Parameter dependence of the controllability set of the Liénard equation (Q1901884)

The author considers the following system which corresponds to a Liénard equation with control: (1) \(\dot x= y\), \(\dot y= - x-\mu p(x, y)- \alpha u\). Here \(\alpha\) and \(\mu\) are parameters, \(p(x, 0)= 0\), \(u\) is the control, (2) \(u\in U= \{u: |u|\leq 1\}\). Piecewise continuous functions \(u(t)\) with values from \(U\) are viewed as admissible controls. The author obtains necessary and sufficient conditions for continuous dependence of the controllability set of (1), (2) on the parameters \(\alpha\), \(\mu\) in terms of existence of a closed loop solution of the system \(\dot x= y\), \(\dot y= -x- \mu p(x, y)- \alpha\text{ sgn } y\).