then satisfies the following equations:
According to the definition of and by direct calculation, we obtain
In the following, we estimate , and :
According to the results of well posedness, we havewhich implies
The proof of Proposition 3 is completed.
Next, we consider the controllability of the nonlinear system. We have the following result.
Proposition 4. Let and , satisfying , then there exists a control , such thathas a solution , satisfying .
Before the proof, we decompose the nonlinear operator:intowhere and , which have the following properties.
Lemma 1. For and , there exists , such that
Proof. Analogously
Lemma 2 (local inverse theorem). Let be two Banach spaces, satisfying . If for and is a surjection. Then, for any satisfying , the equation must have a solution .
With Lemmas 1 and 2, we can prove Proposition 4.
Proof. Taking , with the norm defined asand .
We define the set-valued mapping:Obviously, and .
Let , thenThe conclusion of Proposition 3 shows the fact that is a surjection.
According to Lemma 2, system (106) has a solution andIn this way, null controllability of system (4) is obtained. Equivalently, local exact controllability to the trajectory of system (2) is achieved.
Data Availability
No data were used to support this study.
Conflicts of Interest
The author declares no conflicts of interest.
Acknowledgments
The author would like to thank the editors and the referees for their valuable comments and suggestions which improved the original manuscript. This research was supported by the Philosophy and Social Sciences Project of Jiangsu Province (no. 2019SJA1881).