Continuation by parameter scheme in nonlinear boundary-value problem of maximum principle (Q748860)

For the time-optimal problem \[ (P_{\epsilon})\quad T\to \inf \text{ subject to } \dot x=f(x)+u;\quad x,u\in E^ n,\quad u\in U, \] \[ x(0)=g(\epsilon)=\bar a_ 0+\epsilon (a_ 0-\bar a_ 0)\neq 0,\quad 0\leq \epsilon \leq 1;\quad x(T)=0,\quad f(0)=0 \] necessary optimality conditions have the form \[ \dot x=f(x)+c'(\psi),\quad {\dot \psi}=- f^*_ 1(x)\psi,\quad x(0)=g(\epsilon),\quad x(T)=0, \] where \(c(\psi)=\max_{u\in U}(\psi,u)\) and \(c'(\psi)\) denotes the gradient of the support function c(\(\psi\)) of the smooth convex compact set u. Let the solutions, the parameters \(T=\bar T_ 0\), \(p=\psi (0)=\bar p_ 0\) of the problem \((P_ 0)\) be known. It is shown that the parameters \(T=T(\epsilon)>0\), \(p=\psi (0)=p(\epsilon)\), \(0\leq \epsilon \leq 1\) for the problem \((P_{\epsilon})\) are solutions of the following system of differential equations: \[ \frac{dT}{d\epsilon}=-\frac{(\psi (p,T,\epsilon),y(p,T,\epsilon))}{c(\psi (p,T,\epsilon))}, \] \[ \frac{dp}{d\epsilon}=[pp^*+\Psi^*(p,T,\epsilon)X(p,T,\epsilon)]^{- 1}\Psi^*(p,T,\epsilon)[-y(p,T,\epsilon)-c'(\psi (p,T,\epsilon))\frac{dT}{d\epsilon}], \] where \(\dot y=Ay+B\chi\), \({\dot \chi}=-Ry-A^*\chi\), \(y(0)=g'(\epsilon)\), \(\chi (0)=0\), \(A=f'(x)\), \(B=c''(\psi)\), \(R=(\psi,f(x))''_{xx}\), \(\dot X=AX+B\Psi\), \({\dot \Psi}=-RX-A^*\Psi\), \(X(0)=0\), \(\Psi (0)=E\).