Construction of the condition-extremal Lyapunov functions in studies of the trajectories behavior of continuous dynamic systems on a plane

The paper presents the solution of the problem of constructing a conditionally extremal Lyapunov function for a continuous dynamical system described by a system of second-order differential equations. This is a quadratic Lyapunov function, which guarantees the minimality of the time before the linearization of a trajectory in a neighborhood of the equilibrium state of the system, into the section of the Lyapunov function inscribed in a given band. With respect to the quadratic Lyapunov function, it is assumed that the condition of equality of the ratio of the modulus minimum of the first derivative of the Lyapunov function on the section to the value of the function itself to a given number is fulfilled. A conditional-extremal Lyapunov function is sought as a quadratic Lyapunov function of the class under consideration, the cross-section of which, inscribed in the strip, is the trajectory of a system linearized in a neighborhood of the equilibrium state with a given initial point.