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Stationary solutions for a boundary controlled Burgers’ equation. (English) Zbl 0967.93051

Burger’s equation in the form \[ \Phi(w,w_x, w_{xx}, w_t,\varepsilon)= w_t+ ww_x- \varepsilon w_{xx}= f(x,t), \] was introduced in 1940 by J. Burgers in his study of fluid flow. Extensive research on the properties of Burgers’ equation has been conducted in the last two decades, both on the whole real line and on a closed interval. Much is known about properties of solutions, and the asymptotic behavior for small values of the parameter \(\varepsilon\). C. I. Byrnes and the last two authors published several results on boundary control of Burgers’ equation.
Here the main results are obtained for “stationary” Burgers’ equation: \(ww_x-\varepsilon w_{xx}= f(x)\), with \(x\in (0,1)\). \(\varepsilon\) is apparently regarded as constant, and instead the feedback gain \(k\) is varied on \([0,+\infty)\). However, after the routine Riccati transformation \(w= -2\varepsilon(v',v)\), the importance of \(\varepsilon\), or rather of \(\varepsilon^2\) becomes clear. As usual after this transformation the stationary Burgers’ equation becomes a linear second-order equation \(y''- Fy=\lambda y\), resembling a Sturm-Liouville system. However, the spectral parameter \(\lambda(\varepsilon, k,w(0))\) enters into boundary conditions of this system, making life complicated. \(f(x)\) is the forcing term about which the authors make the assumption \(f(x)= -f(1-x)\), \(u\) is the boundary control input, defined at \(x=0\) and \(x=1\), with \(w_x(0,t)= -u_0(t)\), \(w_x(1,t)= -u_1(t)\), while \(y_0(t)= w(0,t)\), and \(y_1= w(1,t)\) are boundary outputs. Proportional feedback is given by \(u_0(t)= -ky_0(t)\), \(u(t)= -ky_1(t)\). Thus a closed-loop feedback system is defined. The boundary control of the stationary Burgers’ equation is determined by varying the gain \(k\).
The main results are as follows: With zero gain there are infinitely many constant solutions. For a positive gain with small values of \(k\) three independent solutions exist, while for large values of \(k\) there is a single asymptotically stable solution. A graph illustrating the bifurcation phenomenon is displayed for a specific choice of the inhomogeneous term \(f(x)\). This figure and other figures serve to illustrate numerical verifications of their results. The paper, complementing previous work of Gilliam, Shubov, in cooperation with Byrnes and others, is well written and proofread.

MSC:

93C20 Control/observation systems governed by partial differential equations
93D15 Stabilization of systems by feedback
35B37 PDE in connection with control problems (MSC2000)
35Q53 KdV equations (Korteweg-de Vries equations)
35B32 Bifurcations in context of PDEs

Software:

AUTO-86; AUTO
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References:

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