Stabilization of Oscillations of a Controlled Autonomous System

Abstract—

We consider a smooth autonomous system in general form that admits a non-degenerate periodic solution. A global family (with respect to the parameter h) of nondegenerate periodic solutions is constructed, the law of monotonic variation of the period on the family is derived, and the existence of a reduced second-order system is proved. For it, the problem of stabilizing the oscillation of the controlled system, distinguished by the value of the parameter h, is solved. A smooth autonomous control is found, and an attracting cycle is constructed.

APPENDIX

Reduction of a homogeneous system.

We set down the variation equations for the solution (6) as

$$\begin{gathered} \delta \dot {x} = \frac{{\partial X(x,y)}}{{\partial x}}\delta x + \frac{{\partial X(x,y)}}{{\partial y}}\delta y, \\ \delta \dot {y} = \frac{{\partial Y(x,y)}}{{\partial x}}\delta x + \frac{{\partial Y(x,y)}}{{\partial y}}\delta y, \\ \end{gathered} $$

(A.1)

where the partial derivatives are calculated for x = φ(h, t), y = ψ(h, t).

The following matrix gives the fundamental system of solutions in (A.1):

$$\left\| {\begin{array}{*{20}{c}} {\frac{{\partial \varphi (h,t)}}{{\partial t}}}&{\frac{{\partial \varphi (h,t)}}{{\partial h}}} \\ {\frac{{\partial \psi (h,t)}}{{\partial t}}}&{\frac{{\partial \psi (h,t)}}{{\partial h}}} \end{array}} \right\|$$

(A.2)

with the determinant

$$\Delta (h,t) = \Delta (h,0)\exp \int\limits_0^t {\left( {\frac{{\partial X(x,y)}}{{\partial x}} + \frac{{\partial Y(x,y)}}{{\partial y}}} \right)} d\tau .$$

We set down the general solution of the system in (A.1)

$$\begin{gathered} \delta x = {{c}_{1}}\dot {\varphi }(h,\,\,t) + {{c}_{2}}\varphi '(h,\,\,t), \\ \delta y = {{c}_{1}}\dot {\psi }(h,\,\,t) + {{c}_{2}}\psi '(h,\,\,t) \\ \end{gathered} $$

(A.3)

with constants c₁ and c₂. We use resolution of the system (A.3) with regard to c₁ and c₂ to transition to new variables u, ${v}$:

$$u = - (\dot {\psi }\delta x - \dot {\varphi }\delta y){\text{/}}\Delta ,\quad {v} = [\eta (h,\,\,t)\delta x - \xi (h,\,\,t)\delta y]{\text{/}}\Delta .$$

(A.4)

At the same time we assume

$$\xi (h,t) = \frac{{T'(h)}}{{T{\text{*}}}}t\dot {\varphi }(h,\,\,t) + \frac{{\partial \varphi (h,\,\,t)}}{{\partial h}},$$

$$\eta (h,t) = \frac{{T'(h)}}{{T{\text{*}}}}t\dot {\psi }(h,\,\,t) + \frac{{\partial \psi (h,\,\,t)}}{{\partial h}},$$

where the functions ξ(h*, t), η(h*, t) would be T *-periodic. Finally, we obtain the following from the expression for ${v}$ in (A.4):

$${v} = \frac{{T'(h)}}{{T{\text{*}}}}tu + (\psi '(h,\,\,t)\delta x - \varphi '(h,\,\,t)\delta y){\text{/}}\Delta {\kern 1pt} {\kern 1pt} .$$

(A.5)

We substitute the first solution into formulas (A.4) and (A.5) and calculate u = 0, ${v}$ = 1. Correspondingly, we set down the following for the second solution:

$$u = - (\dot {\psi }\varphi {\kern 1pt} '\, - \dot {\varphi }\psi {\kern 1pt} '){\text{/}}\Delta = 1,\quad {v} = \frac{{T'(h{\text{*}})}}{{T{\text{*}}}}tu + u.$$

Therefore, we transition the system (A.1) to the following form:

$$\dot {u} = 0,\quad {\dot {v}} = \frac{{T'(h{\text{*}})}}{{T{\text{*}}}}u.$$