Theory of invariant variational problems in optimization of controlled dynamic systems (Q1320762)

The optimal control problem \(\int^{t_ 1}_{t_ 0} f^ 0(x,u,t)dt\to\min\), \(\dot x= f(x,u,t)\), \(t\in [t_ 0,t_ 1]\), \(x(t_ 0)= x_ 0\), \(x(t_ 1)= x_ 1\), where \(x\in \mathbb{R}^ n\), \(u\in \mathbb{R}^ n\) and \(f^ 0: \mathbb{R}^{n+ m+1}\to \mathbb{R}\), \(f: \mathbb{R}^{n+ m+1}\to \mathbb{R}^ n\) are continuously differentiable with respect to \(x\) and \(u\), is considered. Noether's theory of invariant variational problems is generalized by introducing the notion of invariance of an optimal process in a generalize sense with respect to a one-parameter Lie group. For this kind of invariance necessare and sufficient conditions are derived. A generalization of Noether's theorem to construct new first integrals is given.