Singular characteristics of first order partial differential equations in optimal control and differential games (Q2771543)

This paper deals with a new concept of singular characteristics of the first order partial differential equations in the optimal control and differential games. The singular characteristics represent the equations of singular trajectories in the optimal control problems and differential games. The singular characteristics are effective in construction of singular lines, surfaces and manifolds of non-smooth solutions of partial differential equations. The author considers the regular and singular trajectories of the optimal control problem \(\dot x=f(x,u), u\in U\subset R^{s}, t\in[0,T]\), \(x(0)=x^0,\;x(T)\in M\subset R^{n}\), \(J=T+\Phi(x(T))\to\min_{u}\). Then the notion of viscosity solution of the problem \(F(x,u(x),p(x))=0, x\in\Omega\subset R^{n}; u(x)=w(x), x\in M\subset\partial\Omega, p=\partial u/\partial x\) is introduced and the notion of singular characteristic is defined. The equations of singular characteristics for the cases of universal, equivocal and focal surfaces in optimal control theory are obtained.NEWLINENEWLINEFor the entire collection see [Zbl 0949.00042].