Sensitivity to perturbations in variable structure systems (Q5946096)

A sensitivity matrix is derived for a differential boundary value problem representing a variable structure system. The authors consider a system of ordinary differential equations NEWLINE\[NEWLINEx'(t)= g(x(t), t) x(t_0)= x_0,\quad t_0\leq t<\infty,\quad x\in\mathbb{R}^n,\tag{i}NEWLINE\]NEWLINE NEWLINE\[NEWLINEg= f_i(x(t), t),\qquad i= 1,2,\tag{ii}NEWLINE\]NEWLINE where the index changes when the trajectory crosses a moving boundary \(D(x,t)= 0\). The sensitivity matrix \(S\) is a product of three \((n,n)\) matrices \(S= M_2 PM_1\). The authors describe how these may be used in approximation to the solution of (i), (ii) by a shooting method. Results of computations are presented for the planar system NEWLINE\[NEWLINE\dot r= F(r),\quad \dot\theta= 2- r- 0.75\cos\theta,\tag{iii}NEWLINE\]NEWLINE where \(F(r)= r(1- r)\) when \(r\sin\theta\geq 0\) and \(F(r)= -0.4 r(2-r)\) when \(r\sin\theta< 0\).