Necessary conditions for optimal generalized controls. I, II (Q800627)

The author extends Pontryagin's maximum principle to a system \((1)\quad\dot x=f(x,u,t)+B(u,t)v, x(T_ 0)=x_ 0\), where u is a piecewise continuous control with a fixed range in \(R^ n\) while v ranges in some linear space of generalized functions. In the first part of the paper there are no restrictions on the end points x(T) of solutions of (1). In the second part x(T)\(\in G\), where G is a convex body. A functional \(\sum c_ ix_ i(T)\), where the time T is not fixed, is to be minimized. Any optimal control (u,v) maximizes the integral \(\int^{T}_{T_ 0}H(x(t),p(t),u(t),v(t))dt,\) where H is a Hamiltonian function. This condition is also sufficient for optimality if the system (1) is linear with respect to the variable x.