The construction of optimal modes for exciting oscillation of a two-link physical pendulum (Q2761275)

The paper deals with a plane two-link pendulum whose links are homogeneous thin rods. The authors discuss the problem of optimal oscillation mode of the pendulum. The equations of motion are \(\dot K=L(\varphi_1,\varphi_2)\), \(\dot\varphi_1 = K/A(\varphi_2) - C(\varphi_2)u\), \(\dot\varphi_2 = u\), where \(L(\varphi_1,\varphi_2) = k_1\sin\varphi_1 + k_2\sin(\varphi_1 + \varphi_2)\), \(C(\varphi_2) = B(\varphi_2)/A(\varphi_2)\), \(k_1 = -\frac 12 m_1l_1g - m_2l_1g\), \(k_2 = -\frac 12 m_2l_2g\). The initial conditions are \(K(0) = 0\), \(\varphi_1(0)= \varphi_{10}>0\), \(\varphi_2(0)= \varphi_{20}\). The aim is to construct a control \(u(t)\) satisfying the condition \( |u|<c \) and providing at time \(t=T\) the maximal deviation from the vertical line of the first link, i.e. \(F_0 = -\varphi_1(T)\to\max\), while \( F_1 = K(T)=0\). The authors give necessary conditions for extremum, and construct a numerical algorithm to solve the corresponding boundary value problem.