On the controllability of the Burger equation.(English)Zbl 0897.93034

The author considers the existence of weak solutions to the problem of controllability for the equation: $u_t+ (u^2/2)_x=0 \tag{1}$ in the following sense: Suppose that there exists a weak solution $$\widehat u:\mathbb{R}^+ \times\mathbb{R} \to\mathbb{R}\in L^\infty ((0,\infty), X)$$ such that for any $$z_0$$, $$z_1\in X$$, and for any $$T\in \mathbb{R}^+$$, $$\widehat u(0,.) =z_0$$, $$\widehat u(T,.) =z_1$$, then equation (1) is said to be controllable in $$X$$. If the exact equality is replaced by the requirement that $$\| u(T,.)-z_1 \|< \varepsilon$$ for any given $$\varepsilon$$, then it is said to be approximately controllable in $$X$$. Such solutions are sought in a subset of the class of entropic functions of bounded variation. A function $$f(x)$$ is said to be entropic if for any $$x$$ in its domain (which is a subset of $$\mathbb{R})$$ the limit from the left $$f(x-)$$ is greater or equal than the limit from the right $$f(x+)$$. The author uses J. M. Coron’s idea of considering the linearization of the operator not around the zero solution (which does not work for this system), but instead around trajectories that vanish at times 0 and $$T$$. This is closely related to the “extension technique” originally introduced by D. L. Russell around 1973, which included the idea of variation of the domain.
Reviewer: V.Komkov (Roswell)

MSC:

 93C20 Control/observation systems governed by partial differential equations 93B05 Controllability 35Q30 Navier-Stokes equations 35Q53 KdV equations (Korteweg-de Vries equations)
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References:

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