Controlling the unsteady analogue of saddle stagnation points (Q2841015)

The problem addressed in this paper concerns a system of ordinary differential equations NEWLINE\[NEWLINE\dot x = v_0(x)+\varepsilon v_1(t,x),NEWLINE\]NEWLINE where \(x\in\mathbb R^2\). When \(\varepsilon=0\), the system is time-invariant, and the authors assume that it possesses an equilibrium point \(a\) of saddle type: more precisely, the authors assume that \(v_0(x)\) is differentiable at \(a\) and that the Jacobian matrix \(Dv_0(a)\) has real eigenvalues of opposite sign. When \(\varepsilon\not= 0\) but sufficiently small, the equilibrium point is replaced by a hyperbolic trajectory \(a_\varepsilon(t)\): the behavior of nearby trajectories can be described in terms of exponential dichotomy. In particular, the existence of stable and unstable manifolds can still be recognized.NEWLINENEWLINEIn this paper, the term \(v_1(t,x)\) is interpreted as a control, to be designed in order to assign a prescribed behavior for \(a_\varepsilon(t)\). The authors describe two possible approaches to the problem.NEWLINENEWLINEA natural application of these results is the study of stagnation points in fluid planar dynamics, but other applications are also considered in the paper. The extension of the results to equations of more general form and numerical simulations are finally discussed.