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.
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 , the Adomian decomposition method [5, 6], the direct discontinuous Galerkin method , and the B-spline quasi-interpolation method .
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 , 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:
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
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
According to Gronwall’s inequality, we have
Let us denote as the dual space of and for convenience. Similarly, , , and so on; also, omit for convenience.
According to the above two estimates and (20), we can obtainwhich implies is bounded in .
2.3. Existence of Weak Solutions
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
2.4. Uniqueness of Weak Solutions
If and are two solutions of (8), letting , then satisfies
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 . 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 .
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 ). Step 1: we firstly prove the following estimate: Consider the dual operator: