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Analysis of instantaneous control for the Burgers equation. (English) Zbl 1022.49001
The authors study the distributed control of the one-dimensional Burgers equation by means of instantaneous controls for a tracking type functional. The process consists in choosing at time ${t}_{k}$ a control which minimizes a functional taking into account a tracking type problem involving the state and the target at time ${t}_{k+1}$. The authors prove that this procedure is equivalent to writing the controlled Burgers equation in a closed loop form. In the minimization procedure at time ${t}_{k}$ the choice of the step size for the gradient method is crucial in the analysis of the stability of the closed loop system. The authors establish different stability results for the corresponding closed loop systems and they illustrate their theoretical results by different numerical simulations.
##### MSC:
 49J20 Optimal control problems with PDE (existence) 35Q53 KdV-like (Korteweg-de Vries) equations