A NOTE ON THE SEMIGLOBAL CONTROLLABILITY OF THE SEMILINEAR WAVE EQUATION

作者：Joly Romain^*; Laurent Camille

来源：SIAM Journal on Control and Optimization, 2014, 52(1): 439-450.

DOI：10.1137/120891174

• semiglobal controllability
• wave equation
• internal control
• compact global attractor
• heteroclinic orbits

We study the internal controllability of the semilinear wave equation v(tt)(x, t) - Delta v(x,t) + f(x, v(x, t)) = 1(omega)u(x, t) for some nonlinearities f which can produce several nontrivial steady states. One of the usual hypotheses to get semiglobal controllability is to assume that f(x, v) v >= 0. In this case, a stabilization term u = gamma(x) v(t) makes any solution converging to zero. The semiglobal controllability then follows from a theorem of local controllability and the time reversibility of the equation. In this paper, the nonlinearity f can be more general, so that the solutions of the damped equation may converge to an equilibrium other than 0. To prove semiglobal controllability, we study the controllability inside a compact attractor and show that it is possible to travel from one equilibrium point to another by using the heteroclinic orbits.

• 出版日期2014