Spline-controllability and the approximate determination of the optimal spline-control (Q909607)

The author considers the quasilinear control system \[ \dot x=A(t)x+b(t)u+\epsilon f(t,x);\quad x(t_ 0)=x_ 0,\quad x(T)=x_ T \] where \(x\in {\mathbb{R}}^ n\), \(u\in {\mathbb{R}}^ 1\), \(t\in [t_ 0,T]\) and \(\epsilon\) is a small positive parameter. Admissible controls are the spline functions of order \(\tau\) which are defined from the increasing sequence \(\Delta_ N\) of real numbers \(t_ k\) with \(t_ 0<t_ 1<...<t_ N=T\). Sufficient spline-controllability conditions for this system with small \(\epsilon >0\) are given. The optimal spline-control problem for the above system with the quadratic cost functional \[ I=\int^{T}_{t_ 0}[(P(t)x,x)+q(t)u^ 2]dt \] is studied where (P(t)h,h)\(\geq 0\) and \(q(t)>0\). The spline-controllability problem and the optimal spline-control problem are solved with the help of approximate methods.