Mathematics

We consider the steering problem for the system $\dot x=a(t,x)+ub(t,x)$ to the equilibrium ($x=0$). The initial point $x(0)$ can be expressed as a Volterra-type series of the form $x(0)=\sum v_{m_1\ldots m_k}\xi_{m_1\ldots m_k}$ where $v_{m_1\ldots m_k}$ are constant vectors (depending on $a$ and $b$ and their derivatives at $t=0$, $x=0$) and $\xi_{m_1\ldots m_k}$ are iterated integrals generated by $\int_0^\theta \tau^{m}u(\tau)d\tau$, $m=0,1,2,\ldots$. This control system generates the certain right ideal structure in the algebra of iterated integrals. Besides, the grading structure naturally arises if some restriction on the control (for example, $|u(t)|\le1$) is fixed. The mentioned right ideal is constructed (in accordance with the graduation) by analysis of linear dependence of values at $t=0$, $x=0$ of elements of Lie algebra of vector fields generated by $a$ and $b$. The homogeneous approximation of the initial control system is expressed explicitly in algebraic terms as orthogonal projections of the Lie elements of the algebra of iterated integrals on the orthogonal complement to the ideal. Reference: G.M.Sklyar and S.Yu.Ignatovich, Approximation of time-optimal control problems via nonlinear power moment min-problems, SIAM J. on Control and Optimiz., 42(), 1325-1346.