Design of the oscillatory systems with the extremal dynamic properties

In this article the problem of determination of such coefficients a(1), a(2), . . . , a(n) and eigenvalues s(1), s(2), . . . , s(n) of the characteristic equation which yield required extremal values of the solution x(t) at the extremal value tau of time is solved. The extremal values of x(tau) and tau are treated as functions of the roots s(1), s(2), . . . , s(n). The analytical formulae enable us to design the systems with prescribed dynamic properties. For solution of the problem the properties of symmetrical equations are used. The method is illustrated by an example of the equation of 4-th degree. The regions of the different kinds of the roots: real, with one pair of complex and two pairs of complex roots are illustrated. A practical problem is shown.