Abstract
This paper focuses on the stabilization problem for the linear parabolic system using the backstepping method. The exponentially stability results for considered parabolic system are derived in two cases with Dirichlet and Neumann local terms. Also, the boundary conditions for the problem is assumed to be mixed or Robin-type boundary conditions. The main aim is to achieve the stability of the considered system using the backstepping method with help of Volterra integral transformation. The explicit solutions of kernel functions in integral transformation is obtained by using Laplace transform and designed a boundary control law to the closed-loop system. Finally, the effectiveness and applicability of the derived results are validated through a single-species pattern generation model.
Highlights
Introduction e partial differential equations (PDEs) describe the several mathematical models of real-life physical problems such as propagation of heat or sound, chemical reactors, fluid flows, signal processing, population genetics, and many others. e important idea of this work is to study the exponential stability of parabolic systems with mixed or Robin boundary conditions using the backstepping method and boundary control law. is approach provides significant design freedom, a better insight to the key features such as computable transient performance, and ability to handle uncertainties to a certain level
It is noted that the well-posedness of explicit solution to kernel equation and appropriate design of the stable target system are the main points to be focused in the process. e considered model has wide range of applications including travelling wavetrains model with oscillatory kinetics, linearizing a tubular chemical adiabatic reactor, single-species pattern generation, and so on. e main aim of this work is to derive some novel results on stabilization of parabolic systems with Dirichlet and Neumann local terms. e stabilization of various types of systems got much attention of the researchers in the existing literature; see [1,2,3] and references therein
Β is hyperbolic PDE (53) is similar to (49); using similar lines, the explicit solution of (53) can be achieved. ere, from the above analysis, it is easy to conclude that the parabolic system with Neumann local term (44) is exponentially stable
Summary
(1 − β)w(1, t) + βwx(1, t) 0, where x ∈ (0, 1), t > 0, ε > 0, α ∈ [0, 1], and β ∈ (0, 1), the state variable w(x, t) ∈ R with initial condition w(x, 0) w0(x), c2 is an arbitrary constant which describes the coefficient of local term (that is, Neumann interconnection), and U(t) is the actuation control input. It is noted that we will get a different form of the kernel function and boundary inputs for the above system, which is the main reason to proceed with the following results. For any c2, there exist a function l(·, ·) ∈ C2(0, 1): 0 ≤ x ≤ y ≤ 1 such that w0(·) ∈ L2(0, 1); with compatibility conditions (46) and (47), the closed-loop system (44) with the feedback controller, U(t) αl(0, 0)w(0, t) +(1 − α) l(0, y)w(y, t)dy − α lx(0, y)w(y, t)dy, α ∈ [0, 1]. Substituting the kernel solution in the boundary control law is given by w(0, t) c2 1 − β. The exponential stability of target system (3) for this case followed from eorems 2 and 3 with (1 − β/β) > 0 and (1 − α/α) > 0. Β is hyperbolic PDE (53) is similar to (49); using similar lines, the explicit solution of (53) can be achieved. ere, from the above analysis, it is easy to conclude that the parabolic system with Neumann local term (44) is exponentially stable
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